1. Introduction
The 4-parameter equation
studied in [Reference Anco, da Silva and Freire1, Reference Freire and Silva15, Reference Himonas and Holliman16], is a generalization of the Camassa–Holm equation [Reference Camassa and Holm6]
and the Novikov equation [Reference Hone and Wang20, Reference Novikov25]
that admits certain scaling transformations as symmetries. The equation (1.1) has proven to be an interesting mathematical equation once it is possible to choose $a=k+2,\,b=k+1$
and $c=1$
in order to transform it into a one-parameter family of equations that still unifies (1.2) and (1.3), and also admits the peaked wave solutions $u(t,\,x) = c^{1/k}e^{-|x-ct|},$
called peakon solutions [Reference Camassa and Holm6], where $c$
denotes the wave speed. Despite admitting an infinite number of conservation laws only when the equation is reduced to (1.2) or (1.3), it was not long before the interesting properties of (1.1) attracted attention from researchers. In terms of applied analysis, Himonas and Holliman, in the same paper [Reference Himonas and Holliman16] showed that for any positive integer $k\geq 1$
, $b=a+1$
and $c=1$
, the equation is Hadamard well-posed in $H^{s}(\mathbb {R})$
for $s>3/2$
and, more recently, Barostichi, Himonas and Petronilho [Reference Barostichi, Himonas and Petronilho5] considered the choices $a=k+2,\,b=k+1$
and $c=1$
in (1.1) to extend local well-posedness to global for the resulting equation and also understand the behaviours of global analytic solutions provided that the McKean quantity $m_0=m_0(x):=(1-\partial _x^{2})u(0,\,x)$
does not change sign. For a geometric interpretation of the sign persistance of the McKean quantity and its consequences, see [Reference Constantin7] and discussions in [Reference Constantin and Kolev8, Reference Kolev23].
It is important to observe that the restriction $k\geq 1$
in (1.1) is due to two main reasons: firstly, the Camassa–Holm and Novikov equations are accomplished when we have two particular positive choices of $k$
and, secondly, problems with singularity obviously arise whenever considering $k<1$
. Moreover, the former also explains why the constants $a,\,b,\,c$
are taken as different than zero. However, by allowing $b=0$
and $a=c=1$
in (1.1), one arrives at the equation
where $k$
will be taken as a positive integer and $m=u-u_{xx}$
. In the particular case where $k=1$
, (1.4) is a very particular case of the $b$
-equation $m_t + bmu_x + um_x=0$
considered in [Reference Degasperis, Holm and Hone10] and later shown in [Reference Dullin, Gottwald and Holm11, Reference Holm and Staley19] to have hydrodynamic applications when $b\neq -1$
. Moreover, it can also be obtained from Kodama transformation to describe shallow water elevation [Reference Dullin, Gottwald and Holm12]. In terms of well-posedness, we observe that in [Reference Himonas and Holliman16, Reference Yan26] the authors showed that (1.4) is well-posed for an initial value $u_0\in H^{s}(\mathbb {R}),$
where $s>3/2$
.
It is crucial to observe, however, that although local well-posedness of (1.4) in Sobolev [Reference Himonas and Holliman16], Besov [Reference Yan26] and Gevrey [Reference Barostichi, Himonas and Petronilho3] spaces has been successfully established, not much else has been considered for $k>1$
. In fact, the reasons for this fact are rather simple: the case $k=1$
in (1.4) is only known to conserve the momentum $\int _{\mathbb {R}} m(t,\,x)dx$
for rapidly decreasing solutions, which is equivalent to saying that
is independent of time for the same sort of solution. For the generalized equation (1.4) with $k>1$
, the situation becomes even more drastic as no conservation laws seem to exist [Reference Anco, da Silva and Freire1], which poses a difficulty that perhaps may be impossible to overcome in the attempt to study solutions and their properties.
One of the pioneering works is [Reference da Silva and Freire9], where in the particular case of $k=1$
in (1.4) the authors considered global well-posedness and deduced that, also making the assumption that the McKean quantity does not change sign, in $H^{3}(\mathbb {R})$
, it is possible to extend the local solutions and then the maximal time of existence is infinite. This is indeed a remarkable result once the equation (1.4) lacks the conservation of the $H^{1}(\mathbb {R})$
norm and the construction of a highly non-trivial functional was required to show that the solution could not blow up at a finite time. Following a similar direction, in [Reference Duruk Mutlubas and Freire13] the authors determined global well-posedness for the periodic case and also studied continuation of periodic solutions. For the general case $k>1$
the authors in [Reference da Silva and Freire9] also answered some of the questions raised by Himonas and Thompson [Reference Himonas and Thompson18], giving a characterization of asymptotic behaviour or solutions based on the initial data, and a blow-up criteria has been established in [Reference Yan26]. The determination of global well-posedness for $k>1$
, however, is still an open problem.
In this paper, we are interested in the initial value problem
where
and complementing the results found in [Reference da Silva and Freire9, Reference Duruk Mutlubas and Freire13]. We observe that (1.6)–(1.7) is nothing but the evolution formulation of the Cauchy problem of (1.4) after the inversion of the Helmholtz operator $1-\partial _x^{2}$
.
Consider the $L^{2}_x(\mathbb {R})$
space of square integrable functions endowed with the norm
The main function space of our interest in the present paper is the Gevrey space $G^{\sigma,s}(\mathbb {R})$
, where $\sigma >0$
and $s\in \mathbb {R}$
, of functions in $L^{2}(\mathbb {R})$
such that the norm
is finite, where $\hat {f}$
denotes the Fourier transform
In the particular case where $\sigma \to 0$
, the space $G^{0,s}(\mathbb {R})$
becomes the usual Sobolev space $H^{s}(\mathbb {R})$
. In a result known as Paley–Wiener theorem (see [Reference Katznelson22]), the Gevrey space $G^{\sigma,s}(\mathbb {R})$
is characterized as the restriction to the real line of functions that are analytic on a strip of width $2\sigma$
.
Our main intention is to show that well-posedness of (1.6)–(1.7) goes beyond Sobolev spaces in the sense of a proof for global well-posedness in Gevrey spaces by making use of the Kato–Masuda [Reference Kato and Masuda21] machinery and certain embeddings between spaces.
Before proceeding with our main result, we state a generalization of global well-posedness in Sobolev spaces for equation (1.6)–(1.7) with $k=1$
. We observe that the result proven in [Reference da Silva and Freire9] covers an initial data $u_0\in H^{3}(\mathbb {R})$
, while here we establish the result to $u_0\in H^{s}(\mathbb {R})$
, $s\geq 3$
, which is enough for our purposes.
Proposition 1.1 Given $u_0\in H^{s}(\mathbb {R}),$
$s\geq 3,$
if $m_0 \in L^{1}(\mathbb {R})$
does not change sign, then the unique local solution $u$
for (1.6)–(1.7) with $k=1$
exists globally in $C([0,\,\infty );H^{s}(\mathbb {R})) \cap C^{1}([0,\,\infty );H^{s-1}(\mathbb {R}))$
.
With proposition 1.1 in hand, we enunciate our main result.
Theorem 1.2 Given $u_0\in G^{1,s}(\mathbb {R}),$
with $s>5/2,$
if $m_0(x)$
does not change sign, then the Cauchy problem of (1.6)–(1.7) with $k=1$
has a unique global analytic solution $u\in C^{\omega }([0,\,\infty )\times \mathbb {R})$
.
In the context of hydrodynamic applications, analyticity is a crucial ingredient to prove an intrinsic characterization of symmetric waves, see [Reference Escher and Matioc14] for more details. Here, we observe that our unique space-time analytic solutions provided by theorem 1.2 are not necessarily travelling waves, which then provides an interesting and general result.
The proof of theorem 1.2 relies on the powerful machinery of Kato and Masuda [Reference Kato and Masuda21] and embedding properties of certain spaces, see [Reference Barostichi, Himonas and Petronilho5, Reference Kato and Masuda21]. Another useful space is an adaptation of the Banach spaces proposed by Himonas and Misiolek in [Reference Himonas and Misiolek17]: for $\sigma >0$
and $m$
is a positive integer, the set $E_{\sigma,m}(\mathbb {R})$
of infinitely differentiable functions such that
is a Banach space by its own turn. In order to prove that the lifespan is infinite and the solution is analytic in both variables $t$
and $x$
, we will make use of the auxiliary local well-posedness result.
Proposition 1.3 Given $u_0(x)\in E_{\sigma _0,m}(\mathbb {R}),$
with $m\geq 3$
and for some $\sigma _0\in (0,\,1]$
fixed, for
where
with $c_m>0$
depending only on $m$
, and for every $\sigma \in (0,\,\sigma _0),$
the Cauchy problem for (1.6)–(1.7) has a unique solution $u$
that is analytic in the disc $D(0,\,T(\sigma _0-\sigma ))$
with values in $E_{\sigma,m}(\mathbb {R})$
. Moreover, the bound
holds.
The paper is organized as follows. In § 2 we establish the basic function spaces and auxiliary propositions required for the understanding and proofs of our results. In § 3, we present the proof of proposition 1.3 for any $k\in \mathbb {Z}_+$
, which follows from the technical estimates of § 2. After that, in § 4 we present a proof of proposition 1.1, which can also be found in [Reference da Silva and Freire9] for $s=3$
. Finally, in § 5 we provide a proof for global well-posedness in $H^{\infty }(\mathbb {R})$
and finalize with the extensive and complex proof of theorem 1.2 by making use of proposition 1.3.
2. Function spaces and auxiliary results
In this section we will enunciate the theory behind the function spaces presented in the introduction.
We start recalling that, similarly to Sobolev spaces, one interesting property of Gevrey spaces is that it is possible to continuously embed them based on the parameters $\sigma$
and $s$
, see [Reference Barostichi, Himonas and Petronilho4]:
(i) If $0<\sigma '<\sigma$
and $s\geq 0$, then $\Vert \cdot \Vert _{G^{\sigma ',s}}\leq \Vert \cdot \Vert _{G^{\sigma,s}}$ and $G^{\sigma,s}(\mathbb {R})\hookrightarrow G^{\sigma ',s}(\mathbb {R})$;(ii) If $0< s'< s$
and $\sigma >0$, then $\Vert \cdot \Vert _{G^{\sigma,s'}}\leq \Vert \cdot \Vert _{G^{\sigma,s}}$ and $G^{\sigma,s}(\mathbb {R})\hookrightarrow G^{\sigma,s'}(\mathbb {R})$.
Although our main result involves the use of Gevrey spaces, we will need to consider some auxiliary spaces and their embeddings. Following the work of Kato and Masuda [Reference Kato and Masuda21] about the Korteweg–de Vries equation, for $r>0$
fixed we define the spaces $A(r)$
of functions that can be analytically extended to a function on a strip of width $r$
, endowed with the norm
for $s\geq 0$
and every $\sigma \in \mathbb {R}$
such that $e^{\sigma }< r$
. Observe that if $r\geq r'$
then $f\in A(r)$
implies that $f\in A(r')$
and, therefore, $A(r)\subset A(r')$
.
For $H^{\infty }(\mathbb {R}) := \bigcap \limits _{s\geq 0}H^{s}(\mathbb {R})$
, we have the following sequence of embeddings (see lemma 2.3 and lemma 2.5 in [Reference Barostichi, Himonas and Petronilho4] and lemma 2.2 in [Reference Kato and Masuda21]):
for $\sigma >0$
and $s\geq 0$
.
Similarly to Gevrey spaces, we have $E_{\sigma,m}(\mathbb {R})\hookrightarrow E_{\sigma ',m}(\mathbb {R})$
for $0<\sigma '<\sigma$
and, more importantly, $E_{\sigma,m}(\mathbb {R})$
is also continuously embedded into $H^{\infty }(\mathbb {R})$
for all $m\geq 1$
and $\sigma >0$
, see page 750 of [Reference Barostichi, Himonas and Petronilho5]. Moreover, it is important to emphasize that if $1\leq m\leq m'$
, then $\vert \vert \vert f\vert \vert \vert _{E_{\sigma,m}}\leq \vert \vert \vert f\vert \vert \vert _{E_{\sigma,m'}}$
.
To be able to extend regularity of global solutions, we will need to first consider local well-posedness in $E_{\sigma _0, m}(\mathbb {R})$
for some $\sigma _0\in (0,\,1]$
and $m\geq 3$
, and for that purpose some estimates will be required. The first one we enunciate is the algebra property, which allows us to relate the norm of multiplication to the multiplication of norms.
Lemma 2.1 Algebra property
For any positive integer $m,$
$0<\sigma \leq 1$
and $\varphi,\,\psi \in E_{\sigma,m}(\mathbb {R}),$
there is a positive constant $c_m$
depending only on $m$
such that
Proof. The proof follows closely the lines of [Reference Himonas and Misiolek17].
Consider an equation of the form $m_t = F(u,\,u_x,\,u_{xx},\,u_{xxx})$
and let $g(x) = e^{-|x|}/2$
be the Green function of the equation $(1-\partial _x^{2})u = \delta (x)$
, where $\delta$
denotes de Dirac delta distribution. Then we can write the inverse of the Helmholtz operator $1-\partial _x^{2}$
as
With respect to the spaces $E_{\sigma,m}(\mathbb {R})$
, the Helmholtz operator and its inverse have some important and useful properties that will be necessary to prove local well-posedness.
Lemma 2.2 For $0<\sigma '<\sigma \leq 1,$
$m\geq 1$
and $\varphi \in E_{\sigma,m}(\mathbb {R}),$
then
Proof. The proofs of (2.4) and (2.5) follow immediately from the analogous estimates for Sobolev spaces and will be omitted, while the the proof of (2.3) requires an immediate adaptation of the proof of lemma 2.4 (page 580) of [Reference Himonas and Misiolek17].
In what follows, a function $u$
belongs to the space $C^{\omega }(I;X)$
if it is analytic in the interval $I$
as a function of $t$
and $u(t,\,\cdot )$
belongs to $X$
. We will be interested in $C^{\omega }(I;G^{\sigma,s}(\mathbb {R})),\, C^{\omega }(I;E_{\sigma,m}(\mathbb {R}))$
and $C^{\omega }(I;A(r))$
. In the case where $u\in C^{\omega }(I\times \mathbb {R})$
, then $u(t,\,x)$
is analytic for $(t,\,x)\in I\times \mathbb {R}$
.
The final result of this section, called Autonomous Ovsyannikov Theorem, will be used in the next section to prove proposition 1.3. Its proof uses a very classical fixed point argument and follows closely the ideas in [Reference Barostichi, Himonas and Petronilho4, Reference Luo and Yin24], see also [Reference Baouendi and Goulaouic2].
Proposition 2.3 Autonomous Ovsyannikov Theorem
Let $X_{\delta }$
be a scale of decreasing Banach spaces for $0<\delta \leq 1,$
that is, $X_\delta \subset X_{\delta '},\, \Vert \cdot \Vert _{\delta '}\leq \Vert \cdot \Vert _{\delta },\, 0<\delta '<\delta \leq 1,$
and consider the Cauchy problem
Given $\delta _0\in (0,\,1]$
and $u_0\in X_{\delta _0}$
, assume that $G$
satisfies the following conditions:
(i) For $0<\delta '<\delta <\delta _0$
, $R>0$ and $a>0$, if the function $t\mapsto u(t)$ is holomorphic on $\{t\in \mathbb {C}; 0<|t|< a(\delta _0-\delta )$ with values in $X_{\delta }$ and $\sup \limits _{t< a(\delta _0-\delta )}\Vert u-u_0\Vert _{\delta }< R$, then the function $t\mapsto G(t,\,u(t))$ is holomorphic on the same set with values in $X_{\delta '}$.(ii) $G:X_{\delta }\rightarrow X_{\delta '}$
is well defined for any $0<\delta '<\delta <\delta _0$ and for any $R>0$ and $u,\,v\in B(u_0,\,R)\subset X_{\delta }$, there exist positive constants $L$ and $M$ depending only on $u_0$ and $R$ such that
\[ \Vert G(u) - G(v)\Vert_{\delta'}\leq \frac{L}{\delta-\delta'}\Vert u-v\Vert_{\delta},\quad \Vert G(u_0)\Vert_{\delta}\leq \frac{M}{\delta_0-\delta}, \]
$0<\delta <\delta _0$
. Then for
the initial value problem (2.6) has a unique solution $u\in C^{\omega }([0,\,T(\delta _0-\delta )),\,X_{\delta })$
, for every $\delta \in (0,\,\delta _0)$
, satisfying
3. Local well-posedness in the Himonas–Misiolek space
In this section we want to prove proposition 1.3 by making use of Autonomous Ovsyannikov Theorem. Before doing so in the next subsections, observe that the embeddings $E_{\sigma,m}(\mathbb {R})\hookrightarrow E_{\sigma ',m}(\mathbb {R})$
, for $0<\sigma '<\sigma$
, guarantee that the function
taken as the right-hand side of (1.6), is a well-defined function from $E_{\sigma,m}(\mathbb {R})$
to $E_{\sigma ',m}(\mathbb {R})$
for every choice of $k$
. Moreover, condition 1 for the Autonomous Ovsyannikov Theorem is trivially satisfied. Therefore, it remains to prove condition 2.
The proof of proposition 1.3 will be given in two parts. First we will separately prove the case where $k=1$
, and then proceed to the case $k>1$
. We would like to point out that proposition 1.3 holds for any positive choice of the parameter $k$
, which then recovers the case $b=0$
for the $b$
-equation.
3.1 Proof for $k=1$
Consider the function $F(u)$
given by (1.7) with $k=1$
.
Proposition 3.1 Given $\sigma _0\in (0,\,1],$
$u_0\in E_{\sigma _0,m}(\mathbb {R}),$
with $m\geq 3,$
and $\sigma \in (0,\,\sigma _0),$
there exists a positive constant $M$
that depends only on $m$
and $u_0$
such that
Proof. For $u_0\in E_{\sigma _0,m}(\mathbb {R})$
, write
Using the triangle inequality, lemma 2.2 and lemma 2.1, we obtain
where $M=2c_m \vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}^{2}$
, finishing the proof.
Proposition 3.2 Let $R>0$
and $\sigma _0\in (0,\,1]$
. Given $u_0\in E_{\sigma _0,m}(\mathbb {R}),$
with $m\geq 3,$
and $0<\sigma '<\sigma <\sigma _0,$
if $u,\,v\in E_{\sigma,m}(\mathbb {R})$
are such that
then there exists a positive constant $L$
that depends only on $m,$
$u_0$
and $R$
such that
Proof. From the triangle inequality, we have
By observing that
where in the last inequality we used the fact that $\vert \vert \vert {\partial _x(u-v)}\vert \vert \vert _{E_{{\sigma },{m-2}}}\leq \vert \vert \vert {u-v}\vert \vert \vert _{E_{{\sigma },{m}}}$
and an analogous estimate for $\partial _x(u+v)$
.
Since
we conclude that for $L = 4c_m(R + \vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}})$
the bound
holds for $m\geq 3$
and $0<\sigma '<\sigma <\sigma _0$
, completing the proof.
We are now in conditions to finalize the proof of proposition 1.3 for $k=1$
. For this purpose, observe that in proposition 3.1 we have $M = 2c_m \vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}^{2},$
while in proposition 3.2 we have $L = 4c_m (R+\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}})$
for any $R>0$
. Letting $C = 4c_m,$
then we can write $L = C(R+\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}})$
and $M = \frac {C}{2}\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}^{2}.$
From propositions 3.1 and 3.2, the conditions for the Autonomous Ovsyannikov Theorem are satisfied and, therefore, for $m\geq 3$
and $T$
given by (2.7) there exists a unique solution $u$
to the Cauchy problem (1.6) which for every $\sigma \in (0,\,\sigma _0)$
is a holomorphic function in $D(0,\,T(\sigma _0-\sigma ))$
to $E_{\sigma,m}(\mathbb {R})$
and satisfies (2.8). Taking $R = \vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}$
yields
and the proof of existence and uniqueness of proposition 1.3 is finished for $k=1$
.
3.2 Proof for $k> 1$
For the case $k>1$
, we will make use of the simple algebraic inequality
Consider the function $F(u)$
given by (1.7). Similarly to proposition 3.1 and proposition 3.2 for the case $k=1$
, we will estimate $\vert \vert \vert {F(u_0)}\vert \vert \vert _{E_{{\sigma },{m}}}$
and $\vert \vert \vert {F(u)-F(v)}\vert \vert \vert _{E_{{\sigma '},{m}}}$
for $m\geq 3$
and $0<\sigma '<\sigma <\sigma _0\leq 1$
.
Proposition 3.3 Given $\sigma _0\in (0,\,1],$
$u_0\in E_{\sigma _0,m}(\mathbb {R}),$
with $m\geq 3,$
and $\sigma \in (0,\,\sigma _0),$
there exists a positive constant $M$
that depends only on $m$
and $u_0$
such that
Proof. Given $u_0\in E_{\sigma _0,m}(\mathbb {R})$
, using (1.7) we have
From lemma 2.2 and the algebra property, we can write
Thus,
By letting
we finally conclude that
for $0<\sigma <\sigma _0$
, and the result is proven.
Before proceeding with the next estimate, it is necessary to state a result that only requires the triangle inequality and successive applications of the algebra property.
Lemma 3.4 For $u,\,v\in E_{\sigma,m}(\mathbb {R})$
, with $\sigma >0$
and $m\geq 1$
, let
Then there exists a positive constant $c_m$
depending only on $m$
such that
We shall now proceed with the last crucial estimate required to make use of the Autonomous Ovsyannikov Theorem and finish the proof of proposition 1.3.
Proposition 3.5 Let $R>0$
and $\sigma _0\in (0,\,1]$
. Given $u_0\in E_{\sigma _0,m}(\mathbb {R}),$
with $m\geq 3$
and $0<\sigma '<\sigma <\sigma _0,$
if $u,\,v\in E_{\sigma,m}(\mathbb {R})$
are such that
then there exists a positive constant $L$
that depends only on $m,$
$u_0$
and $R$
such that
Proof. Given $R>0$
, $\sigma _0\in (0,\,1]$
and $u_0\in E_{\sigma _0,m}(\mathbb {R})$
, with $m\geq 3$
, let $0<\sigma '<\sigma <\sigma _0$
. In terms (1.7), we write
Since $u^{k+1}-v^{k+1} = (u-v)f_k(u,\,v)$
, from lemma 2.2 and lemma 3.4 we obtain
For the second term, write
which, together with the triangle inequality, the algebra property and proposition 3.2, yield
From the proof of proposition 3.2 we know that $\vert \vert \vert {u+v}\vert \vert \vert _{E_{{\sigma },{m}}}<2 (R+\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}).$
Moreover, we also have
which tells that
and
To deal with the third and last term on the right-hand side of (3.2), observe that
Thus, we can write
From the estimate (3.3) it is then obtained
Now under substitution of the respective terms in (3.2), we arrive at
Observe now that for $k>1$
we have $2^{k-2}<2^{k-1}$
, $2\leq 2^{k-1}$
and, from (3.1), $3+2^{k-3}<2^{k}$
. It means that the last inequality can be written as
where L = $C(R+\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}})^{k},$
with $C = 2^{k}[\tfrac {1}{k+1}+\tfrac {3k}{2}+\tfrac {k(k-1)}{2}]c_m^{k}$
, and the proof is finished.
We will now proceed with the final part of the proof of proposition 1.3. For $M = \frac {C}{2^{k}}\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}^{k+1}$
and $R=\vert \vert \vert {u_0}\vert \vert \vert _{E_{{\sigma _0},{m}}}$
, from the Autonomous Ovsyannikov Theorem, for
there exists a unique solution $u$
to the Cauchy problem of (1.6) which, for every $\sigma \in (0,\,\sigma _0)$
, is a holomorphic function in $D(0,\,T(\sigma _0-\sigma ))$
into $E_{\sigma _0,m}(\mathbb {R})$
. Therefore, the proof of proposition 1.3 is complete for any positive integer $k$
. Observe that taking $k=1$
will result in the same $T$
obtained last section, which shows that it indeed unifies both cases.
Since lemma 2.1 and a similar lemma 2.2 are still valid for Gevrey spaces $G^{\sigma,s}(\mathbb {R})$
, where $0<\sigma <\sigma '\leq \sigma _0\leq 1$
and $s>1/2$
, see [Reference Barostichi, Himonas and Petronilho4, Reference Luo and Yin24], a repetition of the same calculations for $\sigma _0=1$
provides an analogous result for these spaces, which will be useful and is therefore stated in the next result. For an alternative proof, see theorem 1 in [Reference Barostichi, Himonas and Petronilho3].
Corollary 3.6 Given $u_0(x):=u(0,\,x)\in G^{1,s}(\mathbb {R}),$
with $s\geq 5/2,$
there exists $T>0$
such that for every $\sigma \in (0,\,1)$
the Cauchy problem for (1.6) has a unique solution $u\in C^{\omega }([0,\,T(1-\sigma )); G^{\sigma,s}(\mathbb {R}))$
.
4. Global well-posedness in Sobolev spaces
In this section we will prove proposition 1.1 and the proof will be based on the local well-posedness in Sobolev spaces, a certain estimate for the $H^{3}(\mathbb {R})$
norm of the local solution and the Sobolev embedding theorem. It is worth mentioning that the proof for the case $s=3$
is already proven in [Reference da Silva and Freire9] and, therefore, presented here just for the sake of completeness. Firstly we enuntiate a result due to Yan [Reference Yan26], see also Himonas and Holliman [Reference Himonas and Holliman16].
Lemma 4.1 Suppose that $u_0\in H^{s}(\mathbb {R}),\, s>3/2$
. There exist a maximal time of existence $T>0$
and a unique solution $u\in C([0,\,T);H^{s}(\mathbb {R}))\cap C^{1}([0,\,T);H^{s-1}(\mathbb {R}))$
of (1.6)–(1.7) with $k=1$
. Moreover, the solution $u$
satisfies the following energy estimate:
for some positive constant $c_s$
. Finally, the data-to-solution map $u(0)\mapsto u(t)$
is continuous.
Proof. For the proof of existence and uniqueness of solution, see corollary 2.1 of [Reference Yan26], while the estimate (4.1) is given by (2.29) of [Reference Himonas and Holliman16].
After having the energy estimate (4.1) guaranteed, we enunciate a result stated as part of lemma 5.1 and theorem 3.1 of da Silva and Freire [Reference da Silva and Freire9].
Lemma 4.2 Given $u_0\in H^{3}(\mathbb {R})$
, let $u$
be the corresponding unique solution of (1.6)–(1.7) with $k=1$
.
(a) If there exists $\kappa >0$
such that $u_x>-\kappa$, then $\Vert u\Vert _{H^{3}}\leq e^{\kappa t/2}\Vert u_0\Vert _{H^{3}}$;(b) If $m_0$
does not change sign, then $-u_x \leq \Vert m_0\Vert _{L^{1}}$ for each $(t,\,x)\in [0.T)\times \mathbb {R}$.
As a consequence, we can extend the last lemma to and initial data in $H^{s}(\mathbb {R})$
for $s\geq 3$
as the following corollary states:
Corollary 4.3 Let $u_0\in H^{s}(\mathbb {R}),\, s\geq 3,$
be an initial data with corresponding local solution $u$
. If $m_0$
does not change sign, then
Proof. Since $s\geq 3,$
we have $H^{s}(\mathbb {R})\subset H^{3}(\mathbb {R})$
and $H^{s-1}(\mathbb {R})\subset H^{2}(\mathbb {R})$
. Therefore, $u_0\in H^{3}(\mathbb {R})$
and, from lemma 4.2(b), there exists $\kappa = \Vert m_0\Vert _{L^{1}}$
such that $-u_x < \kappa$
. The result now follows from lemma 4.2(a).
We are now ready to prove theorem 1.1.
Proof of proposition 1.1. For $u_0\in H^{s}(\mathbb {R}),\, s\geq 3,$
let $u\in C([0,\,T);H^{s}(\mathbb {R}))\cap C^{1} ([0,\,T);H^{s-1}(\mathbb {R}))$
be the unique local solution. From lemma 4.1, the solution is such that (4.1) holds for $0\leq t< T$
. From Grönwall's inequality, we have
Since $m_0$
does not change sign, from the Sobolev embedding theorem we have
for $s>1/2$
. Taking $s=2$
and using corollary 4.3, we obtain
Note that $u_0\in H^{s}(\mathbb {R})$
for $s\geq 3$
tells that $u_0\in H^ 3(\mathbb {R})$
and then, under substitution in (4.2), the condition becomes
where $K = \kappa /2$
. This means that $u$
does not blow-up at a finite time and the solution $u$
can be extended globally in time.
5. Global well-posedness and radius of spatial analyticity
In this section we prove theorem 1.2. In what follows, we will consider the initial value problem
and make use of local and global well-posedness in Sobolev spaces to extend regularity. The machinery here presented follows closely the ideas of Kato and Masuda [Reference Kato and Masuda21] and later Barostichi, Himonas and Petronilho [Reference Barostichi, Himonas and Petronilho5]. Since the proof of theorem 1.2 is extremely technical and extensive, we opt to divide the result in several propositions that together will give our desired result. The propositions that will be presented next will be proven in the next subsections. We start with a very important result regarding global well-posedness in $H^{\infty }(\mathbb {R})$
.
Proposition 5.1 If $u_0\in G^{1,s}(\mathbb {R}),$
$s>3/2,$
and $m_0$
does not change sign, then (5.1) has a unique solution $u\in C([0,\,\infty );H^{\infty }(\mathbb {R}))$
.
Once we have the global solution established, we will extend regularity to the Kato–Masuda space. We are able to find $r_1>0$
such that for each fixed arbitrary time $T>0$
the solution will belong to $A(r_1(t))$
as a space function for $t\in [0,\,T]$
. From the definition of the spaces $A(r)$
, this $r_1$
will be the radius of spatial analyticity of the solution.
Proposition 5.2 Given $u_0\in G^{1,s}(\mathbb {R})$
, with $s>5/2$
, suppose that $m_0$
does not change sign and let $u\in C([0,\,\infty );H^{\infty }(\mathbb {R}))$
be the unique solution to the initial value problem of (5.1). Then there exists $r_1>0$
such that $u\in C([0,\,\infty );A(r_1))$
. Moreover, for every $T>0$
an explicit lower bound for the radius of spatial analyticity is given by
where $L_1=\frac {52\sqrt {2}}{7}\Vert {u_0}\Vert _{{\sigma _0},{2}}$
for $\sigma _0<0$
fixed, $L_2 = 112\mu,\, L_3 = r(0)e^{L_1}$
and $\mu = 1+\max \{\Vert {u}\Vert _{H^{2}};t\in [0,\,T]\}$
.
Proposition 5.2 says that the global solution is analytic in $x$
and gives a lower bound for the radius of spatial analyticity. The next step is to extend regularity to $t$
. Our first step towards this goal is to prove that the solution $u$
is locally analytic in time, as enunciated by the next result.
Proposition 5.3 Given $u_0\in G^{1,s}(\mathbb {R}),$
with $s>5/2,$
let $u\in C([0,\,\infty ); A(r_1))$
be the unique solution of (5.1). Then there exist $T>0$
and $\delta (T)>0$
such that the unique solution $u$
belongs to $C^{\omega }([0,\,T];A(\delta (T)))$
.
Once local analyticity is established, we show that the analytic lifespan is infinite.
Proposition 5.4 For the unique solution $u \in C^{\omega }([0,\,T];A(\delta (T)))$
, we have
Finally, to conclude our result, we use a result proved by Barostichi, Himonas and Petronilho in [Reference Barostichi, Himonas and Petronilho5] (see page 752).
Lemma 5.5 If $u\in C^{\omega }([0,\,T]; A(r(T)))$
for all $T>0$
and some $r(T)>0,$
then $u \in C^{\omega }([0,\,\infty )\times \mathbb {R})$
.
Proof of theorem 1.2. The proof is now reduced to a recollection of the previous propositions. Given $u_0\in G^{1,s}(\mathbb {R})$
, if $m_0$
does not change sign, from proposition 5.1 we have a unique solution $u\in C([0,\,\infty ),\, H^{\infty }(\mathbb {R}))$
. From proposition 5.2, we guarantee the existence of $r_1>0$
such that $u \in C([0,\,\infty ),\,A(r_1))$
, which by proposition 5.3 belongs to $C^{\omega }([0,\,T],\,A(\delta (T)))$
for certain $T>0$
and $\delta (T)>0$
. Proposition 5.4 then guarantees that $u\in C^{\omega }([0,\,T],\,A(\delta (T)))$
for every $T>0$
and then lemma 5.5 concludes that the solution $u$
is global analytic for both variables.
Moreover, we observe from proposition 5.2 that given $T>0$
, we have $u(t)\in A(r_1)$
for $t\in [0,\,T]$
and $r_1(t) \geq L_3e^{-L_1e^{L_2t}}$
. By means of the forthcoming expression (5.3) obtained in the proof of proposition 5.1 we can determine the radius of spatial analyticity as
where
and $\mu,\, \sigma _0$
are given as in proposition 5.2.
5.1 Proof of proposition 5.1
For the proof of proposition 5.1, the only ingredients required are proposition 1.1, the embeddings $G^{1,s}(\mathbb {R})\subset H^{\infty }(\mathbb {R})$
and $H^{s}(\mathbb {R})\subset H^{s'}(\mathbb {R})$
for $s>s'$
, as shown next.
Since $u_0\in G^{1,s}(\mathbb {R})$
, from the embedding $G^{1,s}(\mathbb {R})\subset H^{\infty }(\mathbb {R})$
the initial data belongs, in particular, to $H^{s}(\mathbb {R})$
for any $s\geq 3$
. From theorem 1.1, there exists a unique global solution $u$
in $C([0,\,\infty );H^{s}(\mathbb {R}))\cap C^{1}([0,\,\infty );H^{s-1}(\mathbb {R}))$
for $s\geq 3$
, which means that $u(t,\,\cdot ) \in \bigcap \limits _{s\geq 3}H^{s}(\mathbb {R})$
. Now, the embedding $H^{3}(\mathbb {R})\subset H^{s'}(\mathbb {R})$
for $s'\in [0,\,3]$
shows that $u(t,\,\cdot )\in H^{s'}(\mathbb {R})$
and $u(t,\,\cdot )\in H^{\infty }(\mathbb {R})$
, concluding the proof of proposition 5.1
5.2 Proof of proposition 5.2
This is by far the most technical and complicated result. The proof consists of bounding a certain inner product and using properties of dense spaces to find such $r_1$
. For $m\geq 0$
, it will be more convenient to consider an auxiliary norm
in $A(r)$
and recover the usual norm (2.1) as we make $m\to \infty$
. Moreover, we observe that $\Vert {u}\Vert _{{\sigma },2,m}\leq \Vert {u}\Vert _{{\sigma },{2}}$
.
For our initial value problem (5.1), we note that, given $m\geq 0$
, the function $F:H^{m+5}(\mathbb {R})\rightarrow H^{m+2}(\mathbb {R})$
is well-defined and continuous. Therefore, for $Z=H^{m+5}(\mathbb {R})$
and $X= H^{m+2}(\mathbb {R})$
the following result, which will be called Kato–Masuda Theorem, is valid, see theorem 1 in [Reference Kato and Masuda21] or theorem 4.1 in [Reference Barostichi, Himonas and Petronilho5] for more general formulations.
Lemma 5.6 Kato–Masuda
Let $\{\Phi _{\sigma }:-\infty <\sigma <\bar {\sigma }\}$
be a family of real functions defined on an open set $O\subset Z$
for some $\bar {\sigma }\in \mathbb {R}$
. Suppose that $F: O \rightarrow X$
is continuous, where $F$
is the function given by (5.1) and
(a) $D\Phi _{\cdot }(\cdot ):\mathbb {R}\times Z\rightarrow \mathcal {L}(\mathbb {R}\times X; \mathbb {R})$
given by
\[ D\Phi_{\sigma}(v)F(v):=\langle F(v),\quad D \Phi_{\sigma}(v)\rangle \]is continuous, where $D$
denotes the Fréchet derivative;(b) there exists $\bar {r}>0$
such that
\[ D\Phi_{\sigma}(v)F(v) \leq \beta(\Phi_{\sigma}(v)) + \alpha(\Phi_{\sigma}(v))\partial_{\sigma}\Phi_{\sigma}(v), \]for all $v\in O$
and some nonnegative continuous real functions $\alpha (r)$ and $\beta (r)$ well-defined for $-\infty < r<\bar {r}$.
For $T>0,$
let $u\in C([0,\,T];O)\cap C^{1}([0,\,T];X)$
be a solution of the initial value problem (5.1) such that there exists $b<\bar {\sigma }$
with $\Phi _{b}(u_0)<\bar {r}$
. Finally, let $\rho (v)$
be the unique solution of
Then for
where $T_1>0$
is the lifespan of $\rho,$
we have
We observe the complexity of the Kato–Masuda Theorem and the amount of hypothesis required for the final result. It is important to mention as well that the procedure to prove our desired proposition 5.2 goes through Kato–Masuda Theorem and (5.2), see also proposition 4.1 of [Reference Barostichi, Himonas and Petronilho5].
However, one of the main issues is to establish the bound of item $(b)$
. Before doing so, we shall define convenient parameters and functions that will be used from now on.
For $u\in H^{m+5}(\mathbb {R})$
and $m\geq 0$
, let
Given $u_0\in G^{1,s}(\mathbb {R}),\, s>5/2,$
such that $m_0$
does not change sign, let $u\in C([0,\,\infty ),\,H^{\infty }(\mathbb {R}))$
be the unique global solution of (5.1). From the embedding $G^{1,s}(\mathbb {R})\subset A(1),$
we have that $u_0\in A(1)$
. Let $\sigma _0<0=: \bar {\sigma }$
, which means that $e^{\sigma _0}<1$
and, from the definition of $A(1)$
, we have $\Vert u\Vert _{\sigma _0,2}<\infty$
.
For the global solution $u$
, fix $T>0$
and define $\mu = 1+\max \{\Vert {u}\Vert _{H^{2}}; t\in [0,\,T]\}$
and $O= \{v\in H^{m+5}(\mathbb {R}); \Vert v\Vert _{H^{2}}<\mu \}$
. Observe that the family $\{\Psi _{\sigma,m}; -\infty < \sigma < \bar {\sigma }, m\geq 0\}$
is well-defined on $O$
and $F:O\rightarrow X$
is continuous. Moreover, for this same family item $(a)$
is satisfied, see Kato and Masuda [Reference Kato and Masuda21], page 460. For item $(b)$
, we will need the following result.
Proposition 5.7 Given $u\in H^{m+5}(\mathbb {R}),\, m\geq 0,$
for $\sigma \in \mathbb {R}$
we have the bound
where $\bar {K}(p) = 224p$
and $\bar {\alpha }(p,\,q) = 832(1+p)q^{1/2}$
.
Proof. Since $F(u) \!=\! \displaystyle {-\frac {1}{2}\partial _x[u^{2} \!+\! 3(1\!-\!\partial _x^{2})^{-1}u_x^{2}]}$
and $\displaystyle {\frac {1}{2}D\Vert {\partial _x^{j}u}\Vert _{H^{2}}^{2}w = \langle \partial _x^{j}w\,,\, \partial _x^{j} u\rangle _{H^{2}}},$
see [Reference Barostichi, Himonas and Petronilho4, Reference Kato and Masuda21], by making $w = F(u)$
and summing over $j$
according to $\Phi _{\sigma,m}$
we have
From the proof of lemma 4.1 in [Reference Barostichi, Himonas and Petronilho5] (equations (6.14) and (6.16) with $k=1$
), we know that
where $\bar {K_1}(p) = 32p$
and $\alpha _1(p,\,q) = 64(1+p)q^{1/2}$
, and
where $\bar {K_2}(p) = 64p$
and $\alpha _1(p,\,q) = 256(1+p)q^{1/2}$
. Under substitution of the respective terms in the inequality for $D\Phi _{\sigma,m}(u)F(u)$
we obtain
where $\bar {K}(p) = 224p$
and $\bar {\alpha }(p,\,q) = 832(1+p)q^{1/2}$
.
Proof of proposition 5.2. To prove the proposition, we basically need to complete the details for item $(b)$
of the Kato–Masuda Theorem. Therefore, we need to find $\bar {r}>0$
and continuous functions $\alpha (r)$
and $\beta (r)$
for $-\infty < r<\bar {r}$
.
For $\bar {K}$
and $\bar {\alpha }$
given in proposition 5.7, let
Observe that
(i) $\alpha (r)$
and $\beta (r)$ are continuous for $r<\bar {r}$;(ii) $\rho _m(t)\leq \rho (t),$
for $t\in [0,\,T]$ and $\rho _m(t)\to \rho (t)$ uniformly.(iii) $\bar {K}(p)= 224p$
and $\bar {\alpha }(\bar {p},\,q) = 832(1+\bar {p})q^{1/2}$, for fixed $\bar {p}$, are nondecreasing.
From the definition of $\mu$
, $\mu >\Vert u\Vert _{H^{2}}$
and then from observation (iii) above, for all $v\in O$
, we have
The inequality of proposition 5.7 then yields
and item $(b)$
is finally satisfied.
For the same $T>0$
, let $b:=\sigma _0<\bar {\sigma }$
and observe that $\rho _m(t)$
is a solution for the Cauchy problem
Since
Kato–Masuda Theorem says that for
we have
for $t\in [0,\,T]$
. By letting $m\to \infty$
, we obtain
and $u(t)\in A(r_1)$
for $r_1 = e^{\sigma (t)}\leq e^{\sigma (T)}$
for $t\in [0,\,T]$
. Once we have the expressions for $\alpha$
and $\rho$
, we can estimate the radius of spatial analyticity $\sigma (t)$
. In fact, we have
where $A=\frac {26\sqrt {2}}{7\mu }(1+\mu )\Vert {u_0}\Vert _{{\sigma _0},{2}}$
and $B=112\mu$
. Observe that since $\mu \geq 1,$
we have $A\leq \frac {52\sqrt {2}}{7}\Vert {u_0}\Vert _{{\sigma _0},{2}}=:L_1$
. By letting $L_2:=B$
, we have
where $L_3=r(0)e^{L_1}$
.
5.3 Proof of proposition 5.3
Given $u_0 \in G^{1,s}(\mathbb {R}),$
with $s>5/2$
, let $u\in C([0,\,\infty );A(r_1))$
be the unique solution whose existence is guaranteed by proposition 5.2. For the same initial data, from corollary 3.6 there are $\tilde {T}>0$
and a unique solution $\tilde {u}\in C^{\omega }([0,\,\tilde {T} (1-\delta );G^{\delta,s}(\mathbb {R}))$
for $\delta \in (0,\,1)$
. Let $T = \frac {\tilde {T}}{2}(1-\delta )$
, that is, $\delta = \displaystyle {1-2\frac {T}{\tilde {T}} =: \delta (T)},$
and $\tilde {u}\in C^{\omega }([0,\,T];G^{\delta,s}(\mathbb {R}))\subset C^{\omega }([0,\,T];A(\delta (T)))$
once $G^{\delta,s}(\mathbb {R}) \subset A(\delta )$
.
Since $A(r)\hookrightarrow H^{\infty }(\mathbb {R})$
for $r>0$
, then $\tilde {u}\in C^{\omega }([0,\,T];H^{\infty }(\mathbb {R}))\subset C([0,\,T];H^{\infty }(\mathbb {R}))$
. From the uniqueness of the solution, we know that $u=\tilde {u}$
for $t\in [0,\,T]$
, which means that $u\in C^{\omega }([0,\,T];A(\delta (T))),$
for $T = \frac {\tilde {T}}{2}(1-\delta )$
and $\delta (T)>0$
.
5.4 Proof of proposition 5.4
We will make use of the following result from [Reference Barostichi, Himonas and Petronilho5] (lemma 5.1).
Lemma 5.8 Let $\delta >0$
and $m\geq 1$
. Then $E_{\delta,m}(\mathbb {R})$
is continuously embedded in $A(\delta )$
. Conversely, if $f\in A(r)$
for some $r>0$
then $f\in E_{\delta,m}(\mathbb {R})$
for all $\delta < r/e$
.
Proof of proposition 5.4. We will prove the result by contradiction as we assume that $T^{\ast }<\infty$
. For the solution $u \in C^{\omega }([0,\,T];A(\delta (T)))\subset C([0,\,T];H^{\infty }(\mathbb {R}))$
, from the uniqueness of the global solution and the definition of $T^{\ast }$
it is immediate that if $T^{\ast }<\infty$
, then $u(T^{\ast })$
is well-defined. Moreover, from proposition (5.2), we have $u(T^{\ast })\in A(r_1)$
.
Let $\delta _0<\min \{1,\, r_1/e\}$
and then the converse of lemma 5.8 tells that
for all $\delta _0>0.$
From proposition 1.3, there exist $\epsilon >0$
and a unique solution $\tilde {u}\in C^{\omega }([0,\,\epsilon ];E_{\delta,m}(\mathbb {R}))$
for $0<\delta <\delta _0$
such that $\tilde {u}(0) = u(T^{\ast })$
. On the other hand, since $E_{\delta,m}(\mathbb {R})\hookrightarrow A(\delta )\subset H^{\infty }(\mathbb {R}),$
we have
and the uniqueness of the global solution tells that $\tilde {u}(0) = u(T^{\ast })$
, which means that
Let $s=T^{\ast }+t$
. Then $u(s) = \tilde {u}(s-T^{\ast }),$
for $s\in [T^{\ast },\,T^{\ast }+\epsilon ],$
that is,
Based on the definition of $T^{\ast }$
, let $T>0$
be such that $T^{\ast }-\epsilon < T< T^{\ast }$
and the solution $u$
then belongs to $C^{\omega }([0,\,T];A(\delta (T)))$
for some $\delta (T)>0$
.
Observe now that if $\sigma '\geq \sigma,$
then $A(\sigma ')\subset A(\sigma ).$
By defining $\tilde {\delta } = \min \{\delta,\,\delta (T)\}$
, which means in particular that $\delta \geq \tilde {\delta }$
and $\delta (T)\geq \tilde {\delta }$
, then
which says that $T^{\ast }$
cannot be the supremum. As a result of the contradiction, $T^{\ast }$
must be infinite and, for every $T>0$
, there exists $r(T)>0$
such that $u\in C^{\omega }([0,\,T];A(r(T)))$
.
Acknowledgments
This work was supported by FAPESP (grant number 2019/23688-4) and the Royal Society under a Newton International Fellowship (reference number 201625). The author would also like to thank the reviewers for the suggestions that led to the improvement of this paper.
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