2. Mathematical settings and main results

In this section, we describe the mathematical settings and results. We denote the theoretical concentration or cell density at position $x \in \Omega \,:\!=\, [-L , L]$ at time $t\gt 0$ by $\rho = \rho (x,t)$ . We investigate the solution to the following nonlocal Fokker–Planck equation:

(P) \begin{equation} \rho _ t = \rho _{xx} - ( \rho ( W*\rho )_x )_x \ \text{in} \ \Omega \times (0,\infty ), \end{equation}

where the periodic boundary condition

(2.3) \begin{equation} \left \{ \begin{aligned} &\rho (-L,t) = \rho (L, t), \ t\gt 0, \\ &\rho _x(-L, t) = \rho _x( L, t), \ t\gt 0, \end{aligned} \right . \end{equation}

is imposed and the initial datum is given by $\rho (x, 0) \,:\!=\, \rho _0(x)$ . Here, $W*u(x)$ is defined by

\begin{equation*} W*u (x) \,:\!=\, \int _\Omega W(x-y) u(y) dy \end{equation*}

for $W \in L^1_{\mathrm{per}}(\Omega ) \,:\!=\, \{u|_\Omega \in L^1(\Omega ) \ | \ u(x)=u(x+2L), \ x \in {\mathbb{R}} \}$ . Setting $d_j\gt 0, (j=1,\ldots , M)$ which are the diffusion coefficients in ( $\mbox{KS}^{M,\varepsilon }$ ) introduced below, we define the following function

(2.4) \begin{equation} k_j(x)\,:\!=\, \frac {1}{2\sqrt {d_j} \sinh \frac {L}{\sqrt {d_j} } } \cosh \frac {L - \left | x \right |}{\sqrt { d_j } }. \end{equation}

This is actually a fundamental solution to the elliptic equation explained in Lemma 4.1 below. The typical examples of $W$ are as follows:

(2.5) \begin{align} &W(x) = k_1(x) \text{ with any } d_1\gt 0, \end{align}

(2.6) \begin{align} &W(x) = k_1(x) - k_2(x) \text{ with any } d_1\lt d_2, \end{align}

(2.7) \begin{align} &W(x) = (R_0 - |x| )\chi _{B(R_0)}(x), \end{align}

(2.8) \begin{align} &W(x) = ((a_1+a_2)R_0 -a_2R_1 -a_1 |x| )\chi _{B(R_0)}(x) - a_2(R_1-|x|)\chi _{ B(R_1)\backslash B(R_0) }(x) \end{align}

with any $a_1, a_2\gt 0$ , where $R_1\gt R_0\gt 0$ are constants called the sensing radius, $B(R_0)$ is a ball with radius $R_0$ and origin centre, and

\begin{equation*} \chi _{B(R_0)}(x)= \left \{ \begin{aligned} &1 \ & \text{if} \ x \in B(R_0),\\ &0 \ & \text{otherwise}. \end{aligned} \right .\\ \end{equation*}

The profiles of (2.6) and (2.7) are presented in Figures 1 (a) and 2 (a), respectively. The nonlocal Fokker–Planck equation (P) with the integral kernels (2.5) and (2.6) corresponds to the parabolic-elliptic Keller–Segel systems. A linear stability analysis is presented in Section 6. Integral kernels (2.7) and (2.8) were introduced by Carrillo et al. [Reference Carrillo, Murakawa, Sato, Togashi and Trush6] and Murakawa and Togashi [Reference Murakawa and Togashi17]. They have a compact support corresponding to the cell body. If these integral kernels are used for the potential $W$ in (P), this describes the situation in which $\rho$ at each point detects the surrounding cell density in the own cell body and the velocity of the aggregation is determined. This corresponds to the Haptotaxis phenomenon.

Figure 1. Profile of the integral kernel (2.6) and distribution of eigenvalue with $\omega _{n,1}$ with same parameters as those in Figure 5.

Figure 2. Profile of (2.7) and distribution of eigenvalue with $\omega _{n,2}$ with same parameters as those in Figure 6.

First, we have the following existence result. To construct a mild solution to (P), we define a function space with a norm as

\begin{equation*} E_{\tau }\,:\!=\, C([0, \tau ]; H^1(\Omega )), \quad \left \| \cdot \right \|_{ E_{\tau } } \,:\!=\, \left \| \cdot \right \|_{ C([0, \tau ];H^1 (\Omega ))} \end{equation*}

for any time $\tau \gt 0$ . Introducing the following function:

(2.9) \begin{align} G(x,t) \,:\!=\, \frac {1 }{2L } \sum _{n \in \mathbb{Z}} e^{-\sigma _n^2 t} e^{i\sigma _n x}, \end{align}

where $i$ is the imaginary unit and

\begin{equation*} \sigma _n \,:\!=\, \frac {n \pi }{L}, \quad n \in \mathbb{Z}, \end{equation*}

we define the map $\Gamma \,{:}\, E_{\tau } \to E_{\tau }$ as

\begin{align*} \Gamma [u] (x,t) \,:\!=\, (G * \rho _0)(x,t) - \int _0^t \int _\Omega G(x-y, t-s) ( u (W*u)_x )_x(y, s) dy ds, \quad u \in E_{\tau }. \end{align*}

We say that a function $u \in C([0,T]; H^1(\Omega ))$ for any $T\gt 0$ is a mild solution to (P) with $\rho (x,0)=\rho _0(x)$ , provided $ u=\Gamma [u]$ . The following proposition is proven using the standard argument of the fixed point theorem.

Proposition 2.1. Let $R\gt 0$ be an arbitrary real number, and assume that $W \in W_{\mathrm{per}}^{1,1}(\Omega )\,:\!=\,\{u|_\Omega \in W^{1,1}(\Omega ) \ | \ u(x)=u(x+2L), \ x \in {\mathbb{R}} \}$ and

\begin{align*} \rho _0 \in H^1(\Omega ) \ \text{with} \ \left \| \rho _0 \right \|_{H^1(\Omega )} \lt R. \end{align*}

Then, for any $T\gt 0$ , there exists a unique mild solution $\rho$ to ( P ) in $ C( [0, T]; H^1(\Omega ) ) \cap L^2( 0, T; H^2(\Omega ) )$ satisfying

\begin{equation*} \left \| \rho \right \|_{C([0, T], H^1 (\Omega ))} \lt C_0, \end{equation*}

where $C_0 =C_0(L,R,T, \|W_x\|_{L^1(\Omega )} )$ . Moreover, this mild solution satisfies (P) in $L^2(0,T;L^2(\Omega ))$ .

Next, we will approximate the solution to (P) with any integral kernel using that to a Keller–Segel system which is a local dynamics. Introducing the auxiliary factors $v_j^{M,\varepsilon } = v_j^{M,\varepsilon }(x,t), \ (j=1, \ldots , M)$ , we consider the following Keller–Segel system in which the linear sum of $v_j^{M,\varepsilon }$ is imposed in nonlocal term in (P):

\begin{equation*} \left \{ \begin{aligned} \rho ^{M,\varepsilon }_t & = \rho ^{M,\varepsilon }_{xx} - \left( \rho ^{M,\varepsilon } \left( \sum _{j=1}^M a_j v_j^{M,\varepsilon } \right)_x \right)_x,\\ (v_j^{M,\varepsilon } )_t &= \frac {1}{\varepsilon } \left( d_j \left( v_{j}^{M,\varepsilon } \right)_{xx} - v_j^{M,\varepsilon } + \rho ^{M,\varepsilon } \right), \ (j=1,\cdots , M) \end{aligned} \right . \ \text{in} \ \Omega \times (0,\infty ).\qquad\qquad\qquad\quad\ \ \qquad (\text{KS}^{M,\varepsilon }) \end{equation*}

Here, $0 \lt \varepsilon \ll 1$ is a sufficiently small parameter, $d_j \gt 0$ is the diffusion coefficient, and each $a_j \in {\mathbb{R}}$ is a constant that determines whether $v_j^{M,\varepsilon }$ acts as an attractive or repulsive substance in the aggregation process of $\rho ^{M,\varepsilon }$ . Because the solutions to ( $\mbox{KS}^{M,\varepsilon }$ ) depends on $M$ and $\varepsilon$ , we denoted them by $(\rho ^{M,\varepsilon }, v_j^{M,\varepsilon })$ , respectively. The same periodic boundary condition as that in (P) is imposed in the equations of ( $\mbox{KS}^{M,\varepsilon }$ ) as follows:

(2.10) \begin{equation} \left \{ \begin{aligned} &\rho ^{M,\varepsilon }(-L,t) = \rho ^{M,\varepsilon }(L, t), \ t\gt 0, \\ &\rho ^{M,\varepsilon }_x(-L, t) = \rho ^{M,\varepsilon }_x( L, t), \ t\gt 0,\\ &v_j^{M,\varepsilon }(-L,t) = v_j^{M,\varepsilon }(L,t), \ t\gt 0, \\ &\left(v_j^{M,\varepsilon }\right)_x(-L,t) = \left(v_j^{M,\varepsilon }\right)_x(L,t), \ t\gt 0 \\ \end{aligned} \right . \end{equation}

for $j=1, \ldots , M$ . Furthermore, we impose the following initial conditions as

(2.11) \begin{equation} \rho ^{M,\varepsilon } (x, 0) \,:\!=\,\rho _0^{M,\varepsilon }(x) = \rho _0(x), \quad v_j^{M,\varepsilon }(x, 0) \,:\!=\, ( v_{j})_0(x), \quad (j=1, \ldots , M). \end{equation}

( $\mbox{KS}^{M,\varepsilon }$ ) is a Keller–Segel system with multiple components with the linear sensitivity function. The role of $v_j^{M,\varepsilon }$ can be distinguished by the sign of the coefficient $a_j$ . If $a_j\gt 0$ , then $v_j^{M,\varepsilon }$ is an attractive substance for $\rho ^{M,\varepsilon }$ and $\rho ^{M,\varepsilon }$ aggregates toward to the region in which the gradient of $v_j^{M,\varepsilon }$ is high independently on the value of its concentration. In contrast, if $a_j\lt 0$ , $v_j^{M,\varepsilon }$ acts as a repulsive substance for $\rho ^{M,\varepsilon }$ , and $\rho ^{M,\varepsilon }$ migrates away from the region in which the gradient of $v_j^{M,\varepsilon }$ is high.

Introducing the following function:

(2.12) \begin{align} &G^\varepsilon _j(x,t) \,:\!=\, \frac {1 }{2L } \sum _{n \in \mathbb{Z}} e^{ - \frac { d_j \sigma _n^2 + 1 }{ \varepsilon }t } e^{i\sigma _n x}, \end{align}

we define the maps $\Psi _j: E_{\tau } \to E_{\tau }$ and $\Phi : E_{\tau } \to E_{\tau }$ as

(2.13) \begin{align} &\Psi _j[u](x,t) \,:\!=\, (G_j^\varepsilon * (v_j)_0)(x,t) + \frac {1}{\varepsilon } \int _0^t \int _\Omega G_j^\varepsilon ( x -y, t-s ) u (y, s) dy ds, \quad u \in E_{\tau },\\[-10pt]\nonumber \end{align}

(2.14) \begin{align} &\Phi [u] (x,t) \,:\!=\, (G * \rho _0)(x,t) - \int _0^t \int _\Omega G(x-y, t-s) \left( u \left( \sum _{j=1}^M a_j \Psi _j[u] \right)_x \right)_x (y, s) dy ds, \quad u \in E_{\tau }, \end{align}

respectively. We now define the scalar integral equation of ( $\mbox{KS}^{M,\varepsilon }$ ) as $\rho ^{M,\varepsilon } = \Phi [\rho ^{M,\varepsilon }]$ . We say that a function $(u,\{v_j\}_j) \in C([0,T]; H^1(\Omega ))\times C( [0, T]; C^2(\Omega ) )$ for any $T\gt 0$ is a mild solution to ( $\mbox{KS}^{M,\varepsilon }$ ) with (2.11), provided $ u=\Phi [u]$ and $v_j=\Psi _j[u]$ for $j=1,\ldots ,M$ .

With the above settings, we obtain a unique solution to ( $\mbox{KS}^{M,\varepsilon }$ ) using the fixed point argument.

Theorem 2.2. Let $R\gt 0$ be an arbitrary real number and assume

(2.15) \begin{align} & \rho _0\in H^1(\Omega ) \ \text{with } \left \| \rho _0 \right \|_{H^1(\Omega )} \lt R,\\[-10pt]\nonumber \end{align}

(2.16) \begin{align} & ( v_{j})_0\in C^2(\Omega ), \quad j=1, \ldots , M. \end{align}

Then, there exists a real positive number $\tau _0=\tau _0(M, \{a_j,d_j\}_{j=1}^M, L, R)$ such that for any $\varepsilon \gt 0$ there exists a unique mild solution $(\rho ^{M,\varepsilon }, \{ v_j^{M,\varepsilon } \}_{j=1}^M)$ to ( $\mbox{KS}^{M,\varepsilon }$ ) in $ C( [0, \tau _0]; H^1(\Omega ) ) \times C( [0, \tau _0]; C^2(\Omega ) )$ satisfying

\begin{equation*} \left \| \rho ^{M,\varepsilon } \right \|_{ C([0, \tau _0]; H^1 (\Omega ))} \lt 2R. \end{equation*}

We can extend the existence time of the solution to an arbitrary time $T$ as follows:

Corollary 2.3. Suppose the same assumption of Theorem 2.2 . For any $T\gt 0$ , there exists a positive constant $\tilde {C}_0=\tilde {C}_0(\tau _0,T)$ such that there exists a unique mild solution $(\rho ^{M,\varepsilon }, \{v_j^{M,\varepsilon }\}_{j=1}^M)$ to ( $\mbox{KS}^{M,\varepsilon }$ ) in $ C( [0, T]; H^1(\Omega ) ) \cap L^2( 0, T; H^2(\Omega ) ) \cap H^1(0,T;L^2(\Omega )) \times C( [0, T]; C^2(\Omega ) )\cap L^2( 0, T; H^3(\Omega ) )\cap H^1( 0, T; L^2(\Omega ) )$ satisfying

\begin{equation*} \left \| \rho ^{M,\varepsilon } \right \|_{ C([0,T]; H^1 (\Omega ))} \lt \tilde {C}_0. \end{equation*}

Moreover, this mild solution $(\rho ^{M,\varepsilon }, \{v_j^{M,\varepsilon }\}_{j=1}^M)$ satisfies the system ( $\mbox{KS}^{M,\varepsilon }$ ) in $L^2(0,T;L^2(\Omega ))\times L^2(0,T;C(\Omega ))$ .

In order to show the relationship between the solutions of nonlocal Fokker–Planck equation (P) with any potential $W$ and the Keller–Segel system ( $\mbox{KS}^{M,\varepsilon }$ ), we first investigate the relationship between the solution to (P) with the potential provided by $W= \sum _{j=1}^M a_j k_j$ and the solution to ( $\mbox{KS}^{M,\varepsilon }$ ).

Theorem 2.4. Let $M$ be an arbitrary fixed natural number, and $\rho$ be a solution to (P) equipped with $W = \sum _{j=1}^M a_j k_j$ and the initial value $\rho _0 \in C^2(\Omega )$ . Let $\rho ^{M,\varepsilon }$ be a solution to ( $\mbox{KS}^{M,\varepsilon }$ ) equipped with

(2.17) \begin{align} & \rho ^{M,\varepsilon } _0 = \rho _0,\\[-10pt]\nonumber\end{align}

(2.18) \begin{align} &( (v_{1})_0, \ldots , (v_{M})_0 ) = ( k_1*\rho _0, \ldots , k_M*\rho _0) . \end{align}

Then, for any $T\gt 0$ , there exist positive constants $C_1$ and $C_2$ that depend on $M$ , $\{a_j,d_j\}_{j=1}^M$ , $L$ , $R$ and $T$ such that for any $\varepsilon \gt 0$

(2.19) \begin{align} &\left \| \rho ^{M,\varepsilon } - \rho \right \|_{ C([0,T]; H^1(\Omega ))} + \left \| \rho ^{M,\varepsilon } - \rho \right \|_{ L^2(0,T; H^2(\Omega ))} \le C_1\varepsilon ,\\[-10pt]\nonumber \end{align}

(2.20) \begin{align} &\left \| v_j^{M,\varepsilon } - k_j*\rho \right \|_{ C([0,T]; H^1(\Omega ))} + \left \| v_j^{M,\varepsilon } - k_j*\rho \right \|_{ L^2(0,T; H^2(\Omega ))}\le C_2 \varepsilon . \end{align}

Remark 2.5. We note that $C_1$ and $ C_2$ are independent of $\varepsilon$ . Moreover, using the Sobolev embedding theorem for the first terms on the left-hand sides of (2.19) and (2.20) implies that

\begin{align*} &\left \| \rho ^{M,\varepsilon } - \rho \right \|_{ C([0,T]; C(\Omega ))} \le \tilde {C}_1\varepsilon ,\\[4pt] &\left \| v_j^{M,\varepsilon } - k_j*\rho \right \|_{ C([0,T]; C(\Omega ))} \le \tilde {C}_2 \varepsilon , \end{align*}

where $\tilde {C}_1$ and $\tilde {C}_2$ are obtained by multiplying the constants of the Sobolev embedding theorem by $C_1$ and $C_2$ , respectively.

The first convergence in Theorem 2.4 shows not only that the solution $\rho ^{M,\varepsilon }$ to ( $\mbox{KS}^{M,\varepsilon }$ ) is sufficiently close to that to (P) with $W= \sum _{j=1}^M a_j k_j$ when $\varepsilon$ is very small, but also that the convergence rate is of the order of $\varepsilon$ . The second convergence shows that the solution of the auxiliary substances $v_j^{M,\varepsilon }$ is also extremely close to $k_j*\rho$ as $\varepsilon$ tends to $0$ . The proof of this theorem is presented in Section 4.

Using the convergence of Theorem 2.4, we can approximate the solution to (P) with any even smooth kernel $W$ as that to ( $\mbox{KS}^{M,\varepsilon }$ ) with the specified parameters $\{d_j, a_j \}_{j=1}^M$ . Indeed, the parameters $\{ a_j\}_{j=1}^M$ are determined by the shape of $W$ using Theorem 5.3 in Section 5. Using the interpolation polynomial with the Chebyshev nodes, we can demonstrate the convergence in Theorem 5.3. The explicit formula for the coefficients in the Lagrange interpolation polynomial for an arbitrarily given function is constructed in Proposition 5.2 for the proof of Theorem 5.3. Although Theorem 5.3 and Proposition 5.2 are some of our main results, they are presented in Section 5 for convenience. Setting the diffusion coefficients of $v_j^{M,\varepsilon }$ as

(2.21) \begin{equation} d_1:\text{ sufficiently large}, \quad d_j = \frac {1}{(j-1)^2}, \ j=2,\cdots , M \end{equation}

and $d_1$ may be limit of infinity, we obtain

\begin{equation*} k_1(x)=\frac {1}{2L}, (d_1\to \infty ), \quad k_j(x) = \frac { j-1}{2 \sinh (j-1) L } \cosh (j-1)(L - \left | x \right |). \end{equation*}

The choice of $d_j$ is same as that in [Reference Ninomiya, Tanaka and Yamamoto19]. Because the profile of the fundamental solution $k_j$ is unimodal even if the value of $j$ changes, it seems to be difficult to approximate any potential $W$ by the linear sum of $k_j$ . However, we can obtain the following corollary. Let $f$ be

(2.22) \begin{equation} f(x) \,:\!=\, W\big(L - \log \big(x+\sqrt {x^2-1}\big)\big) = W( L- \cosh ^{-1}(x)), \end{equation}

and we set the assumption that for $n \in \mathbb{N}$

(2.23) \begin{equation} \lim _{x\to 1+0}f^{(n+1)}(x)=f^{(n+1)}(1)\lt \infty , \end{equation}

and there exists a positive constant independent of $n$ such that

(2.24) \begin{equation} \max _{x\in [1,\cosh L] } | f^{(n+1)}(x) | \le C \end{equation}

for any $n \in \mathbb{N}$ . We note that the function $ L- \cosh ^{-1}(x)$ is the inverse function of $\cosh (L-x)$ on $[0,L]$ . Since $(L- \cosh ^{-1}(x))'=-1/\sqrt {x^2-1}\,=\!:\,\mathscr{F}$ and it satisfies the recurrence formula $(x^2-1)\mathscr{F}^{(n+2)} = -x(2n+1)\mathscr{F}^{(n+1)} -n^2\mathscr{F}^{(n)}$ , we see that $f \in C^{n+1}((1, \cosh L])$ . Thus, from the assumption (2.23), we obtain that $f \in C^{n+1}([1, \cosh L])$ .

Corollary 2.6. Assume that $W \in C^\infty (\Omega )$ , $W$ is even, (2.23) and (2.24) for any $n\in \mathbb{N}$ . Then, for any $M \in \mathbb{N}$ , there exist constants $\{a_j\}_{j=1}^M$ and a positive constant $C_{W}$ that is independent of $M$ such that

(2.25) \begin{equation} \left\| W - \sum _{j=1}^M a_j k_j \right\|_{C^1(\Omega )} \le C_W \frac { M }{2^{M-1} (M-1)!} \left( \frac {\cosh L -1 }{2} \right)^{M-1} . \end{equation}

Remark 2.7. Although the convergence rate becomes worse, we can relax the condition (2.24) as

\begin{equation*} \begin{cases} {\displaystyle }\max _{x\in [1,\cosh L] } | f^{(n+1)}(x) | = O(n!), \ &(0\lt L\lt \log (5+2\sqrt {6})),\\[3pt] \text{There exists a constant} \ C\gt 0 \ \text{such that} {\displaystyle }\max _{x\in [1,\cosh L] } | f^{(n+1)}(x) | = O(C^n), \ &( \log (5+2\sqrt {6}) \le L). \end{cases} \end{equation*}

The evaluation of uniform convergence up to derivatives of the Lagrange interpolating polynomials with Chebyshev nodes for $L$ and $n$ , such as using the Lebesgue constant, is an open problem. Examples of the potential $W$ are $\cosh j(L-|x|)$ and $-(\!\cosh ^2(L-|x|)-1)\cos (\!\cosh (L-x))$ . In the former case, $f(x) = T_j(x)$ , and in the latter case, $f(x) = -(x^2-1) \cos x$ for $x \in [1,\cosh L]$ .

There are degrees of freedom in the choices of $d_j$ and $k_j$ , that is, the second equation of the approximation equation ( $\mbox{KS}^{M,\varepsilon }$ ). It is not known whether the above choices are optimal in terms of convergence of approximation errors and dependence on the number of approximation equations, and is an open question. On the other hand, as for the second equation of ( $\mbox{KS}^{M,\varepsilon }$ ), the superposition of fundamental solutions as in Corollary 2.6 can approximate even functions in $C^1(\Omega )$ , and in this sense, ( $\mbox{KS}^{M,\varepsilon }$ ) is suitable for approximating nonlocal Fokker–Planck equation.

Because we estimate the error of the solutions of the two nonlocal Fokker–Planck equations with $W=\sum _{j=1}^M a_j k_j$ and any given potential $W(x)$ , we prepare the following lemma.

Lemma 2.8. Suppose that $w_1, w_2 \in W_{\mathrm{per}}^{1,1}(\Omega )$ and let $\rho _j, \ (j=1,2)$ denote the solution to

\begin{align*} {(\mbox{P}_j)} \qquad \left \{ \begin{aligned} & (\rho _j)_t = ( \rho _j )_{xx} - ( \rho _j (w_j*\rho _j)_x )_x \ \text{in} \ \Omega \times (0,\infty ),\\ &\rho _j(x, 0)=\rho _0\in H^1(\Omega )\\ \end{aligned} \right . \end{align*}

with the periodic boundary condition

\begin{equation*} \left \{ \begin{aligned} &\rho _j(-L,t) = \rho _j(L, t), \ t\gt 0, \\ &(\rho _j)_x(-L, t) = (\rho _j)_x( L, t), \ t\gt 0, \end{aligned} \right . \end{equation*}

respectively. Then for any $T\gt 0$ , there exists a positive constant ${\tilde C_T} = {\tilde C_T} (\rho _0, L, T, \|w_{1,x} \|_{L^1(\Omega )}, \|w_{2,x} \|_{L^1(\Omega )})$ such that

(2.26) \begin{equation} \left \| \rho _1 - \rho _2 \right \|_{ C([0,T];H^1 (\Omega ))}^2 + \left \| \rho _1 - \rho _2 \right \|_{ L^2 (0,T;H^2(\Omega ))}^2 \le {\tilde C_T} \left \| w_1 - w_2 \right \|_{ W^{1,1} (\Omega )}^2. \end{equation}

This lemma shows that the difference between the solutions to the two Fokker–Planck equations is bounded by the difference between the two potentials. The proof is presented in Subsection 4.3

Referring to $\{ \alpha _j^{M-1}\}$ in Theorem 5.3, we put

(2.27) \begin{equation} a_{j}=\left \{ \begin{aligned} &2L\alpha _0^{M-1}, \quad (j=1),\\[3pt] &2 \alpha _{j-1}^{M-1} (\sinh ((j-1) L) )/ (j-1), \quad (j=2,\ldots M). \end{aligned} \right . \end{equation}

The approximation of $W$ in Theorem 5.3 requires a constant term with $j=0$ in $\sum _{j=0}^n\alpha _j^n \cosh j(L-|x|)$ . From this requirement, it is necessary to define $d_1$ as a sufficiently large value. We estimate the differences between the two solutions to (P) with an arbitrary even $W$ in $C^\infty (\Omega )$ and $W = \sum _{j=1}^M a_j k_j$ by using (2.25) in Corollary 2.6 and (2.26) in Lemma 2.8. Moreover, because we can estimate the solutions to ( $\mbox{KS}^{M,\varepsilon }$ ) and (P) with $W = \sum _{j=1}^M a_j k_j$ from Theorem 2.4, we obtain the following main result.

Theorem 2.9. For any even 2L-periodic function $W$ in $C^\infty (\Omega )$ with (2.23) and (2.24) for any $n \in \mathbb{N}$ , any time $T\gt 0$ , any $M \in \mathbb{N}$ and any $\varepsilon \gt 0$ , there exists a Keller–Segel system ( $\mbox{KS}^{M,\varepsilon }$ ) with $M+1$ component, a positive constant $C_T^{(1)}$ that is independent of $M$ and $\varepsilon$ , and a positive constant $C_T^{(2)}=C_T^{(2)}(M)$ that is independent of $\varepsilon$ such that

(2.28) \begin{equation} \left \| \rho ^{M,\varepsilon } - \rho \right \|_{ C([0,T]; H^1 (\Omega ))} \le C_T^{(1)} \frac { M }{2^{M-1} (M-1)!} \left( \frac {\cosh L -1 }{2} \right)^{M-1} + C_T^{(2)}(M)\varepsilon , \end{equation}

where $\rho$ is the solution to (P) equipped with $\rho _0 \in C^2(\Omega )$ and $\rho ^{M,\varepsilon }$ is the first component of the solution to ( $\mbox{KS}^{M,\varepsilon }$ ) equipped with (2.17) and (2.18).

This theorem shows that the solution to the nonlocal Fokker–Planck equation with any potential can be approximated by that to the multiple components of the Keller–Segel system with specified parameters. This convergence result shows a relationship between any advective nonlocal interactions and local dynamics. Furthermore, the convergence rate is specified with respect to $M$ and $\varepsilon$ . Since the right-hand side of (2.28) includes the power and factorial terms, this indicates that the error in $({\rm{KS}}^{M,0})$ can converge to $0$ rapidly.

3. Mild solution

To show Theorem 2.2, we present some lemmas. Throughout this section, we set $d_j\gt 0$ , and $a_j$ are constants for $j=1,\ldots ,M$ .

3.1. Fundamental solution and boundedness

First, we provide the following lemma:

Lemma 3.1. Functions $G$ of (2.9) and $G_j^{\varepsilon }$ of (2.12) are the fundamental solution for $u_t = u_{xx}$ and

(3.29) \begin{equation} u_t = ( d_j u_{xx} - u)/ \varepsilon \end{equation}

with periodic boundary condition of (2.3) type, respectively.

The proof of this lemma is obtained by substituting and calculating.

Next, we have the following lemma.

Lemma 3.2. Assume $\phi \in E_{\tau }$ . Then (2.13) satisfies

\begin{equation*} \left \{ \begin{aligned} & ( \Psi _j[\phi ] )_t = \frac {1}{\varepsilon } \left( d_j (\Psi _j[\phi ])_{xx} - \Psi _j[\phi ] +\phi \right), \quad x\in \Omega , \ t\gt 0, \\ &\Psi _j[\phi ](x, 0) = (v_{j})_0(x), \quad x\in \Omega . \end{aligned} \right . \end{equation*}

The proof of this lemma is provided by substituting (2.13) into the equation and calculating. Next, we estimate the boundedness of $\Psi _j[\phi ]$ in the following lemma.

Lemma 3.3. Assume that $ \phi \in E_{\tau }$ and (2.16), and let $C_3$ be a positive constant given by $C_3 \,:\!=\,\sqrt { L} /(d_j \sqrt {6})$ . Then, (2.13) satisfies the following estimates

(3.30) \begin{align} & \left \| ( \Psi _j[\phi ] )_{x} \right \|_{C([0, \tau ]; C(\Omega ))} \le \left \| ( v_{j})_{0,x} \right \|_{ C (\Omega )} + C_3 \left \| \phi \right \|_{ C([0, \tau ]; L^2 (\Omega ) ) } ,\\[-10pt]\nonumber \end{align}

(3.31) \begin{align} & \left \| ( \Psi _j[\phi ] )_{xx} \right \|_{C([0, \tau ]; C(\Omega ))} \le \left \| ( v_{j})_{0,xx} \right \|_{ C (\Omega )} + C_3 \left \| \phi _x \right \|_{ C([0, \tau ]; L^2 (\Omega )) } . \end{align}

Proof of Lemma 3.3. From the triangular inequality, we have

(3.32) \begin{equation} \begin{aligned} \left \| ( \Psi _j[\phi ] )_{x} \right \|_{C([0, \tau ]; C(\Omega ))} &\le \left \| G_j^\varepsilon *( v_{j})_{0,x} \right \|_{C([0, \tau ]; C (\Omega ))} + \frac {1}{\varepsilon } \left \| \int _0^t \int _{\Omega } (G_{j}^\varepsilon )_x (\cdot -y, \cdot -s) \phi (y,s) dy ds \right \|_{ C([0, \tau ]; C (\Omega ))}. \end{aligned} \end{equation}

By the maximum principle for the heat equation (3.29), we compute the first term.

\begin{align*} \left \| G_j^\varepsilon *( v_{j})_{0,x} \right \|_{C([0, \tau ]; C (\Omega ))} &\le \left \| \int _{\Omega } G_j^\varepsilon (\cdot -y, 0)( v_{j})_{0,x} (y) dy \right \|_{ C (\Omega )} = \left \| ( v_{j})_{0,x} \right \|_{ C (\Omega )}. \end{align*}

The last equality is based on the Fourier series expansion. Next, we denote the Fourier coefficient of $\phi$ by

\begin{equation*} p_n(t) \,:\!=\, \frac {1}{\sqrt {2L}}\int _\Omega \phi (x,t) e^{-i\sigma _n x} dx. \end{equation*}

Before estimating the second term of (3.32), we can compute that

\begin{align*} \sum _{n \in \mathbb{Z}} \left( \frac { | \sigma _n| }{d_j\sigma _n^2 + 1} \right)^2 & =\sum _{n \ne 0} \left( \frac { | \sigma _n| }{d_j\sigma _n^2 + 1} \right)^2 = 2 \sum _{n = 1}^\infty \left( \frac { \sigma _n }{d_j\sigma _n^2 + 1} \right)^2 \\ &\le 2 \sum _{n = 1}^\infty \left( \frac { 1 }{d_j\sigma _n } \right)^2 = \frac {2}{d_j^2} \frac { L^2 }{\pi ^2} \sum _{n = 1}^\infty \frac {1}{n^2} = \frac { L^2 }{3 d_j^2 }. \end{align*}

Then, we see that

\begin{align*} &\frac {1}{\varepsilon } \left \| \int _0^t \int _{\Omega } \big(G_{j}^\varepsilon \big)_x (\cdot -y, \cdot -s) \phi (y,s) dy ds \right \|_{C([0, \tau ]; C (\Omega ))} \\ &= \frac {1}{\varepsilon } \sup _{\substack { t\in [0, \tau ], \\ x \in \Omega }} \left | \frac {1}{2L} \sum _{n \in \mathbb{Z}} i\sigma _n e^{ - \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }t } e^{ i\sigma _n x } \int _0^t e^{ \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }s } \int _\Omega e^{- i \sigma _n y} \phi (y,s) dy ds \right |\\ &= \frac {1}{\varepsilon } \sup _{\substack { t\in [0, \tau ], \\ x \in \Omega }} \left | \frac {1}{\sqrt {2L}} \sum _{n \in \mathbb{Z}} i\sigma _n e^{ - \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }t } e^{ i\sigma _n x } \int _0^t e^{ \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }s } p_n(s) ds \right |\end{align*}

\begin{align*} &\le \frac {1}{\sqrt {2L}} \sup _{\substack { t\in [0, \tau ], \\ x \in \Omega }} \left | \sum _{n \in \mathbb{Z}} i\sigma _n e^{ i\sigma _n x } \frac { 1 }{d_j\sigma _n^2 + 1} \left( 1 - e^{ - \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }t } \right) \sup _{s \in [0,t]} | p_n(s) | \right | \\ &\le \frac {1}{\sqrt {2L}} \sup _{ t\in [0, \tau ] } \sum _{n \in \mathbb{Z}} \frac { | \sigma _n | }{d_j\sigma _n^2 + 1} \left( 1 - e^{ - \frac { d_j\sigma _n^2 + 1 }{ \varepsilon }t } \right) | p_n(t) | \le \frac {1}{\sqrt {2L}} \sum _{n \in \mathbb{Z}} \frac { | \sigma _n | }{d_j\sigma _n^2 + 1} \sup _{ t\in [0, \tau ] } | p_n(t) | \\ & \le \frac {1}{\sqrt {2L}} \sqrt { \sum _{n \in \mathbb{Z}} \left( \frac { | \sigma _n| }{d_j\sigma _n^2 + 1} \right)^2 } \sup _{ t\in [0, \tau ] }\sqrt { \sum _{n \in \mathbb{Z}} | p_n(t) |^2 } \le C_3 \left \| \phi \right \|_{ C( [0, \tau ]; L^2 (\Omega ))}. \end{align*}

Thus, we obtain the first estimation. Replacing the functions $(v_{j})_{0,x}$ and $ \phi$ with $(v_{j})_{0,xx}$ and $ \phi _{x}$ in (3.32), respectively, we have also the second assertion of the boundedness by the same calculation as above.

Because we will use the following boundedness several times, we give the following lemma:

Lemma 3.4. Assume $f=f(x) \in L^2(\Omega )$ and $ g = g(x,t)\in C( [0,T];L^2(\Omega ))$ for any $T\gt 0$ , and let $C_4$ be a positive constant given by $C_4\,:\!=\,\pi /(d_jL)$ . Then, we obtain that, for all $t\in (0,T]$ ,

(3.33) \begin{align} &\left \| G(\cdot , t)*f \right \|_{ L^2 (\Omega )} \le \left \| f \right \|_{ L^2 (\Omega ) },\\[-10pt]\nonumber \end{align}

(3.34) \begin{align} &\left \| \int _0^t \int _\Omega G( \cdot -y , t-s) g(y,s) dy ds \right \|_{ L^2 (\Omega )} \le t \left \| g \right \|_{ C([0,T]; L^2 (\Omega )) },\\[-10pt]\nonumber \end{align}

(3.35) \begin{align} &\left \| \int _0^t \int _\Omega G_x(\! \cdot -y , t-s) g(y,s) dy ds \right \|_{ L^2 (\Omega )} \le \sqrt {t} \left \| g \right \|_{ C([0,T]; L^2 (\Omega )) },\\[-10pt]\nonumber \end{align}

(3.36) \begin{align} &\frac {1}{\varepsilon } \left \| \int _0^t \int _\Omega G_j^\varepsilon ( \cdot -y , t-s) g(y,s) dy ds \right \|_{ L^2 (\Omega )} \le \left \| g \right \|_{ C([0,T]; L^2 (\Omega )) },\\[-10pt]\nonumber \end{align}

(3.37) \begin{align} &\frac {1}{\varepsilon } \left \| \int _0^t \int _\Omega (G_{j}^\varepsilon )_x(\! \cdot -y , t-s) g(y,s) dy ds \right \|_{ L^2 (\Omega )} \le C_4 \left \| g \right \|_{ C([0,T]; L^2 (\Omega )) }. \end{align}

We note that the right-hand sides of (3.36) and (3.37) do not depend on $\varepsilon$ . The proof is based on the estimation of the Fourier coefficients of $f$ and $g$ . As this is a straightforward calculation, we present the proof in A.

3.2. Contraction map

We show that the map $\Phi$ becomes a contraction map from

\begin{equation*} B_R \,:\!=\, \{ v \in E_{\tau } \ | \ \left \| v \right \|_{ E_{\tau } } \lt 2R \} \end{equation*}

to $B_R$ taking the sufficiently small time $\tau \gt 0$ .

Lemma 3.5. Assume that $\phi \in B_R$ and (2.16). Then, (2.13) satisfies the following estimate

\begin{equation*} \sup _{ \left \| \phi \right \|_{ E_{\tau } } \lt 2R } \left \| \Big ( \phi \sum _{j=1}^M a_j ( \Psi _j[\phi ] )_x \Big )_x \right \|_{ L^2 (\Omega )} \le M_R, \end{equation*}

where

\begin{equation*} M_R \,:\!=\, 2R \sum _{j=1}^M | a_j | \Big ( \left \|( v_{j})_{0,x} \right \|_{ C(\Omega )} + \left \|( v_{j})_{0,xx} \right \|_{ C(\Omega )} +4C_3R \Big ). \end{equation*}

Since this proof is straightforward, we put it in B. Next, we have the following lemma.

Lemma 3.6. Assume that $\phi \in B_R$ and (2.15). Then, (2.14) satisfies

\begin{equation*} \left \| \Phi [ \phi ] \right \|_{ H^1 (\Omega )}(t) \le R +tM_R +\sqrt { t } M_R. \end{equation*}

Proof of Lemma 3.6. From the Minkowski inequality, we see that

(3.38) \begin{align} \left \| \Phi [ \phi ] \right \|_{ H^1 (\Omega )} (t) &\le \left \| G * \rho _0 \right \| _{ H^1 (\Omega )} (t)\notag \\ & \qquad +\left \| \int _0^t \int _\Omega G( \cdot -y , t-s) \left( \phi \left( \sum _{j=1}^M a_j \Psi _j[\phi ] \right)_x \right)_x (y,s) dy ds \right \| _{ H^1 (\Omega )} . \end{align}

Then, by using (3.33) in Lemma 3.4, we can compute the first term of (3.38) as follows:

\begin{align*} \left \| G * \rho _0 \right \| _{ H^1 (\Omega )} ^2 (t) &= \left \| G * \rho _0 \right \|_{ L^2 (\Omega )} ^2 (t) + \left \| G * (\rho _{0})_x \right \|_{ L^2 (\Omega )} ^2 (t) \\ &= \left \| \rho _0 \right \|_{ L^2 (\Omega )}^2+\left \| (\rho _{0})_x \right \|_{ L^2 (\Omega )}^2 =\left \| \rho _0 \right \|_{ H^1 (\Omega )}^2 \lt R^2. \end{align*}

Next, we estimate the second term of (3.38). Utilising (3.34) in Lemmas 3.4 and 3.5, we compute that

\begin{align*} &\left \| \int _0^t \int _\Omega G( \cdot -y , t-s) \left( \phi (\sum _{j=1}^M a_j \Psi _j[\phi ])_x \right)_x (y,s)dy ds \right \| _{ L^2 (\Omega )}^2 \le t^2 \left \| \left( \phi \sum _{j=1}^M a_j ( \Psi _j[\phi ] )_x \right)_x \right \|_{ C([0,\tau ]; L^2 (\Omega ))}^2 \le t^2 M_R^2. \end{align*}

Similarly, (3.35) in Lemma 3.4 yields that

\begin{align*} &\left \| \int _0^t \int _\Omega G_x(\! \cdot -y , t-s) \left( \phi (\sum _{j=1}^M a_j \Psi _j[\phi ])_x \right)_x (y,s) dy ds \right \| _{ L^2 (\Omega )}^2 \le t \left \| \left( \phi \sum _{j=1}^M a_j ( \Psi _j[\phi ] )_x \right)_x \right \|_{C([0,\tau ]; L^2 (\Omega ))}^2 \le tM_R^2. \end{align*}

Consequently, we observe that $\Phi : E_{\tau } \to E_{\tau }$ , and that there exists $\tau _1 \gt 0$ independent of $\varepsilon$ such that for $\tau \lt \tau _1$ $\Phi$ is a map from $B_R$ to $B_R$ .

Lemma 3.7. Assume that $\phi , \psi \in E_{\tau }$ . Then, (2.13) satisfies that

(3.39) \begin{align} &\left \| ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x \right \|_{ C([0, \tau ]; L^2 (\Omega ))} \le \left \| \phi _x - \psi _x \right \|_{ C([0, \tau ];L^2 (\Omega ) )},\\[-10pt]\nonumber \end{align}

(3.40) \begin{align} & \left \| ( \Psi _j[\phi ] )_{xx} - ( \Psi _j[ \psi ] )_{xx} \right \|_{ C([0, \tau ]; L^2 (\Omega ))} \le C_4 \left \| \phi _x - \psi _x \right \|_{ C([0, \tau ]; L^2 (\Omega ) ) }, \end{align}

where $C_4$ denotes the constant defined in Lemma 3.4 .

Since the proof is simply based on using Lemma 3.4, it is put in B. Finally, we show that map $\Phi$ becomes a contraction map by taking a sufficiently small $\tau _2\gt 0$ and setting $\tau = \tau _2$ .

Lemma 3.8. Assume that $\phi , \psi \in E_{\tau }$ . Then there exists a positive constant $C_5$ independent of $\varepsilon$ such that

\begin{equation*} \left \| \Phi [\phi ] - \Phi [\psi ] \right \|_{ H^1 (\Omega )} (t) \le C_5 \sqrt {t} \left \| \phi - \psi \right \|_{ C([0,\tau ]; H^1 (\Omega ))}. \end{equation*}

Proof of Lemma 3.8. Since

(3.41) \begin{equation} \left \| \Phi [\phi ] - \Phi [\psi ] \right \|_{ H^1 (\Omega )} ^2 (t) = \left \| \Phi [\phi ] - \Phi [\psi ] \right \|_{ L^2 (\Omega )} ^2 (t) + \left \| ( \Phi [\phi ] - \Phi [\psi ] )_x \right \|_{ L^2 (\Omega )} ^2 (t), \end{equation}

we estimate each term on right-hand side. We can compute that

\begin{align*} \left \| \Phi [\phi ] - \Phi [\psi ] \right \|_{ L^2 (\Omega )} (t) &= \left \| \int _0^t \int _\Omega G( \cdot - y, t-s ) \left( \phi \sum _{j=1}^M a_j ( \Psi _j[\phi ] )_x - \psi \sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right)_x (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ &\le \left \| \int _0^t \int _\Omega G( \cdot - y, t-s ) \left( \phi \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x ) \right)_x (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ &\quad + \left \| \int _0^t \int _\Omega G( \cdot - y, t-s ) \left( ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right)_x (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ &\,=\!:\, K_1(t) + K_2(t). \end{align*}

Using (3.35) in Lemma 3.4, we estimate that

\begin{align*} K_1(t)&= \left \| \int _0^t \int _\Omega G_x(\! \cdot -y , t-s)\left( \phi \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x ) \right) (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ &\le \sqrt { t} \left \| \phi \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x ) \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \\ & \le \sqrt { t} \left \| \phi \right \|_{ C([0,\tau ]; C (\Omega ))} \left \| \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x ) \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \\ & \le \sqrt { t} \left \| \phi \right \|_{ C([0,\tau ]; C (\Omega ))} \sum _{j=1}^M |a_j| \left \| ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \\ & \le \sqrt { t} \left \| \phi \right \|_{ C([0,\tau ]; C (\Omega ))} \sum _{j=1}^M |a_j| \left \| \phi _x - \psi _x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))}, \end{align*}

where the boundary integration from the integration by parts in the first equality becomes $0$ because of the periodicity, we used $\phi \in C(\Omega \times [0,\tau ])$ from the Sobolev embedding theorem, the Minkowski inequality, and (3.39) in Lemma 3.7.

Similarly, to this estimation, (3.34) in Lemma 3.4 yields that

\begin{align*} K_2(t) &= \left \| \int _0^t G( \cdot - y, t-s ) \left( ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right)_x (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ & \le t \left \| \left( ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right)_x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \\ & \le t \left( \left\| ( \phi _x - \psi _x )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} + \left \| ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_{xx} \right\|_{ C( [0, \tau ]; L^2 (\Omega ))} \right) \\ &\le t \left( \sum _{j=1}^M | a_j | \left \| ( \Psi _j[ \psi ] )_x \right \|_{ C([0,\tau ]; C (\Omega ))} \left \| \phi _x - \psi _x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \right.\\ &\quad \quad \left. + \sum _{j=1}^M | a_j | \left \| ( \Psi _j[ \psi ] )_{xx} \right \|_{ C([0,\tau ]; C (\Omega ))} \left \| \phi - \psi \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \right)\\ &\le t \sqrt {2} C_6 \left \| \phi - \psi \right \|_{ C( [0, \tau ]; H^1 (\Omega ))}, \end{align*}

where we utilised the Minkowski inequality, the boundedness (3.30) and (3.31) in Lemma 3.3, and

(3.42) \begin{equation} C_6\,:\!=\, \max \left\{ \sum _{j=1}^M | a_j | \left \| ( \Psi _j[ \psi ] )_x \right \|_{ C([0,\tau ];C (\Omega ))}, \sum _{j=1}^M | a_j | \left \| ( \Psi _j[ \psi ] )_{xx} \right \|_{ C([0,\tau ];C (\Omega ))} \right\}. \end{equation}

We note that $C_6$ does not depend on $\varepsilon$ .

Next, we estimate the second term of (3.41) on right-hand side. First, we write as

\begin{align*} \left \| ( \Phi [\phi ] - \Phi [\psi ] )_x \right \|_{ L^2 (\Omega )} (t) & \le \left\| \int _0^t \int _\Omega G_x(\! \cdot -y, t-s ) \left( \phi \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x ) \right)_x(y,s) dy ds \right\|_{ L^2 (\Omega )} \\ &+ \left\| \int _0^t \int _\Omega G_x(\! \cdot -y, t-s ) \left( ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right)_x (y,s) dy ds \right \|_{ L^2 (\Omega )} \\ &\,=\!:\, \mathscr{K}_1(t) + \mathscr{K}_2(t). \end{align*}

Similarly, to the previous estimations, using (3.35) in Lemma 3.4, we obtain that

\begin{align*} \mathscr{K}_1(t) &\le \sqrt {t} \left( \left \| \phi _x \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x )\right\|_{ C( [0, \tau ]; L^2 (\Omega ))} + \left \| \phi \sum _{j=1}^M a_j ( ( \Psi _j[\phi ] )_{xx} - ( \Psi _j[ \psi ] )_{xx} )\right\|_{ C( [0, \tau ]; L^2 (\Omega ))} \right) \\ & \le \sqrt {t} \left( \sum _{j=1}^M | a_j | \left \| ( \Psi _j[\phi ] )_x - ( \Psi _j[ \psi ] )_x \right \|_{C( [0, \tau ]; C (\Omega ))} \left \| \phi _x \right \|_{C( [0, \tau ]; L^2 (\Omega ))} \right.\\ & \qquad \qquad \left. + \sum _{j=1}^M | a_j | \left \| \phi \right \|_{ C( [0, \tau ]; C (\Omega ))} \left \| ( \Psi _j[\phi ] )_{xx} - ( \Psi _j[ \psi ] )_{xx}\right \|_{C( [0, \tau ]; L^2 (\Omega ))} \right)\\ & \le \sqrt {t} \sum _{j=1}^M | a_j | \left( C_3 \left \| \phi _x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \left \| \phi - \psi \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} +C_4 \left \| \phi \right \|_{C( [0, \tau ]; C (\Omega ))} \left \| \phi _x - \psi _x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))}\right)\\ &\le \sqrt {t} \sqrt {2} C_7 \left \| \phi - \psi \right \|_{C( [0, \tau ]; H^1 (\Omega ))}, \end{align*}

where we used the boundedness (3.30) in Lemma 3.3, the boundedness that $\phi \in C([0,\tau ]; C(\Omega ))$ , (3.39) and (3.40) in Lemma 3.7, and we put

\begin{equation*} C_7 \,:\!=\, \max \left\{ \left \| \phi _x \right \|_{C([0,\tau ]; L^2 (\Omega ))} \left( \sum _{j=1}^M | a_j |C_3 \right), \ \left \| \phi \right \|_{ C([0,\tau ]; C (\Omega )) } \left( \sum _{j=1}^M | a_j | C_4 \right) \right\}. \end{equation*}

It should also be noted that $C_7$ does not depend on $\varepsilon$ .

Finally, using (3.35) in Lemma 3.4, we obtain that

\begin{align*} \mathscr{K}_2(t) &\le \sqrt {t} \left( \left\| ( \phi _x - \psi _x )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} + \left \| ( \phi - \psi )\sum _{j=1}^M a_j ( \Psi _j[ \psi ] )_{xx} \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \right)\\ &\le \sqrt {t} \sum _{j=1}^M |a_j| \Big ( \left \| ( \Psi _j[ \psi ] )_x \right \|_{ C( [0, \tau ]; C (\Omega ))} \left \| \phi _x - \psi _x \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \\ &\qquad \qquad \qquad + \left \| ( \Psi _j[ \psi ] )_{xx} \right \|_{C( [0, \tau ]; C (\Omega ))} \left \| \phi - \psi \right \|_{ C( [0, \tau ]; L^2 (\Omega ))} \Big )\\ &\le \sqrt {t} \sqrt {2} C_6 \left \| \phi - \psi \right \|_{C( [0, \tau ]; H^1 (\Omega ))}, \end{align*}

where $C_6$ is as defined in (3.42). Putting

\begin{equation*} C_5\,:\!=\,\left \| \phi \right \|_{ C([0,\tau ]; C (\Omega ))} \sum _{j=1}^M |a_j| +\sqrt {2} ( C_6\sqrt {\tau }+C_6+C_7), \end{equation*}

we complete the proof.

Consequently, taking a sufficiently small value $\tau =\tau _2\gt 0$ which is independent of $\varepsilon$ , we obtain

\begin{equation*} \left \| \Phi [\phi ] - \Phi [\psi ] \right \|_{ C([0, \tau _2]; H^1 (\Omega ))} \le \frac 1 2 \left \| \phi - \psi \right \|_{ C([0,\tau _2]; H^1 (\Omega ))}. \end{equation*}

Thus, the map $\Phi \,:\, B_R \to B_R$ is a contraction map.

Proof of Theorem 2.2. By setting $\tau _0 \,:\!=\, \min \{ \tau _1, \tau _2\}$ , we see that the map $\Phi : B_R \to B_R$ is a contraction map. From the Banach fixed-point theorem, the equation $\rho ^{M,\varepsilon } = \Phi [\rho ^{M,\varepsilon }]$ has a unique solution in $C([0,\tau _0]; H^1(\Omega ))$ . Since $ \Phi [\rho ^{M,\varepsilon }](-L,t)=\Phi [\rho ^{M,\varepsilon }](L,t)$ and $\Phi _x[\rho ^{M,\varepsilon }](-L,t)=\Phi _x[\rho ^{M,\varepsilon }](L,t)$ for $t\gt 0$ , this mild solution satisfies the periodic boundary condition.

Proof of Corollary 2.3. Repeating to use Theorem 2.2, we can connect the mild solutions for $t \in [0,T]$ at any time $T\gt 0$ . Using the term-by-term of the weak derivative for the integral equations with respect to $x$ and $t$ , we observe that the solutions $\rho ^{M,\varepsilon }$ and $v_j^{M,\varepsilon }$ satisfy ( $\mbox{KS}^{M,\varepsilon }$ ) in $L^2(0,T;L^2(\Omega ))$ and $L^2(0,T;C(\Omega ))$ , respectively.

Differentiating the right-hand sides of $\rho ^{M,\varepsilon } = \Phi [\rho ^{M,\varepsilon } ]$ and $v_j^{M,\varepsilon } = \Psi _j[ \rho ^{M,\varepsilon } ]$ with respect to $x$ by applying the term-by-term differentiation, respectively, we can show that $\rho ^{M,\varepsilon } \in L^2( 0, T; H^2(\Omega ) )$ and $v_j^{M,\varepsilon } \in L^2( 0, T; H^3(\Omega ) )$ . Similarly, differentiating the right-hand side of $\rho ^{M,\varepsilon } = \Phi [\rho ^{M,\varepsilon }]$ with respect to $t$ by applying the term-by-term differentiation, we see that $\rho ^{M,\varepsilon } \in H^1(0,T; L^2(\Omega ))$ .

4. Singular limit analysis

To demonstrate Theorem 2.4, we prepare the lemmas in the following subsections. Throughout this section, we set $d_j\gt 0$ , and $a_j$ are constants for $j=1,\ldots ,M$ .

4.1. Fundamental solution

First, we have the following lemma.

Lemma 4.1. $k_j$ defined in (2.4) is the fundamental solution to

\begin{equation*} \left \{ \begin{aligned} &-d_jv_{xx} + v = \delta (x),\\ &v(-L) = v(L), \quad v_{x}(-L) = v_{x}(L), \end{aligned} \right . \end{equation*}

where $\delta$ denotes the Dirac delta function. Moreover,

\begin{align*} & (k_j)_x(x) = \left \{ \begin{aligned} &\frac { -c_k(j) }{\sqrt {d_j}} \sinh \frac { L-x }{\sqrt {d_j}},\quad x \in (0,L],\\[5pt] &\frac { c_k(j) }{\sqrt {d_j}} \sinh \frac { L+x }{\sqrt {d_j}},\quad x \in [-L,0), \end{aligned}\right . \qquad c_k(j) \,:\!=\, \frac {1}{2\sqrt {d_j} \sinh ( L / \sqrt {d_j}) } \end{align*}

holds in the weak sense.

Proof. For the proof of the second assertion, we can directly obtain the weak derivatives by multiplying $k_j$ by the test function $\varphi _x \in C_0^\infty (\Omega )$ , and the integration by parts. For any $\varphi \in C_0^\infty (\Omega )$ , we can compute that

\begin{equation*} \int _\Omega k_j(x)\varphi _{xx}(x) dx= -\frac {\varphi (0)}{d_j} + \frac {1}{d_j} \int _\Omega k_j(x) \varphi (x) dx. \end{equation*}

This implies that

\begin{equation*} \int _\Omega k_j(x)(-d_j \varphi _{xx}+\varphi )(x)dx = \varphi (0). \end{equation*}

Lemma 4.2. Let $C_8$ be a positive constant given by

\begin{equation*} C_8 \,:\!=\, 2c_k(j) \left( \cosh \frac {L}{ \sqrt { d_j} } - 1 \right). \end{equation*}

Then, $k_j$ satisfies that $\| k_j \|_{ L^1 (\Omega )} = 1,\ \| (k_j)_x \|_{ L^1 (\Omega )} = C_8$ , and

(4.43) \begin{equation} \left \| (k_j)_x \right \|_{ C (\Omega )} = \frac {1}{2 d_j}. \end{equation}

As the proof is elementary, we omit it.

4.2. Boundedness of auxiliary factors

Next, we estimate the boundedness of the solutions $(\rho ^{M,\varepsilon },\{v_j^{M,\varepsilon }\}_{j=1}^M)$ to ( $\mbox{KS}^{M,\varepsilon }$ ) in this subsection. First, we obtain the following lemma.

Lemma 4.3. Let $\rho ^{M,\varepsilon }$ be the solution to the first equation of ( $\mbox{KS}^{M,\varepsilon }$ ) with $\rho _0 \in C^2(\Omega )$ . Then, $\rho ^{M,\varepsilon } \in C^1([0,T];L^2(\Omega ))$ and there exist positive constants $C_9$ and $C_{10}$ that depend on $(\rho _0)_{xx}, M, \{a_j,d_j\}_{j=1}^M, L, R$ and $T$ but are independent of $\varepsilon$ such that

\begin{align*} &\left \| \rho _t^{M,\varepsilon } \right \|_{ C([0,T];L^2 (\Omega ))} \le C_9,\qquad \left \| k_j * \rho _t^{M,\varepsilon } \right \|_{ C([0,T];L^2 (\Omega ))} \le C_9,\\ &\left \| (k_j)_x * \rho _t^{M,\varepsilon } \right \|_{ C([0,T];L^2 (\Omega ))} \le C_{10}. \end{align*}

Although we should explicitly write the dependence of $t$ on the norm of the spatial direction for functions depending on the position $x$ and time $t$ , for example, $\|\cdot \|_{L^2(\Omega )}(t)$ , we abbreviate the symbol of $(t)$ for the simple descriptions from here.

Proof of Lemma 4.3. First, we denote the Fourier coefficient of $ ( \rho ^{M,\varepsilon } (\sum _{j=1}^M a_j v_j^{M,\varepsilon })_x )_x$ by

\begin{equation*} q_n(t) \,:\!=\, \frac {1}{\sqrt {2L}} \int _\Omega \left( \rho ^{M,\varepsilon } \left( \sum _{j=1}^M a_j v_j^{M,\varepsilon } \right)_x \right)_x e^{ - i \sigma _n x} dx. \end{equation*}

Since $\rho ^{M,\varepsilon }$ is the mild solution, we have

\begin{equation*} \rho ^{M,\varepsilon }_t = \frac { \partial }{ \partial t } \Phi [\rho ^{M,\varepsilon }](x,t) =( G_t*\rho _0 )(x,t) + \frac {1}{\sqrt {2L}} \sum _{n \in \mathbb{Z}} e^{ i \sigma _n x} \left( q_n(t) - \int _0^t \sigma _n^2 e^{ -\sigma _n^2( t - s ) } q_n(s) ds \right). \end{equation*}

We will estimate each term. Since $G_t*\rho _0 = G_{xx} * \rho _0 = G* (\rho _0)_{xx}$ , the maximum principle yeilds that

\begin{equation*} \|G_t*\rho _0 \|_{ C([0,T];L^2 (\Omega )) } = \| G* (\rho _0)_{xx} \|_{ C([0,T];L^2 (\Omega )) } \le \| (\rho _0)_{xx} \|_{ L^2 (\Omega ) }. \end{equation*}

From the Fourier series expansion, we obtain that

\begin{equation*} \left \| \frac {1}{\sqrt {2L}} \sum _{n \in \mathbb{Z}} e^{ i \sigma _n \cdot } q_n(\cdot ) \right \|^2_{ C([0,T];L^2 (\Omega ) ) } = \sup _{t\in [0,T]} \sum _{n \in \mathbb{Z}} q_n^2(t) = \left \| \left ( \rho ^{M,\varepsilon } \left( \sum _{j=1}^M a_j v_j^{M,\varepsilon }\right)_x \right)_x \right\|^2_{ C([0,T];L^2 (\Omega ) ) }, \end{equation*}

where the last term is bounded by Lemma 3.5. Finally, we compute that

\begin{align*} &\left \| \frac {1}{\sqrt {2L}} \sum _{n \in \mathbb{Z}} \sigma _n^2 e^{ i \sigma _n \cdot } \int _0^t e^{ -\sigma _n^2( \cdot - s ) } q_n(s) ds \right \|^2_{ C([0,T];L^2 (\Omega ) )} \\ & = \sup _{t\in [0,T]} \sum _{n \in \mathbb{Z}} \sigma _n^4 \left( \int _0^t e^{ -\sigma _n^2( t - s ) } q_n(s) ds \right)^2 \le \sup _{t\in [0,T]} \sum _{n \in \mathbb{Z}} | q_n(s) |^2 = \left \| \left( \rho ^{M,\varepsilon } \left( \sum _{j=1}^M a_j v_j^{M,\varepsilon } \right)_x \right)_x \right \|^2_{ C([0,T];L^2 (\Omega ) ) }. \end{align*}

Putting

\begin{equation*} C_9\,:\!=\, \| (\rho _0)_{xx} \|_{ C([0,T];L^2 (\Omega )) } + 2M_R, \end{equation*}

we obtain the first and the second assertions. The Young inequality and Lemma 4.2 implies the third and last assertions, where we put

\begin{equation*} C_{10}\,:\!=\, C_8C_9. \end{equation*}

Now, we estimate the difference between the solutions in the following auxiliary equations

(4.44) \begin{align} \varepsilon \big(v_{j}^{M,\varepsilon }\big)_t& = d_j \big(v_{j}^{M,\varepsilon }\big)_{xx} - v_j^{M,\varepsilon } + \rho ^{M,\varepsilon }, \end{align}

(4.45) \begin{align} 0 & = d_j (v_{j})_{xx} - v_j + \rho ^{M,\varepsilon } \end{align}

for $j= 1, \ldots , M$ , where the third and fourth conditions of (2.10) are imposed, respectively, and $\rho ^{M,\varepsilon }$ denotes the solution to ( $\mbox{KS}^{M,\varepsilon }$ ). We note that the solution to (4.45) is given by $v_j = k_j * \rho ^{M,\varepsilon }$ . We set the difference as

\begin{equation*} V_j^\varepsilon (x,t) \,:\!=\, v_j^{M,\varepsilon }(x,t) - v_j(x,t). \end{equation*}

Lemma 4.4. Let $(\rho ^{M,\varepsilon },v_j^{M,\varepsilon })$ and $v_j$ be the solutions to ( $\mbox{KS}^{M,\varepsilon }$ ) with $\rho _0\in C^2(\Omega )$ and (2.18) and (4.45), respectively, and let $C_9$ and $C_{10}$ be positive constants in Lemma 4.3 . Then, for any time $T\gt 0$ the following estimates hold:

\begin{align*} & \left \| V_j^\varepsilon \right \|_{ C([0,T]; L^2 (\Omega ))}^2 + d_j \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le C_9^2 \varepsilon ^2 \left( 1 + \frac { T}{2} \right),\\[4pt] & \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{C([0,T]; L^2 (\Omega ))}^2 + d_j \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le C_{10}^2 \varepsilon ^2 \left( 1 + \frac { T}{2} \right), \\[4pt] & \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ C([0,T]; L^2 (\Omega ))}^2 + d_j \left \| \big(V_{j}^\varepsilon \big)_{xxx} \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le \frac {C_{10}^2 \varepsilon ^2}{d_j} \left( \frac 1 2 + T \right). \end{align*}

When excluding the terms multiplied by $d_j$ on the left-hand sides, the above inequality holds without $T/2$ on the right-hand sides.

Proof of Lemma 4.4. Taking the difference between equations of (4.44) and (4.45), we see that

(4.46) \begin{align} \varepsilon (V_j^\varepsilon )_t &= - \varepsilon (v_{j})_t + d_j \big(V_{j}^\varepsilon \big)_{xx} - V_j^\varepsilon \notag \\[4pt] & = - \varepsilon k_j * \rho ^{M,\varepsilon }_t + d_j \big(V_{j}^\varepsilon \big)_{xx} - V_j^\varepsilon . \end{align}

Multiplying this equation by $V_j^\varepsilon$ , integrating over $\Omega$ and using Lemma 4.3, we have

(4.47) \begin{align} \frac {\varepsilon }{2} \frac {d}{dt} \left \| V_j^\varepsilon \right \|_{ L^2 (\Omega )}^2 &= -\varepsilon \int _\Omega \big(k_j * \rho ^{M,\varepsilon }_t V_j^\varepsilon \big) - d_j \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2 - \left \| V_{j}^\varepsilon \right \|_{ L^2 (\Omega )}^2 \notag \\[3pt] &\le \varepsilon \left \| k_j * \rho ^{M,\varepsilon }_t \right \|_{ L^2 (\Omega )} \left \| V_j^\varepsilon \right \|_{ L^2 (\Omega )} - d_j\left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2 - \left \| V_{j}^\varepsilon \right \|_{ L^2 (\Omega )}^2 \notag \\[3pt] &\le \frac {\varepsilon ^2}{2} \left \| k_j * \rho _t^{M,\varepsilon } \right \|_{ L^2 (\Omega )}^2 - d_j\left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2 - \frac {1}{2} \left \| V_{j}^\varepsilon \right \|_{ L^2 (\Omega )}^2 \notag \\[3pt] &\le \frac {\varepsilon ^2}{2} C_9^2 - d_j\left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2 - \frac {1}{2} \left \| V_{j}^\varepsilon \right \|_{ L^2 (\Omega )}^2 . \end{align}

Applying to the classical Gronwall lemma to

\begin{equation*} \frac {d}{dt} \left \| V_j^\varepsilon \right \|_{ L^2 (\Omega )}^2 \le \varepsilon C_9^2 - \frac {1}{\varepsilon } \left \| V_{j}^\varepsilon \right \|_{ L^2 (\Omega )}^2, \end{equation*}

we have that

(4.48) \begin{equation} \left \| V_j^\varepsilon \right \|_{ L^2 (\Omega )}^2 \le \left \| V_j^\varepsilon (\cdot , 0) \right \|_{ L^2 (\Omega )}^2 e^{-t/\varepsilon } + \varepsilon ^2 C_9^2 ( 1 - e^{-t/\varepsilon } ) \le \varepsilon ^2 C_9^2 \end{equation}

from the initial conditions given in (2.18). Furthermore, integrating (4.47) over $(0,T)$ , we see that

(4.49) \begin{align} d_j \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le \frac {\varepsilon }{2} \left \| V_j^\varepsilon \right \|_{ L^2 (\Omega )}^2 + d_j \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le \frac {\varepsilon ^2 C_9^2 T}{2}. \end{align}

Applying $\sup _{t \in [0,T]}$ to (4.48) and adding it to (4.49), we obtain the first assertion.

Similarly, multiplying (4.46) by $-\big(V_{j}^\varepsilon \big)_{xx}$ and integrating over $\Omega$ , we have

(4.50) \begin{align} \frac {\varepsilon }{2} \frac {d}{dt} \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )} ^2 & = -\varepsilon \int _\Omega ((k_j)_x * \rho ^{M,\varepsilon }_t \big(V_{j}^\varepsilon \big)_x) - d_j \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )}^2 - \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2\notag \\ & \le \frac {\varepsilon ^2}{2} C_{10}^2 - d_j\left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )}^2 - \frac {1}{2} \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2, \end{align}

where we used Lemma 4.3. From the Gronwall inequality, we have

(4.51) \begin{equation} \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )}^2 \le \left \| \big(V_{j}^\varepsilon \big)_x(\cdot , 0) \right \|_{ L^2 (\Omega )}^2 e^{-t/\varepsilon } + \varepsilon ^2 C_{10}^2 ( 1 - e^{-t/\varepsilon } ) \le \varepsilon ^2 C_{10}^2 \end{equation}

by the initial condition (2.18). Integrating (4.50) over $(0,T)$ , we see that

(4.52) \begin{align} d_j \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2(0,T; L^2 (\Omega ))}^2\le \frac {\varepsilon }{2} \left \| \big(V_{j}^\varepsilon \big)_x (\cdot , T)\right \|_{ L^2 (\Omega )}^2 + d_j \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2(0,T; L^2 (\Omega ))}^2 \le \frac {\varepsilon ^2 C_{10}^2 T}{2}. \end{align}

Applying $\sup _{t \in [0,T]}$ to (4.51) and adding it to (4.52), we have the second assertion.

Because of $v_j^{M,\varepsilon } \in L^2(0,T,H^3(\Omega ))$ , the similar calculation can be applied. Differentiating (4.46) with respect to $x$ and multiplying it by $-(V_j^\varepsilon )_{xxx}$ , we see that

(4.53) \begin{equation} \frac {\varepsilon }{2} \frac {d}{dt} \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )} ^2 \le \frac {\varepsilon ^2}{2} \frac {C_{10}^2}{d_j} - \frac { d_j }{2} \left \| \big(V_{j}^\varepsilon \big)_{xxx} \right \|_{ L^2 (\Omega )}^2 - \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )}^2. \end{equation}

Thus, the Gronwall lemma yields that

(4.54) \begin{equation} \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )}^2 (t) \le \left \| \big(V_{j}^\varepsilon \big)_{xx}(\cdot , 0) \right \|_{ L^2 (\Omega )}^2 e^{-2t/\varepsilon } + \frac {\varepsilon ^2C_{10}^2}{2 d_j} ( 1 - e^{-2t/\varepsilon } ) \le \frac {\varepsilon ^2C_{10}^2}{ 2 d_j} \end{equation}

from the initial condition given in (2.18). Integrating (4.53) over $(0,T)$ with respect to $t$ and (4.54) implies the final assertion of this lemma.

4.3. Order estimation

Under the above preparation, we estimate the difference of solutions. Set the difference between the solutions to the first component of ( $\mbox{KS}^{M,\varepsilon }$ ) and (P) with $W= \sum _{j=1}^M a_j k_j$ as

\begin{equation*} U^\varepsilon (x,t) \,:\!=\, \rho ^{M,\varepsilon } (x,t) - \rho (x,t). \end{equation*}

We will show the following convergence.

Lemma 4.5. Suppose that $M$ is an arbitrarily fixed natural number. Let $\rho$ be the solution to (P) equipped with $W= \sum _{j=1}^M a_j k_j$ and the initial value $\rho _0 \in C^2(\Omega )$ , and let $\rho ^{M,\varepsilon }$ be the solution to the first component of ( $\mbox{KS}^{M,\varepsilon }$ ) with (2.17) and (2.18). Then, for any $T\gt 0$ , there exists a positive constant $C_{11} = C_{11}((\rho _0)_{xx}, M, \{a_j, d_j\}_{j=1}^M, L, R, T )$ such that for any $\varepsilon \gt 0$

(4.55) \begin{equation} \left \| U^\varepsilon \right \|_{C([0, T]; L^2 (\Omega ))}^2 +\left \| U^\varepsilon _x \right \|_{ L^2( 0,T; L^2 (\Omega ))}^2 \le C_{11} \varepsilon ^2. \end{equation}

We note that $C_{11}$ is independent of $\varepsilon$ .

Proof of Lemma 4.5. Taking the difference between the first equation of ( $\mbox{KS}^{M,\varepsilon }$ ) and the equation of (P), we have

(4.56) \begin{equation} U^\varepsilon _t = U^\varepsilon _{xx} - \sum _{j=1}^M a_j \big( \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_x + U^\varepsilon (k_j)_x *\rho ^{M,\varepsilon } + \rho (k_j)_x*U^\varepsilon \big)_x. \end{equation}

Subsequently, multiplying (4.56) by $U^\varepsilon$ and integrating over $\Omega$ , we obtain

\begin{align*} \frac {1}{2} \frac {d}{dt} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 (t) &= -\left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 (t) + \sum _{j=1}^M a_j \int _\Omega U^\varepsilon _x \big( \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_x + U^\varepsilon (k_j)_x *\rho ^{M,\varepsilon } + \rho (k_j)_x*U^\varepsilon \big)(x,t) dx\\ & = -\left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 (t)+I_1(t) +I_2(t) +I_3(t), \end{align*}

where each term of the integral is set as

\begin{align*} I_1(t)&\,:\!=\,\sum _{j=1}^M a_j \int _\Omega \big(U^\varepsilon _x \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_x\big) (x,t) dx, \\ I_2(t) &\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _x U^\varepsilon (k_j)_x *\rho ^{M,\varepsilon } \big) (x,t) dx, \\ I_3(t)&\,:\!=\,\sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _x\rho (k_j)_x*U^\varepsilon \big) (x,t) dx, \end{align*}

respectively. First, we compute $I_1$ . Using the Cauchy–Schwartz inequality, Sobolev embedding theorem and Lemma 4.4, we have

\begin{align*} I_1 &\le \sum _{j=1}^M |a_j| \int _\Omega | U^\varepsilon _x| | \rho ^{M,\varepsilon } | | \big(V_{j}^\varepsilon \big)_x | dx \\ & \le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M |a_j| \left \| U^\varepsilon _x\right \|_{ L^2 (\Omega )} \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ L^2 (\Omega )} \\ & \le \varepsilon \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M ( C_{10} |a_j| )\left \| U^\varepsilon _x\right \|_{ L^2 (\Omega )} \le \frac 1 2 \left \| U^\varepsilon _x\right \|_{ L^2 (\Omega )}^2 + \frac {\varepsilon ^2 C_{12} }{2} , \end{align*}

where

\begin{equation*} C_{12} \,:\!=\, \left( C_{\mbox s}\tilde {C}_0 \sum _{j=1}^M ( C_{10} |a_j| ) \right)^2, \end{equation*}

and $C_{\mbox s}$ is the constant from the Sobolev embedding theorem. Next, we compute that

\begin{align*} I_2 & \le \sum _{j=1}^M | a_j | \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \left \| (k_j)_x \right \|_{ L^1 (\Omega )} \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \\ & \le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M ( C_8| a_j |) \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \le \frac {1}{4} \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )}^2 + C_{13}^2 \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2, \end{align*}

where we used the estimate in Lemma 4.2 and

\begin{equation*} C_{13} \,:\!=\, C_{\mbox s}\tilde {C}_0 \sum _{j=1}^M ( C_8 | a_j | ) . \end{equation*}

Finally, the Sobolev embedding theorem, the Young inequality and the boundedness in Lemma 4.2 yield that

\begin{align*} I_3 & \le \sum _{j=1}^M | a_j | \left \| \rho \right \|_{ C (\Omega )} \int _\Omega | U^\varepsilon _x| | (k_j)_x*U^\varepsilon | dx \\ & \le \left \| \rho \right \|_{ C (\Omega )} \sum _{j=1}^M( C_8 | a_j | ) \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \le \frac {1}{8} \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )}^2 + 2C_{14}^2 \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2, \end{align*}

where the constant is defined as

\begin{equation*} C_{14} \,:\!=\, C_{\mbox s} C_0 \sum _{j=1}^M ( C_8 | a_j | ) . \end{equation*}

Summarising these estimations, we have

(4.57) \begin{align} \frac {1}{2} \frac {d}{dt} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 &\le - \frac {1}{8} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 + (C_{13}^2 + 2C_{14}^2 ) \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \frac {\varepsilon ^2}{2} C_{12}\notag \\ &= - \frac {1}{8} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 + \frac {C_{15}}{2} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \frac {\varepsilon ^2}{2} C_{12}, \end{align}

where we put $C_{15}\,:\!=\, 2(C_{13}^2 + 2C_{14}^2 )$ . Applying the classical Gronwall inequality with the initial condition (2.17) to

\begin{equation*} \frac {d}{dt} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 \le C_{15} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \varepsilon ^2 C_{12}, \end{equation*}

we have

(4.58) \begin{equation} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 (t)\le \frac {C_{12}}{C_{15}} (e^{ C_{15} t } - 1) \varepsilon ^2. \end{equation}

Furthermore, integrating (4.57) over $(0,T)$ , we also obtain that

(4.59) \begin{align} \frac {1}{4} \left \| U^\varepsilon _x \right \|_{ L^2( 0,T; L^2(\Omega ))}^2 &\le \frac {1}{4} \left \| U^\varepsilon _x \right \|_{ L^2( 0,T; L^2(\Omega ))}^2 +\left \| U^\varepsilon (\cdot , T) \right \|_{ L^2 (\Omega )}^2 \notag \\ &\le C_{15} \int ^T_0 \left \| U^\varepsilon (\cdot , t)\right \|_{ L^2 (\Omega )}^2 dt + C_{12} T \varepsilon ^2 \notag \\ &\le \varepsilon ^2 C_{12} e^{C_{15} T} T. \end{align}

Defining $C_{11} \,:\!=\, C_{12} (e^{ C_{15} T } - 1)/ C_{15} + 4 C_{12} e^{C_{15} T} T$ and adding (4.58) and (4.59) imply the assertion of this lemma.

Similarly to this lemma, we can obtain the following estimate.

Lemma 4.6. Suppose the same assumptions as Lemma 4.5 . Then, for any $T\gt 0$ , there exists a positive constant $C_{16} = C_{16}((\rho _0)_{xx}, M, \{a_j, d_j\}_{j=1}^M, L, R, T)$ such that for any $\varepsilon \gt 0$

\begin{equation*} \left \| U_x^\varepsilon \right \|_{ C([0, T]; L^2 (\Omega ))}^2 +\left \| U^\varepsilon _{xx} \right \|_{ L^2( 0,T; L^2(\Omega ))}^2 \le C_{16} \varepsilon ^2. \end{equation*}

We note that $C_{16}$ is independent of $\varepsilon$ .

Proof of Lemma 4.6. Similarly to Lemma 4.5, multiplying (4.56) by $-U^\varepsilon _{xx}$ and integrating over $\Omega$ , we have

\begin{align*} \frac {1}{2} \frac {d}{dt} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 (t) &= -\left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 (t)+ \sum _{j=1}^M a_j \int _\Omega U^\varepsilon _{xx} \big( \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_x + U^\varepsilon (k_j)_x *\rho ^{M,\varepsilon } + \rho (k_j)_x*U^\varepsilon \big)_x (x,t)dx\\ &= -\left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 (t) + \sum _{j=1}^M a_j \int _\Omega U^\varepsilon _{xx} \big( \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_x \big)_x (x,t)dx \\ &\quad + \sum _{j=1}^M a_j \int _\Omega U^\varepsilon _{xx} \big( U^\varepsilon (k_j)_x *\rho ^{M,\varepsilon } \big)_x (x,t)dx + \sum _{j=1}^M a_j \int _\Omega U^\varepsilon _{xx} ( \rho (k_j)_x*U^\varepsilon )_x (x,t)dx \\ & \,=\!:\, -\left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 (t)+\mathscr{I}_1(t) +\mathscr{I}_2(t) +\mathscr{I}_3(t) +\mathscr{I}_4(t) +\mathscr{I}_5(t) +\mathscr{I}_6(t), \end{align*}

where we defined each term of energy term by the integral as

\begin{align*} \begin{array}{ll} \displaystyle\mathscr{I}_1(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} \rho ^{M,\varepsilon }_x \big(V_{j}^\varepsilon \big)_x \big) (x,t) dx, & \displaystyle\mathscr{I}_2(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} \rho ^{M,\varepsilon } \big(V_{j}^\varepsilon \big)_{xx} \big) (x,t) dx,\\ \displaystyle\mathscr{I}_3(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} U^\varepsilon _x (k_j)_x *\rho ^{M,\varepsilon } \big) (x,t) dx, &\displaystyle\mathscr{I}_4(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} U^\varepsilon (k_j)_{xx} *\rho ^{M,\varepsilon } \big) (x,t) dx,\\ \displaystyle\mathscr{I}_5(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} \rho _x (k_j)_x*U^\varepsilon \big) (x,t) dx, &\displaystyle\mathscr{I}_6(t)\,:\!=\, \sum _{j=1}^M a_j \int _\Omega \big( U^\varepsilon _{xx} \rho (k_j)_{xx}*U^\varepsilon \big) (x,t) dx.\\ \end{array} \end{align*}

First, we estimate $\mathscr{I}_1$ . From Lemma 4.4 and the Sobolev embedding theorem, there exists a positive constant $C_{\mbox s}$ such that $\left \| V_{j,x} \right \|_{ C (\Omega )} \le C_{\mbox s} \left \| V_{j,x} \right \|_{ H^1 (\Omega )}$ . Then, we see that

\begin{align*} {\displaystyle }\mathscr{I}_1 &\le \sum _{j=1}^M \left \| \big(V_{j}^\varepsilon \big)_x \right \|_{ C (\Omega )} |a_j| \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \left \| \rho ^{M,\varepsilon }_x \right \|_{ L^2 (\Omega )} \\ &\le \varepsilon \sum _{j=1}^M C_{\mbox s} \left(C_{10} + \frac {C_{10} }{\sqrt {2 d_j} } \right) |a_j| \left \| \rho ^{M,\varepsilon }_x \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \le \varepsilon ^2 C_{17} + \frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} ^2, \end{align*}

where

\begin{equation*} C_{17} \,:\!=\, 2\left( \tilde {C}_0 \sum _{j=1}^M C_{\mbox s} \left(C_{10} + \frac {C_{10} }{\sqrt {2 d_j} } \right) |a_j| \right)^2 \end{equation*}

Next, we compute $ \mathscr{I}_2$ as

\begin{align*} \mathscr{I}_2 &\le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M |a_j| \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \left \| \big(V_{j}^\varepsilon \big)_{xx} \right \|_{ L^2 (\Omega )} \\ & \le \varepsilon \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M \left( \frac {C_{10}}{\sqrt {2 d_j}} |a_j| \right) \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \le \varepsilon ^2 C_{18} + \frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2, \end{align*}

where

\begin{equation*} C_{18} \,:\!=\, 2 \left( C_{\mbox s}\tilde {C}_0 \sum _{j=1}^M \left( \frac {C_{10}}{\sqrt {2 d_j} } |a_j| \right)\right)^2. \end{equation*}

From $\rho ^{M,\varepsilon } \in C(\Omega )$ and Lemma 4.2, we see that

\begin{align*} \mathscr{I}_3 &\le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M |a_j| \left \| (k_j)_x \right \|_{ L^1 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )} \\ & \le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M ( C_8 |a_j| ) \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \le \frac { C_{19} }{2} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 +\frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2, \end{align*}

where we put

\begin{equation*} C_{19} \,:\!=\, 4\left( C_{\mbox s} \tilde {C}_0 \sum _{j=1}^M \big( C_8 |a_j| \big) \right)^2. \end{equation*}

Similarly to that of $\mathscr{I}_3$ , we obtain that

\begin{align*} \mathscr{I}_4 &\le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M |a_j| \left \| (k_j)_{xx} \right \|_{ L^1 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}\\ & \le \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M \frac { |a_j| }{ d_j } \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \\ & \le 2 \left( \left \| \rho ^{M,\varepsilon } \right \|_{ C (\Omega )} \sum _{j=1}^M \frac { |a_j| }{ d_j } \right)^2 \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \frac 1 8 \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 \le C_{20} \varepsilon ^2+ \frac 1 8 \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2, \end{align*}

where we used the estimate (4.55) in Lemma 4.5 and we put

\begin{equation*} C_{20} \,:\!=\, 2 \left( C_{\mbox s} \tilde {C}_0 \sum _{j=1}^M \frac { |a_j| }{ d_j } \right)^2 C_{11}. \end{equation*}

Hereafter, we often use the estimate (4.55) in Lemma 4.5. We compute $\mathscr{I}_5$ as

\begin{align*} \mathscr{I}_5 & \le \sum _{j=1}^M |a_j| \left \| (k_j)_x \right \|_{ C (\Omega )} \left \| U^\varepsilon \right \|_{ L^1 (\Omega )} \left \| \rho _x \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \\ & \le \left \| U^\varepsilon \right \|_{ L^1 (\Omega )} \sum _{j=1}^M \frac {|a_j|}{2d_j} \left \| \rho _x \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \le C_{21} \varepsilon ^2 + \frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2, \end{align*}

where we used (4.43) in Lemma 4.2 and (4.55) in Lemma 4.5, and we put

\begin{equation*} C_{21} \,:\!=\, 4L \left( C_0 \sum _{j=1}^M \frac {|a_j|}{2d_j} \right)^2 C_{11}. \end{equation*}

Similarly, we see that

\begin{align*} \mathscr{I}_6 &\le \left \| \rho \right \|_{ C (\Omega )} \sum _{j=1}^M |a_j| \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )} \left \| (k_j)_{xx} \right \|_{ L^1 (\Omega )} \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \\ & \le \left \| \rho \right \|_{ C (\Omega )} \sum _{j=1}^M \frac { |a_j| }{ d_j } \left \| U^\varepsilon \right \|_{ L^2 (\Omega )} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}\\ & \le 2\left( \left \| \rho \right \|_{ C (\Omega )} \sum _{j=1}^M \frac { |a_j| }{ d_j } \right)^2 \left \| U^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 \le C_{22}\varepsilon ^2 + \frac {1}{8} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2, \end{align*}

where

\begin{equation*} C_{22} \,:\!=\, 2\left( C_{\mbox s} C_0 \sum _{j=1}^M \frac { |a_j| }{ d_j } \right)^2 C_{11}. \end{equation*}

Combining these estimation and setting a positive constant as

\begin{equation*} C_{23} \,:\!=\, 2( C_{17} + C_{18} + C_{20} + C_{21} + C_{22} ), \end{equation*}

we have

(4.60) \begin{equation} \frac {1}{2} \frac {d}{dt} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 \le -\frac {1}{4} \left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2 + \frac {C_{19}}{2} \left \| U_x^\varepsilon \right \|_{ L^2 (\Omega )}^2 + \frac {C_{23} }{2} \varepsilon ^2. \end{equation}

Applying to Gronwall inequality to (4.60) without $-\left \| U^\varepsilon _{xx} \right \|_{ L^2 (\Omega )}^2/4$ , we obtain that

(4.61) \begin{equation} \left \| U^\varepsilon _x \right \|_{ L^2 (\Omega )}^2 (t) \le e^{C_{19}t}\left \| U^\varepsilon _x (\cdot , 0) \right \|_{ L^2 (\Omega )}^2 +\frac {C_{23}}{C_{19}} \big( e^{C_{19} t } -1\big)\varepsilon ^2 = \frac {C_{23}}{C_{19}} \big( e^{C_{19} t } -1\big)\varepsilon ^2 \end{equation}

by the initial condition in (2.17).

Finally, integrating (4.60) over $(0,T)$ , we obtain that

(4.62) \begin{equation} \frac 1 2 \left \| U^\varepsilon _{xx} \right \|_{ L^2 (0,T; L^2(\Omega ))}^2 \le \left \| U^\varepsilon _x (\cdot , T)\right \|_{ L^2 (\Omega )}^2 + \frac 1 2 \left \| U^\varepsilon _{xx} \right \|_{ L^2 (0,T; L^2(\Omega ))}^2 \le \varepsilon ^2 C_{23} e^{C_{19} T} T. \end{equation}

Adding (4.61) and (4.62) yields the assertion of this lemma, where we put

\begin{equation*} C_{16} \,:\!=\, \max \Big \{ \frac {C_{23}}{C_{19}} \big( e^{C_{19} t } -1\big), 2C_{23} e^{C_{19} T} T \Big \}. \end{equation*}

Then, we obtain the following proof.

Proof of Theorem 2.4. Putting

\begin{align*} C_1 \,:\!=\, \sqrt {2}\big ( C_{11} + C_{11}T + C_{16} \big )^{1/2}, \quad C_2 \,:\!=\, C_1 + \big ( C_9^2 +C_{10}^2 \big )^{1/2} + \sqrt { \frac { T }{2d_j} } \big ( 2d_j C_9^2 + C_9^2 + C_{10}^2 \big )^{1/2}, \end{align*}

where we used Lemma 4.5, Lemma 4.6 for $C_1$ and Lemma 4.4 for $C_2$ . We obtain the convergence of the assertion in this theorem.

Next, we prove Lemma 2.8.

Proof of Lemma 2.8. Set the differences between the solutions and between the kernels as

\begin{equation*} U(x,t) \,:\!=\, \rho _1(x,t) - \rho _2(x,t), \quad W_{\mbox{e}} (x) \,:\!=\, w_1(x)-w_2(x), \end{equation*}

respectively. The method for this proof is similar to that of Lemmas 4.5 and 4.6. Taking the difference between the equations $(\mbox{P}_1)$ and $(\mbox{P}_2)$ , we have

(4.63) \begin{align} U_t = U_{xx} - ( \rho _1w_{1,x} *U + \rho _1W_{\mbox{e},x}*\rho _2 +Uw_{2,x}*\rho _2 )_x. \end{align}

Multiplying it by $U$ and integrating it over $\Omega$ , we have

\begin{align*} \frac {1}{2} \frac {d}{dt} \left \| U \right \|_{ L^2 (\Omega )} ^2 = -\left \| U_x \right \|_{ L^2 (\Omega )} ^2 + \int _{\Omega } ( U_x( \rho _1w_{1,x} *U + \rho _1W_{\mbox{e},x}*\rho _2 +Uw_{2,x}*\rho _2) ) . \end{align*}

Since

\begin{align*} \int _{\Omega } ( U_x \rho _1w_{1,x} *U ) &\le \left \| \rho _1 \right \|_{ C (\Omega )} \left \| w_{1,x} \right \|_{ L^1 (\Omega )}\left \| U \right \|_{ L^2 (\Omega )} \left \| U_x \right \|_{ L^2 (\Omega )}, \\ \int _{\Omega } ( U_x \rho _1W_{\mbox{e},x}*\rho _2 ) &\le \left \| \rho _1 \right \|_{ C (\Omega )} \left \| W_{\mbox{e}} \right \|_{ L^1 (\Omega )}\left \| \rho _{2,x} \right \|_{ L^2 (\Omega )} \left \| U_x \right \|_{ L^2 (\Omega )},\\ \int _{\Omega } ( U_xUw_{2,x}*\rho _2 ) & \le \left \| \rho _{2} \right \|_{ C (\Omega )} \left \| w_{2,x} \right \|_{ L^1 (\Omega )} \left \| U \right \|_{ L^2 (\Omega )} \left \| U_x \right \|_{ L^2 (\Omega )} , \end{align*}

we can compute that

(4.64) \begin{align} \frac {1}{2} \frac {d}{dt} \left \| U \right \|_{ L^2 (\Omega )} ^2 \le -\frac {1}{4} \left \| U_x \right \|_{ L^2 (\Omega )}^2 + \frac {C_{24}}{2} \left \| U \right \|_{ L^2 (\Omega )} ^2 + \frac {C_{25}}{2} \left \| W_{\mbox{e}} \right \|_{ L^1 (\Omega )}^2, \end{align}

where we put

\begin{align*} C_{24} &\,:\!=\, 2C_{\mbox s}^2\left( \left(C_0^{(1)} \| w_{1,x} \|_{L^1(\Omega )}\right)^2 + \left(C_0^{(2)} \| w_{2,x} \|_{L^1(\Omega )}\right)^2 \right),\\ C_{25} &\,:\!=\, 2 \left(C_{\mbox s}C_0^{(1)}C_0^{(2)}\right)^2, \end{align*}

and $C_{\mbox s}$ is the coefficient from the Sobolev embedding theorem, and $C_0^{(1)}$ and $C_0^{(2)}$ correspond to the constant $C_0$ in Proposition 2.1 for $\rho _1$ and $\rho _2$ , respectively. Applying the Gronwall inequality to this, we have

(4.65) \begin{equation} \left \| U \right \|_{ C([0,T]; L^2 (\Omega ))}^2 \le \frac {C_{25}}{C_{24}} \big(e^{C_{24} T} -1 \big) \left \| W_{\mbox{e}} \right \|_{ L^1 (\Omega )}^2 \,=\!:\,{\tilde C_T^{(1)}} \left \| W_{\mbox{e}} \right \|_{ L^1 (\Omega )}^2. \end{equation}

Integrating (4.64) over $[0,T]$ and adding it and (4.65), we have

\begin{equation*} \left \| U \right \|_{ C\left([0,T];L^2 (\Omega )\right)}^2 + \left \| U \right \|_{ L^2 (0,T;H^1(\Omega ))}^2 \le {\tilde C_T^{(2)}} \left \| w_1 - w_2 \right \|_{ L^{1} (\Omega )}^2, \quad {\tilde C_T^{(2)}}\,:\!=\, \max \Big \{ {\tilde C_T^{(1)}}, T{\tilde C_T^{(1)}},2C_{25}T e^{C_{24} T} \Big \}. \end{equation*}

Next, multiplying (4.63) by $-U_{xx}$ and integrating it over $\Omega$ , we consider the following equation of energy

\begin{align*} \frac {1}{2} \frac {d}{dt} \left \| U_{x} \right \|_{ L^2 (\Omega )} ^2 =-\left \| U_{xx}\right \|_{ L^2 (\Omega )} ^2 + \int _{\Omega } ( U_{xx} ( \rho _1w_{1,x} *U + \rho _1W_{\mbox{e},x}*\rho _2 +Uw_{2,x}*\rho _2)_x ). \end{align*}

Since

\begin{align*} & \int _{\Omega } \left( U_{xx} \left( \rho _{1,x}w_{1,x} *U + \rho _1w_{1,x} *U_{x}\right) \right) \\ &\le \left \| U \right \|_{ C (\Omega )} \left \| w_{1,x} \right \|_{ L^1 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )} \left \| \rho _{1,x} \right \|_{ L^2 (\Omega )} + \left \| \rho _1\right \|_{ C (\Omega )} \left \| w_{1,x}\right \|_{ L^1 (\Omega )} \left \| U_{x} \right \|_{ L^2 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )}\\ &\le C_{\mbox s} \left \| U \right \|_{ H^1 (\Omega )} \left \| w_{1,x} \right \|_{ L^1 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )} \left \| \rho _{1,x} \right \|_{ L^2 (\Omega )} + \left \| \rho _1\right \|_{ C (\Omega )} \left \| w_{1,x}\right \|_{ L^1 (\Omega )} \left \| U_{x} \right \|_{ L^2 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )},\\ &\int _{\Omega } \left( U_{xx} \left( \rho _{1,x}W_{\mbox{e},x}*\rho _2 + \rho _1W_{\mbox{e},x}*\rho _{2,x}\right) \right) \\ & \le \left \| \rho _2 \right \|_{ C (\Omega )} \left \| W_{\mbox{e},x} \right \|_{ L^1 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )} \left \| \rho _{1,x} \right \|_{ L^2 (\Omega )} + \left \| \rho _1 \right \|_{ C (\Omega )} \left \| W_{\mbox{e},x} \right \|_{ L^1 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )} \left \| \rho _{2,x} \right \|_{ L^2 (\Omega )},\\ &\int _{\Omega } \left( U_{xx} \left( U_xw_{2,x}*\rho _2 + U w_{2,x}*\rho _{2,x} \right) \right) \\ &\le \left \| \rho _2 \right \|_{ C (\Omega )} \left \|w_{2,x} \right \|_{ L^1 (\Omega )} \left \| U_x \right \|_{ L^2 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )} + C_{\mbox s} \left \| U \right \|_{ H^1 (\Omega )} \left \| \rho _{2,x} \right \|_{ L^2 (\Omega )} \left \|w_{2,x} \right \|_{ L^1 (\Omega )} \left \| U_{xx} \right \|_{ L^2 (\Omega )}, \end{align*}

we can obtain that

(4.66) \begin{align} \frac {1}{2} \frac {d}{dt} \left \| U_{x} \right \|_{ L^2 (\Omega )} ^2 &\le -\frac {1}{4} \left \| U_{xx} \right \|_{ L^2 (\Omega )}^2 + \frac {C_{26}}{2} \left \| U \right \|_{ L^2 (\Omega )}^2 + \frac {C_{27}}{2} \left \| U_x \right \|_{ L^2 (\Omega )}^2 + \frac {C_{28}}{2} \left \| W_{\mbox{e},x} \right \|_{ L^1 (\Omega )}^2 \notag \\ &\le -\frac {1}{4} \left \| U_{xx} \right \|_{ L^2 (\Omega )}^2 + \frac {C_{27}}{2} \left \| U_x \right \|_{ L^2 (\Omega )}^2 + \frac {C_{29}}{2} \left \| W_{\mbox{e}} \right \|_{ W^{1,1} (\Omega )}^2 \end{align}

with suitable positive constants from $C_{26}$ to $C_{29}$ , where we put

\begin{align*} C_{26} &\,:\!=\, 4C_{\mbox s}^2 \left( \left( C_0^{(1)} \left \| w_{1,x} \right \|_{ L^1 (\Omega )} \right)^2 + \left( C_0^{(2)} \left \| w_{2,x} \right \|_{ L^1 (\Omega )} \right)^2 \right),\\ &C_{27} \,:\!=\, C_{26} + 4C_{\mbox s}^2\left( \left(C_0^{(1)} \left \| w_{1,x}\right \|_{ L^1 (\Omega )}\right)^2 + \left(C_0^{(2)} \left \| w_{2,x}\right \|_{ L^1 (\Omega )}\right)^2 \right),\\ C_{28} &\,:\!=\, 8\left(C_{\mbox s} C_0^{(1)} C_0^{(2)}\right)^2,\\ C_{29} &\,:\!=\, \max \left\{ \frac {C_{25} C_{26} }{C_{24}} \big(e^{C_{24} T} -1 \big), C_{28} \right\}. \end{align*}

Therefore, the Gronwall inequality yields that

(4.67) \begin{equation} \left \| U_x \right \|_{ C\left( [0,T]; L^2 (\Omega )\right)}^2 \le \frac {C_{29}}{C_{27}} (e^{C_{27} T} -1 )\left \| W_{\mbox{e}} \right \|_{ W^{1,1} (\Omega )}^2 \,=\!:\,{\tilde C_T^{(3)}} \left \| W_{\mbox{e}} \right \|_{ W^{1,1} (\Omega )}^2. \end{equation}

Finally integrating (4.64) and (4.66) over $(0,T)$ and adding them and (4.65) and (4.67), we obtain the assertion of Lemma 2.8 by putting

\begin{equation*} {\tilde C_T} \,:\!=\, \max \left \{ {\tilde C_T^{(2)}}, {\tilde C_T^{(3)}}, 2C_{29}Te^{C_{27}T} \right\}. \end{equation*}

5. Coefficients of linear sum

We now explain the method used for determining the coefficient $\{ a_j \}_{j=1}^M$ of the linear sum of the fundamental solution for a given even potential function $W$ . Furthermore, we will perform the numerical simulations of the approximation of $W$ by sum of $\cosh j(L-|x|)$ , and numerical simulations of (P) and ( $\mbox{KS}^{M,\varepsilon }$ ) with this series expansion. Since $W$ is even, we only consider $[0,L]$ . Throughout this section, we set $d_1$ is sufficiently large and $d_j=1/(j-1)^2$ for $j=2,\ldots ,M$ .

First, we provide the following lemma with respect to the $n$ degree Chebyshev polynomial $T_n(x) \,:\!=\, \cos (n\arccos x)$ . We set coefficients as

\begin{align*} C^n_k \,:\!=\, (-1)^k 2^{n-2k-1} \frac {n}{n-k} \binom {n-k}{k}, \quad \left(k=0, \cdots , \Big [\frac n 2 \Big ] \right) \end{align*}

for $n \in \mathbb{N}$ , where $[\!\cdot\!]$ denotes the Gauss symbol. By this constant, the Chebyshev polynomial of the first kind of $n$ degree $T_n$ can be expressed as $T_n(x) = \sum _{k=0}^{[n/2]}C^n_k x^{n-2k}$ for $x\in [-1,1]$ . The properties of the Chebyshev polynomial can be referred from [Reference Mason and Handscomb16] by Mason and Handscomb. Utilising this equation, we have the following Lemma regarding the change of the variable for $T_n$ .

Lemma 5.1. Setting

\begin{equation*} \mu ^n_{k,j} \,:\!=\, C^n_k \left( \frac {2}{b-a} \right)^{n-2k} \left( -\frac {b+a}{2} \right)^j \binom {n-2k}{j} \end{equation*}

for $n \in \mathbb{N}$ , we define the coefficient as

\begin{equation*} \xi ^n_k \,:\!=\, \left \{ \begin{aligned} &\sum _{l=0}^{[n/2]-[(k+1)/2]} \mu ^n_{l,n-2l-k}, \quad \text{if $n$ is even}, \\[4pt] &\sum _{l=0}^{[n/2]-[k/2]} \mu ^n_{l,n-2l-k}, \quad \text{otherwise}. \\ \end{aligned} \right . \end{equation*}

Then,

\begin{equation*} T_n\left ( \frac {2x - (b+a)}{b-a} \right ) = \sum _{k=0}^n \xi ^n_k x^k, \quad x \in [a,b] \end{equation*}

holds.

Proof of Lemma 5.1. We compute that

\begin{align*} T_n\left( \frac {2x - (b+a)}{b-a} \right) &= \sum _{k=0}^{[n/2]}C^n_k \left( \frac {2x -(b+a) }{ b-a } \right)^{n-2k} =\sum _{k=0}^{[n/2]}C^n_k \left( \frac {2}{b-a} \right)^{n-2k} \left( x - \frac {b+a}{2}\right)^{n-2k}\\ & = \sum _{k=0}^{[n/2]}C^n_k \left( \frac {2}{b-a} \right)^{n-2k} \sum _{j=0}^{n-2k} \binom {n-2k} { j} x^{n-2k-j} \left( -\frac {b+a}{2} \right)^j\\ & = \sum _{k=0}^{[n/2]} \sum _{j=0}^{n-2k} \mu ^n_{k,j} x^{n-2k-j} = \sum _{k=0}^n \xi ^n_k x^k, \end{align*}

where we used the binomial expansion in the third equality.

Next, we explicitly provide the coefficient of the linear sum of the $n$ degree Lagrange interpolation polynomial with the Chebyshev nodes for the arbitrary function $F=F(x)$ for $x\in [a,b]$ . We will replace the arbitrary function $F$ with the function $f$ defined in (2.22) to prove Theorem 5.3. The root of the $n$ degree Chebyshev polynomial, called Chebyshev nodes, in an arbitrary interval $[a,b]$ is given by

\begin{equation*} r^n_j \,:\!=\, \frac {a+b}{2} + \frac {b-a}{2}\cos \frac {2j+1}{2n}\pi , \quad (j=0,\ldots , n-1). \end{equation*}

We have that

(5.68) \begin{equation} \prod _{j=0}^{n}\big(x-r_j^{n+1}\big) =\frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} T_{n+1 }\left( \frac {2x - (b+a)}{b-a} \right), \ n \in \mathbb{N}. \end{equation}

Moreover, setting the coefficient as

(5.69) \begin{equation} \zeta ^n_j \,:\!=\, \frac {F\big(r^{n+1}_j\big)}{\prod _{k=0,k\neq j}^n\big(r^{n+1}_j - r^{n+1}_k\big)}, \quad (j=0,\ldots , n) \end{equation}

for $n \in \mathbb{N}$ , we see that the $n$ degree Chebyshev polynomial for the function $F$ is given by

\begin{equation*} L_n(x)\,:\!=\, \sum _{j=0}^n \zeta ^n_j \prod _{k=0,k\neq j}^n \big(x - r^{n+1}_k\big). \end{equation*}

Note that $L_n( r^{n+1}_j ) = F ( r^{n+1}_j )$ . Then, we obtain the following proposition.

Proposition 5.2. Set

(5.70) \begin{align} &\beta _{l,j}^{n+1} \,:\!=\, \sum _{k=l}^{n} (r^{n+1}_j)^{k-l} \xi ^{n+1}_{k+1} \quad (l=0,\ldots ,n), \ (j=0,\ldots ,n),\notag \\ b^n_l &\,:\!=\, \frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} \sum _{j=0}^n \zeta ^n_j \beta _{l,j}^{n+1} \quad (l=0,\ldots ,n) \end{align}

for $n \in \mathbb{N}$ . Then, the $n$ degree Lagrange interpolation polynomial $L_n$ for an arbitrary function $F$ on $[a,b]$ can be described as

\begin{equation*} L_n(x)= \sum _{l=0}^n b^n_l x^l, \quad x \in [a,b]. \end{equation*}

Proof of Proposition 5.2. Using Lemma 5.1 and (5.68), we can compute that

\begin{align*} L_n(x) &=\sum _{j=0}^n \zeta ^n_j \prod _{k=0,k\neq j}^n \big(x - r^{n+1}_k\big)\\ &=\sum _{j=0}^n \zeta ^n_j \frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} T_{n+1 }\left( \frac {2x - (b+a)}{b-a} \right) \frac {1}{x - r^{n+1}_j}\\ &=\frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} \sum _{j=0}^n \zeta ^n_j \frac { \sum _{k=0}^{n+1} \xi ^{n+1}_k x^k}{x - r^{n+1}_j}. \end{align*}

As $x - r^{n+1}_j$ for $j = 0,\ldots ,n$ are the factors of $T_{n+1}( (2x - (b+a) )/(b-a) )$ , respectively, $\sum _{k=0}^{n+1} \xi ^{n+1}_k x^k/(x - r^{n+1}_j )$ must be divisible from the factor theorem. Thus, we obtain that

\begin{align*} L_n(x) &= \frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} \sum _{j=0}^n\zeta ^n_j \sum _{l=0}^{n} \beta ^{n+1}_{l,j} x^l\\ &=\sum _{l=0}^{n} \left( \frac {1}{2^n} \left( \frac {b-a}{2} \right)^{n+1} \sum _{j=0}^n \zeta ^n_j \beta ^{n+1}_{l,j} \right)x^l = \sum _{l=0}^n b^n_l x^l. \end{align*}

Before the proof of Theorem 5.3, we introduce the following constant for $n \in \mathbb{N}$ and $j \equiv n\pmod 2$ :

\begin{equation*} \delta ^n_j \,:\!=\, \left \{ \begin{aligned} &\frac {1}{2^{n-1} } \binom n { \frac {n-j}{2} } \quad \text{if $j \neq 0$},\\[4pt] &\frac {1}{2^{ n } } \binom n { \frac {n-j}{2} } \quad \text{if $j=0$}.\\ \end{aligned} \right . \end{equation*}

Using this constant, we can describe the formula as $x^n = \sum _{j=0, j \equiv n\pmod 2}^n \delta ^n_j T_j(x),$ [Reference Mason and Handscomb16]. Differentiating it, we then obtain that $x^n = \sum _{j=0, j \equiv n\pmod 2}^n (j+1)\delta ^{n+1}_{j+1} U_j(x)/(n+1)$ , where $U_n(x)\,:\!=\, (\sin (n+1)\theta )/\sin \theta , \ (x=\cos \theta )$ is the Chebyshev polynomial of the second kind. In addition, from the induction, we note that $T_n(\!\cosh (L-|x|))= \cosh n(L-|x|)$ for $n \in \mathbb{N}$ holds. Using $f$ in (2.22) instead of $F$ in (5.69), we reconsider the coefficients $\zeta ^n_j$ in (5.69) and $b^n_l$ in (5.70) in Theorem 5.3.

Theorem 5.3. Assume that $W \in C^m([0,L])$ for given $2 \le m \in \mathbb{N}$ and ( 2.23 ) for $n \le m-1$ . Let $\zeta ^n_j$ and $b^n_l$ be ( 5.69 ) with $f$ and ( 5.70 ) with a natural number $n \le m-1$ for $a=1$ and $b=\cosh L$ , respectively. Set the coefficient $\alpha ^{n}_j$ as

\begin{equation*} \alpha ^n_j = \sum _{\substack { k=j \\ k \equiv j \pmod 2}}^n b^n_k\delta ^k_j. \end{equation*}

Then, there exists a constant $C_L$ that is independent of $n$ such that for any $n \le m-1$

\begin{align*} \left\| W - \sum _{j=0}^{n} \alpha ^{n}_j \cosh j(L-\cdot ) \right\|_{ C^1([0,L])} \le C_L \frac {n+1}{2^{n}n!} \left(\frac {\cosh L-1}{2} \right)^{n} \max _{y\in [1,\cosh L ]}|f^{(n+1)}(y) | \end{align*}

holds.

We note that this theorem can be applicable to the smooth even potential $W$ with (2.23) on $\Omega$ since $\cosh j(L-|x|)$ is even.

Proof of Theorem 5.3. Recall that $f(y) = W(L- \cosh ^{-1} y)$ . From the property of the Lagrange polynomial for $f(y)$ defined in (2.22) for any $y \in [1, \cosh L]$ , there exists a constant $\min _i\{ r^{n+1}_i \} \lt c \lt \max _i\{ r^{n+1}_i\}$ such that

(5.71) \begin{equation} f(y) - L_n(y) = \frac { f^{(n+1)}( c ) }{(n+1)!} \prod _{j=1}^{n+1}\big(y-r_j^{n+1}\big) = \frac {f^{(n+1)} ( c ) }{2^n (n+1)!} \left( \frac { b - a }{2} \right)^{n+1} T_{n+1 }\left( \frac {2y - (b+a)}{b-a} \right) \end{equation}

with $b=\cosh L$ and $a=1$ and thus,

\begin{equation*} \frac {1}{2^n(n+1)!} \left(\frac {\cosh L-1}{2} \right)^{n+1} \max _{ y \in [1, \cosh L] }| f^{(n+1)}(y) | \ge \| f - L_n \|_{C([1,\cosh L])}. \end{equation*}

Here, we used $\| T_n \|_{C[0,1]}=1$ because of the definition. Using Proposition 5.2 and changing the variable to $y = \cosh (L-x)$ for $x \in [0,L]$ , we can compute the right-hand side of the above inequality as

\begin{align*} \| f - L_n \|_{C([1,\cosh L])} &= \sup _{y \in [1,\cosh L]} | f(y) - \sum _{j=0}^n b_j^n y^j| = \left\| f - \sum _{j=0}^n b_j^n \sum _{\substack {k=0 \\ k \equiv j\pmod 2}}^j \delta ^j_k T_k\right\|_{C([1,\cosh L])}\\ &=\left\| f(\!\cosh (L-\cdot ) ) - \sum _{j=0}^n b_j^n \sum _{\substack {k=0 \\ k \equiv j\pmod 2}}^j \delta ^j_k T_k(\!\cosh (L-\cdot )) \right\|_{C([0, L])}\\ &=\left\| W - \sum _{j=0}^n b_j^n \sum _{\substack {k=0 \\ k \equiv j\pmod 2}}^j \delta ^j_k \cosh k(L-\cdot ) \right\|_{C([0, L])}\\ &=\left\| W - \sum _{j=0}^n \sum _{\substack {k=j \\ k \equiv j\pmod 2}}^n b_k^n \delta ^k_j \cosh j(L-\cdot ) \right\|_{C([0, L])}\\ &= \left \| W - \sum _{j=0}^n \alpha ^n_j \cosh j(L-\cdot ) \right\|_{C([0, L])}. \end{align*}

Next, differentiating (5.71), we see that

\begin{align*} f'(y) - L'_n(y) = \frac {f^{(n+1)} ( c ) }{2^n n!} \Big ( \frac { b - a }{2} \Big )^{n} U_{ n }\Big ( \frac {2y - (b+a)}{b-a} \Big ) \end{align*}

because of $T'_{n+1}(x) = (n+1) U_n(x),$ [Reference Mason and Handscomb16]. Setting the function as $\mathscr{G} = \sqrt {y^2-1}$ for $y \in [1,\cosh L]$ , we see that $\mathscr{G}(\!\cosh (L-x)) = \sinh (L-x)$ for $x \in [0,L]$ . Then, we can compute that

\begin{align*} \| \mathscr{G}(f' - L'_n) \|_{C([1,\cosh L])} &=\sup _{y \in [1,\cosh L]} \left| \mathscr{G}(y) \left( f'(y) - \sum _{j=1}^n b_j^n j y^{j-1} \right) \right| \\ &= \left \| \mathscr{G}\left( f' - \sum _{j=1}^n b_j^n \sum _{\substack {k=0 \\ k \equiv j-1\pmod 2}}^{j-1} (k+1) \delta ^j_{k+1} U_k \right) \right\|_{C([1,\cosh L])}\\ &= \left \| W' + \sum _{j=1}^n b_j^n \sum _{\substack {k=1 \\ k \equiv j\pmod 2}}^{j-1} k \delta ^j_{k} \sinh k(L-\cdot ) \right\|_{C([0, L])}\\ &=\left\| W' + \sum _{ j = 1 }^n j \alpha _j^n \sinh j(L-\cdot ) \right \|_{C([0, L])}, \end{align*}

where we used $x^n = \sum _{j=0, j \equiv n\pmod 2}^n (j+1)\delta ^{n+1}_{j+1} U_j(x)/(n+1)$ in the second equality. Thus, we obtain that

\begin{align*} \left\| W' + \sum _{ j = 1 }^n j \alpha _j^n \sinh j(L-\cdot ) \right \|_{C([0, L])} &= \| \mathscr{G} ( f' - L'_n) \|_{C([1,\cosh L])}\\ &\le \sinh L \frac { (n+1) }{2^n n!} \left( \frac { \cosh L - 1 }{2} \right)^{n} \max _{ y \in [1, \cosh L] }| f^{(n+1)} ( y ) |. \end{align*}

Here, we used $\| U_n \|_{C([0,1])} = U_n(0)=n+1$ because the value of $(\sin (n+1)\theta )/\sin \theta$ on extreme points in $\theta \in [0,\pi /2]$ is decreasing. Putting $C_L\,:\!=\,\max \{ (\!\cosh L -1)/4, \sinh L \}$ implies the assertion of this theorem.

As any continuous function can be approximated by the sum of $(\!\cosh (L-x))^j$ by Theorem 5 in [Reference Ninomiya, Tanaka and Yamamoto19], if the coefficients of $(\!\cosh (L-x))^j$ , that is $b^n_j$ , are obtained, it was conceived that the error between $W$ and $\sum _{j=1}^n a_j k_j$ could be estimated. Indeed, the smooth function can be approximated by the Lagrange interpolation polynomial. Thus, it was considered that it should be shown that $L_n$ can be expressed by the form of $\sum _{j=0}^n b_j y^j$ . This is generally unknown. However, when the Chebyshev nodes are given to the Lagrange interpolation polynomial, computing $b^n_j$ resulted in the proof since $T_n(x) = \sum _{k=0}^{[n/2]}C^n_k x^{n-2k}$ holds and the numerator of the Lagrange polynomial must be divisible by $x-r^{n+1}_j$ . In this sense, Proposition 5.2 is essential.

Next, we prove Corollary 2.6

Then, we explain the proof of Theorem 2.9.

Proof of Theorem 2.9. According to Theorem 5.3, for any even function $W$ in $C^\infty ([0,L])$ with (2.23) and (2.24) for any $n\in \mathbb{N}$ , and arbitrary $ M\in \mathbb{N}$ , there exist constants $\{ \alpha _j^{M-1}\}_{j=0}^{M-1}$ such that

\begin{equation*} \left\| W - \sum _{j=0}^{M-1} \alpha _j^{M-1} \cosh j(L-|\cdot |) \right \|_{C^1(\Omega )} \lt C_L \frac { M }{2^{M-1} (M-1)!} \left( \frac {\cosh L -1 }{2} \right)^{M-1} \,=\!:\,C_L C(M). \end{equation*}

Here, we set $n=M-1$ in Theorem 5.3. Putting the parameters as (2.21) and (2.27), then we have

(5.72) \begin{equation} \left\| W - \sum _{j=1}^M a_j k_j\right\|_{C^1(\Omega )} \lt C_L C(M). \end{equation}

Let $\bar { \rho }(x,t)$ be the solution to (P) with the integral kernel $\sum _{j=1}^M a_j k_j$ . Since (5.72) holds and $C(M)$ converges to zero as $M \to \infty$ , there exists a positive constant $\delta$ that is independent of $M$ such that $\|\sum _{j=1}^M a_j k_j\|_{W^{1,1} (\Omega )} \le \delta + \|W \|_{W^{1,1}(\Omega )}$ . Thus, we see that $\tilde C_T$ in Lemma 2.8 does not depend on $M$ . Then, for this fixed $M$ , Theorem 2.4 and Lemma 2.8 yield that for any $\varepsilon \gt 0$

\begin{align*} \left \| \rho - \rho ^{M,\varepsilon } \right \|_{ C( [0,T]; H^1 (\Omega ))} &\le \left \| \rho - \bar { \rho } \right \|_{ C( [0,T]; H^1 (\Omega ))} + \left \| \bar { \rho } - \rho ^{M,\varepsilon } \right \|_{ C( [0,T]; H^1 (\Omega ))} \\ &\le \sqrt { {\tilde C_T} } \left\| W - \sum _{j=1}^M a_j k_j \right\|_{ W^{1,1} (\Omega )} + C_1 \varepsilon \\ &\le 2L C_L \sqrt { {\tilde C_T} } C(M) +C_1 \varepsilon . \end{align*}

Thus, we put $C_T^{(1)}\,:\!=\, 2L C_L \sqrt { {\tilde C_T} }$ and $C_T^{(2)}(M)\,:\!=\, C_1$ , respectively.

We performed a numerical simulation of the approximation for the potential $W$ by the linear combination of $\cosh j(L-|x|)$ . The results are shown in Figure 3. The linear combination of $\cosh j(L-|x|)$ covers a potential $W$ . The longer the length of the interval $L$ becomes, the worse the rate of convergence becomes from the numerical simulations. However, as the rate of convergence can be exponential as given by Theorem 5.3, the method for determining the coefficient of $a_j$ is compatible with and useful for numerical simulations.

Figure 3. Results of a numerical simulation of the approximation for $W$ by the linear combination of $\cosh j(L-|x|)$ . We set $W(x)=e^{-5x^2}(\cos (3\pi ) x- 1/2 \cos (2\pi x))$ , and $L=2$ . (a) Profiles of $W$ and the linear sum of $\cosh j(L-|x|)$ . (b) Profiles of $f$ and the Lagrange interpolation polynomial on $[1, \cosh L]$ . (c) Distribution of $\{ \alpha ^9_j \}_{j=0}^9$ .

Figures 4 and 8 shows the numerical results of (P) with the potential $W(x)=e^{-5x^2}$ and ( $\mbox{KS}^{M,\varepsilon }$ ) with parameters $\varepsilon =0.001$ and $\{\alpha _j^6\}^6_{j=0}$ specified by Theorem 5.3. We can observe that the solution $\rho$ in (P) is approximated by that $\rho ^{7,\varepsilon }$ in ( $\mbox{KS}^{M,\varepsilon }$ ), even though there are seven auxiliary factors $v_j^{7,\varepsilon }$ . In Figure 4, (c) shows the profiles of both $W(x)=e^{-5x^2}$ and $\sum _{j=0}^6 \alpha _j^6 \cosh (j(L-|x|))$ . Since $\sum _{j=0}^6 \alpha _j^6 \cosh (j(L-|x|))$ has good accuracy for the approximation of $e^{-5x^2}$ , both curves are seen to overlap.

Figure 4. Results of numerical simulations for (P) with a potential $W(x) = e^{-5x^2}$ and $\mu$ defined in ( $\mbox{P}_\mu$ ), and ( $\mbox{KS}^{M,\varepsilon }$ ) with $M=7$ . The parameters are given by $L=1$ , $\varepsilon =0.001$ , $d_1=1000000$ , $\mu =5$ and $d_j$ and $a_j$ are provided by (2.21) and (2.27), respectively. (a) Profiles of the numerical result of (P) at $t=200.0$ . The horizontal and vertical axes correspond to the position $x$ and $\rho$ , respectively. The red curve is the numerical result of $\rho$ . (b) Profiles of the numerical result of ( $\mbox{KS}^{M,\varepsilon }$ ) at $t=200.0$ . We impose the same initial data for $\rho ^{7,\varepsilon }$ as that of $\rho$ and $(v_j)_0= k_j*\rho _0, \ (j=1,\ldots ,M)$ . The axes are set same as that of (a). The red and the other colour curves correspond to $(\rho ^{7,\varepsilon },\{v_j^{7,\varepsilon }\}_{j=1}^7)$ , respectively. (c) Profiles of $W$ and $\sum _{j=0}^6 \alpha _j^6 \cosh (j(L-|x|))$ . The orange dashed and blue curves corresponding to $W$ and $\sum _{j=0}^6 \alpha _j^6 \cosh (j(L-|x|))$ , respectively, are drawn in a same plane. (d) The distribution of $\{\alpha _j^6\}_{j=0}^6$ .

6. Linear stability analysis

In this section, we perform a linear stability analysis around the equilibrium point for (P) and ( $\mbox{KS}^{M,\varepsilon }$ ) with three components to specify the role of advective nonlocal interactions in pattern formations. We demonstrate that the eigenvalue of the linearised operator of ( $\mbox{KS}^{M,\varepsilon }$ ) converges to that of (P) as $\varepsilon \to 0$ when the integral kernel is given by $k_j$ of (2.4). We analyse the following equation with the parameter $\mu \gt 0$ :

\begin{equation*} \qquad\qquad\qquad\qquad\qquad\qquad\rho _ t = \rho _{ xx } - \mu ( \rho ( W*\rho )_x )_x \ \text{in} \ \Omega \times (0,\infty ). \qquad\qquad\qquad\qquad\qquad (\mbox{P}_\mu)\end{equation*}

We explain the instability of the solution near the equilibrium point. Let $\rho ^*\gt 0$ and $ \xi =\xi (x,t)$ be an arbitrary constant and a small perturbation, respectively. $\rho = \rho ^*$ becomes a constant stationary solution of (P). Putting $\rho (x,t) = \rho ^* + \xi (x,t)$ and substituting it for ( $\mbox{P}_\mu$ ), we have

\begin{align*} \xi _t & = \xi _{xx} - \mu \left( (\xi + \rho ^*) W * (\xi + \rho ^*)_x \right)_x = \xi _{xx} - \mu ( \rho ^* W*\xi _{xx} + \xi _xW*\xi _x + \xi W*\xi _{xx}). \end{align*}

Focusing on the linear part of above, we denote the linear operator $\mathscr L$ by

\begin{equation*} \mathscr{L}[u]\,:\!=\, u_{xx} - \mu \rho ^* W*u_{xx}. \end{equation*}

Because this linearised operator has $\mu \rho ^*$ , the effects of the strength of aggregation and the mass volume on the pattern formation around the constant stationary solution are equivalent. Therefore, we replace $ \mu \rho ^*$ with $\mu$ . Defining the Fourier coefficient of $W$ as

\begin{equation*} \omega _n \,:\!=\, \frac {1}{ \sqrt {2L} }\int _\Omega W(x) e^{-i\sigma _n x} dx, \quad n \in \mathbb{Z}, \end{equation*}

we have the following lemma with respect to the eigenvalues and eigenfunctions:

Lemma 6.1. Setting the eigenvalues

\begin{equation*} \lambda ( n ) = -\sigma _n^2( 1 - \sqrt {2L} \mu \omega _n), \end{equation*}

then we have

\begin{equation*} \mathscr L[ e^{i \sigma _n x } ] = \lambda ( n ) e^{i \sigma _n x }, \quad n \in \mathbb{Z}. \end{equation*}

Proof. The proof follows from a direct calculation

\begin{align*} &(e^{ i \sigma _n x })_{xx} = -\sigma _n^2 e^{i \sigma _n x }, \\ &W*(e^{ i \sigma _n \cdot })_{xx} = -\sigma _n^2 \int _\Omega W(y) e^{i \sigma _n (x-y) } dy= - \sqrt {2L} \sigma _n^2 \omega _n e^{i \sigma _n x }. \end{align*}

Using this lemma, we find the solution to $\xi _t = \mathscr L [\xi ]$ around $\rho ^*$ in the form of $ \sum _{n\in \mathbb{Z}} \hat {\xi }_n e^{\lambda _n t} e^{i \sigma _n x }$ , where $\{ \hat {\xi }_n \}$ is the Fourier coefficient.

Figure 5. Results of a numerical simulation for ( $\mbox{P}_\mu$ ) with (2.6). The parameters are $\mu =5.0$ , $d_1=0.1$ and $d_2=3.0$ , and the initial data are given by $1.0$ with small perturbations. The horizontal and vertical axes correspond to the position $x$ and value of solution $\rho$ , respectively. The red curve corresponds to the solution $\rho$ . The left, middle left, middle right and right pictures exhibit the profiles of solutions of ( $\mbox{P}_\mu$ ) with (2.6) in the interval $[0, 10]$ at $t = 0, 0.5, 1.0$ and $3.0$ , respectively.

Figure 6. Results of a numerical simulation for ( $\mbox{P}_\mu$ ) with (2.7). The parameters are $\mu =4.0$ and $R=1.0$ , and the initial data are given by $1.0$ with small perturbations. The horizontal and vertical axes correspond to the position $x$ and value of solution $\rho$ , respectively. The red curve corresponds to the solution $\rho$ . The left, middle left, middle right and right pictures exhibit the profiles of solutions of ( $\mbox{P}_\mu$ ) with (2.7) in the interval $[0, 10]$ at $t = 0, 0.8, 2.0$ and $12.0$ , respectively.

Here, we recall the concept of the diffusion-driven instability in pattern formations proposed by Turing [Reference Turing21]. Diffusion-driven instability is a paradox where diffusion, typically leading to concentration homogenisation, destabilises the uniform stationary solution and induces nonuniformity due to the difference in the diffusion coefficients. By using the eigenvalue $\lambda =\lambda (n)$ for the linear operator of the reaction-diffusion system, the diffusion-driven instability can be defined as the eigenvalue $\lambda$ satisfies $\lambda (0)\lt 0$ and there exists $n\in \mathbb{Z}$ such that $\lambda (n)\gt 0$ . For the model ( $\mbox{P}_\mu$ ), the similar situation occurs. If $W$ satisfies that $\lim _{n \to \pm \infty } \omega _n=0$ and that there exists $0 \neq n_1 \in \mathbb{N}$ such that ${\displaystyle } Re \omega _{n_1} \gt 0$ , the maximum eigenvalue may be attained around $n=n_1$ . In that case, the unstable mode around a stable equilibrium point is given by $e^{i \sigma _{n_1} x }$ .

We performed numerical simulations of ( $\mbox{P}_\mu$ ) with the integral kernel (2.6) and (2.7) with the finite volume method. Figures 5 and 6 present the results. The Fourier coefficients for the integral kernels (2.6) and (2.7) are given by

\begin{align*} &\omega _{n,1} = \frac {1}{\sqrt {2 L}} \left (\frac { 1 }{d_1 \sigma _n^2+1} - \frac { 1 }{d_2 \sigma _n^2 +1}\right ), \\[4pt] &\omega _{n,2} = \frac {\sqrt {2} }{ \sqrt { L} } \frac { 1}{ \sigma _n^2 } \left(1-\cos (R_0 \sigma _n ) \right), \end{align*}

respectively. Figures 1 (b) and 2 (b) show the distributions of the eigenvalues with $\omega _{n,1}$ and $\omega _{n,2}$ . The number of peaks of the solution at the beginning of the pattern formation corresponds to the maximum wave number in Figures 5 and 6, respectively.

For (2.6), by introducing $v^\varepsilon _1=k_1*\rho$ and $v^\varepsilon _2=k_2*\rho$ into $W$ , the solution of ( $\mbox{P}_\mu$ ) can be approximated by that of the 3-component Keller–Segel system from Theorem 2.4:

(6.73) \begin{equation} \left \{ \begin{aligned} \rho ^{\varepsilon }_ t & = ( \rho ^{\varepsilon })_{xx} - \mu \big ( \rho ^{\varepsilon } ( v_1^\varepsilon - v_2^\varepsilon )_x \big )_x,\\ (v_1^\varepsilon )_t &= \frac {1}{\varepsilon } \big ( d_1 ( v_1^\varepsilon )_{xx} - v_1^\varepsilon + \rho ^{\varepsilon } \big ),\\ (v_2^\varepsilon )_t &= \frac {1}{\varepsilon } \big ( d_2 ( v_{2}^\varepsilon ) _{xx} - v_2^\varepsilon + \rho ^{\varepsilon } \big ) \end{aligned} \right .\ \text{in} \ \Omega \times (0,\infty ) \end{equation}

with $0 \lt \varepsilon \ll 1$ . In (6.73), $v_1^\varepsilon$ and $v_2^\varepsilon$ represent the attractive and repulsive substances in chemotactic process, respectively. By determining the solution as a Fourier series expansion, the linearised problem is provided by the following system:

\begin{equation*} \boldsymbol{\varphi }_t = \begin{pmatrix} -\sigma _n^2 & \mu \sigma _n^2 & - \mu \sigma _n^2 \\[6pt] \dfrac {1}{\varepsilon } & \dfrac {-d_1 \sigma _n^2 -1 }{\varepsilon } & 0\\[6pt] \dfrac {1}{\varepsilon } & 0 & \dfrac {-d_2 \sigma _n^2 -1 }{\varepsilon } \end{pmatrix} \boldsymbol{\varphi }, \quad \boldsymbol{\varphi } \,:\!=\, \begin{pmatrix} (\hat {\varphi _1})_n\\[2mm] (\hat {\varphi _2})_n\\[2mm] (\hat {\varphi _3})_n \end{pmatrix}, \end{equation*}

where $(\hat {\varphi _3})_n$ is the Fourier coefficient for the perturbations of $v_2^\varepsilon$ . The characteristic polynomial is given by

\begin{align*} P_2(\lambda ,\varepsilon )&= -\lambda ^3 - \Big ( C_{31} + \frac { C_{33} }{ \varepsilon } \Big ) \lambda ^2 - \Big ( \frac { C_{34} }{\varepsilon ^2} +\frac { C_{35} }{\varepsilon } \Big )\lambda - \frac { C_{36} }{\varepsilon ^2},\\ C_{33}&\,:\!=\, 2+d_1\sigma _n^2+d_2\sigma _n^2,\\ C_{34}&\,:\!=\, \left(1+d_1\sigma _n^2\right)\left(1+d_2\sigma _n^2 \right),\\ C_{35}&\,:\!=\,\sigma _n^2\left( 2+ d_1\sigma _n^2 + d_2\sigma _n^2 \right),\\ C_{36}&\,:\!=\,\sigma _n^2\left(1+d_1\sigma _n^2\right)\left(1+d_2\sigma _n^2 \right) + \mu \left( d_1 -d_2 \right)\sigma _n^4. \end{align*}

Then, we can see that

\begin{align*} \varepsilon ^2\frac {\partial P_2}{\partial \lambda } &= -3\varepsilon ^2\lambda ^2 -2\varepsilon ( \varepsilon C_{31} + C_{33} ) \lambda -( C_{34} +\varepsilon C_{35}) \\ &\to - (1+d_1\sigma _n^2)(1+d_2\sigma _n^2 ) \lt 0 \quad (\varepsilon \to 0+0). \end{align*}

Only one eigenvalue converges to a bounded value when $\varepsilon \to 0+0$ from the implicit function theorem. Setting the eigenvalue as $\lambda _\varepsilon$ , we can solve that

\begin{equation*} \lim _{\varepsilon \to 0+0}\varepsilon ^2 P_2(\lambda _\varepsilon ,\varepsilon ) = - C_{34} \lambda _0 - C_{36} =0, \end{equation*}

and thus

\begin{equation*} \lambda _{0} = -\sigma _n^2 + \frac { \mu ( d_2 -d_1 )\sigma _n^4 }{ \left(1+d_1\sigma _n^2\right)\left(1+d_2\sigma _n^2 \right) } = -\sigma _n^2\left( 1- \mu \sqrt {2L} \omega _{n,1} \right). \end{equation*}

This implies that the solution to (6.73) and ( $\mbox{P}_\mu$ ) with (2.6) are sufficiently close, but also that the Fourier mode of ( $\mbox{P}_\mu$ ) when the pattern forms around an equilibrium point is also extremely close to that of the 3-component attraction-repulsion Keller–Segel system (6.73). Because the constant stationary solution is destabilised by the auxiliary factors $v_1^\varepsilon$ and $v_2^\varepsilon$ , the mechanism of the pattern formation is almost the same as that of diffusion-driven instability. In other words, if the integral kernel is provided by (2.6), the solution to (P) is sufficiently close to that of the Keller–Segel system, which can cause the diffusion-driven instability, and thereby suggesting that the kernel $W$ is crucial in generating the diffusion-driven instability in the nonlocal Fokker–Planck equation (P) in linear sense.

We performed a numerical simulation of (6.73) with $\varepsilon =0.001$ . The profile of the solution $\rho ^\varepsilon$ at each time point in Figure 7 is similar to that of $\rho$ in Figure 5. These figures also indicate that stationary solutions to (P) may be approximated by those to ( $\mbox{KS}^{M,\varepsilon }$ ). As explained above, by approximating the dynamics of nonlocal evolution equations using Keller–Segel systems, we can describe the nonlocal dynamics within the framework of local dynamics and identify both mechanisms.

Figure 7. The results of a numerical simulation for (6.73). The parameters $\mu , d_1, d_2$ and the initial data are same as that in Figure 5, and $\varepsilon =0.001$ and $((v_1)_0,(v_2)_0)=( k_1*\rho _0, k_2*\rho _0 )$ . The horizontal and vertical axes correspond to the position $x$ and value of solutions $\rho ^\varepsilon$ , $v_1^\varepsilon$ and $v_2^\varepsilon$ , respectively. The red, green and blue curves correspond to the solution $\rho ^\varepsilon$ , $v_1^\varepsilon$ and $v_2^\varepsilon$ , respectively. The left, middle left, middle right and right pictures exhibit the profiles of solutions of (6.73) in the interval $[0, 10]$ at $t = 0, 0.5, 1.0$ and $3.0$ , respectively.

Figure 8. Comparison of the time evolutions of numerical results given in Figure 4 (a) and (b) until $t=1.0$ .