Well-posedness for nonlinear SPDEs with strongly continuous perturbation

Abstract We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.


Statement of the problem
Let (Ω, F, P ) be a complete, countably generated probability space (e.g. the classical Wiener space), D ⊂ R d be a bounded Lipschitz domain, T > 0, Q T := (0, T ) × D. We are interested in a result of existence and uniqueness of the solution to du − div(a(x, u, ∇u) + F (u)) dt = Φ dW (t) in Ω × Q T , u = 0 on Ω × (0, T ) × ∂, u(0, ·) = u 0 ∈ L 2 (D). (1.1) The nonlinear diffusion-convection operator of Leray-Lions' type is defined as the sum of a Carathéodory function a : D × R d+1 → R d satisfying appropriate growth and coercivity assumptions which will be given below and F : R → R d Lipschitz continuous with Lipschitz constant L > 0, such that F (0) = 0. On the right-hand side, c The Author(s), 2020. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Φ ∈ L 2 (Ω; C ([0, T ]; HS(L 2 (D))) is progressively measurable, where HS(L 2 (D)) is the space of Hilbert-Schmidt operators from L 2 (D) to L 2 (D). W is a cylindrical Wiener process in L 2 (D) with respect to a filtration (F t ) t∈ [0,T ] , denoted (F t ) in the sequel, satisfying the usual assumptions. More precisely, W is defined in the following sense (see, e.g. [3]): for a fixed orthonormal basis (e n ) n∈N * of L 2 (D) and a sequence of independent, real-valued (F t )-Brownian motions (β n ) n∈N * , we define for any t ∈ [0, T ] and u ∈ L 2 (D), W (t), u := ∞ n=1 (e n , u) L 2 (D) β n (t). One can check (see, e.g. [3, p. 85]) that for any u ∈ L 2 (D), W, u is a real-valued, (F t )-Wiener process such that E [ W (t), u W (s), v ] = min(t, s)(u, v) L 2 (D) for any u, v ∈ L 2 (D), for all s, t ∈ [0, T ]. W can be represented in the following way: for any sequence (a n ) ∈ l 1 (R + ) it is easy to check that W (t) := ∞ n=1 √ a n (e n / √ a n )β n (t) is a Q-Wiener process with positive definite, symmetric and nuclear covariance operator Q = diag(a n ) in the (bigger) Hilbert space U := Q −1/2 (L 2 (D)) (1.2) which is obtained as the completion of L 2 (D) with respect to the norm · U induced by the scalar product (u, v) U := (Q 1/2 u, Q 1/2 v) L 2 (D) . However, the stochastic integral t 0 Φ dW (s), t ∈ [0, T ], can be defined independently of the representation of W and the choice of Q by Φ dW (t) := ∞ n=1 Φ(e n ) dβ n (t).

Motivation and former results
The technical novelties of this contribution arise from the stochastic forcing of the nonlinear diffusion-convection operator u → − div(a(x, u, ∇u) + F (u)). The diffusion part u → − div a (x, u, ∇u) is a monotone operator with p-growth and coercivity conditions for 2d/(d + 1) < p < ∞ where d ∈ N * is the space dimension (see § 1.3 for more details). In particular, the assumptions include a class of p-Laplacian operators with 1 < p < 2, p appropriately bounded away from 1. In contrast, the convection part u → − div F (u) is, in general, not monotone but strongly continuous. It is well-known that a semigroup representation of the solution in the sense of [3] is not available in this nonlinear case. Moreover, classical well-posedness theory for monotone SPDEs (see, e.g. [11]) and for locally monotone SPDEs (see [10]) does not apply for a general perturbation of the type − div F (u) for Lipschitz continuous F : R → R d . Therefore, our aim is to show well-posedness results by using a semi-implicit Euler-Maruyama time discretization (i.e. implicit in the operator part and explicit in the noise one). From the a priori estimates, we get weak convergences of approximate solutions with respect to (ω, t, x), but compactness arguments can be applied uniquely with respect to the variables (t, x). It is therefore not possible to identify the limit of the convection part of the operator. However, it is possible to show the existence of a martingale solution by adapting argumentations based on convergence in law and Skorokhod's representation theorem. Then, since a pathwise uniqueness result can be obtained by using a L 1contraction principle, the existence of a strong solution follows by the argument of convergence in probability of Gyöngy and Krylov [8]. Those techniques are wellknown for evolution problems with a multiplicative stochastic perturbation and have been applied in, e.g. [4,6] and by many other authors in the last few decades.
In our case, this method allows to pass to the limit in the nonlinear convection operator. Taking advantage of the additive character of the stochastic perturbation, we can recover the stochastic integral by using a result detailed by Debussche et al. in [4]. Since the stochastic integral takes values in L 2 (D), it does not inherit spatial Sobolev regularity of the solution. Therefore, the compactness arguments are more subtle than in the case of a multiplicative stochastic perturbation as considered in [17] for the operator u → − div(|∇u| p−2 ∇u + F (u)) with p > 2.

Assumptions on the nonlinear operator
The vector field a : is a Carathéodory function in the sense that the mapping (λ, ξ) → a(x, λ, ξ) is continuous for almost every x ∈ D and x → a(·, λ, ξ) is measurable for every (λ, ξ) ∈ R × R d . In the following, let 2d/(d + 1) < p < ∞ and p = p/(p − 1). We impose the following conditions: (A1) a is monotone with respect to its last variable only, for all λ ∈ R, ξ ∈ R d and almost every x ∈ D.
(A3) There exists a constant C 4 a 0 and a nonnegative function h ∈ L p (D) such that From assumptions (A1)-(A2) it follows that the nonlinear operator is well-defined, hemicontinuous and pseudomonotone (see [12, lemma 2.32]). From assumption (1.3) of (A2), it follows that A is coercive.

Strong solutions
Let us recall the notion of strong solution to (1.1):
The proof of theorem 1.1 is contained in the following section. It is based on an approximation procedure by a time discretization of (1.1) introduced in § 2.1. Since there is a lack of compactness with respect to ω ∈ Ω, we use the theorems of Prokhorov and Skorokhod to obtain a.s. the convergence of the sequence of approximate solutions [2].
It is left to show that A −1 τ : W −1,p (D) → W 1,p 0 (D) is demi-continuous. For f ∈ W −1,p (D) and u such that A τ (u) = f , using Gauss-Green's theorem on the convection term, we get f, u W −1,p (D),W 1,p 0 (D) = u 2 2 + τ D a(x, u, ∇u) · ∇u dx. By (A2), D a(x, u, ∇u) · ∇u dx − κ 1 + C 1 a ∇u p p and therefore, using Young's inequality, for any δ > 0 we get . For all n ∈ N * , we define From (2.6) it follows that there exists a not relabelled subsequence of (u n ), u ∈ W 1,p 0 (D) and B in L p (D) d such that u n u in W 1,p 0 (D), u n → u in L 2 (D) thanks to p * > 2 and a(x, u n , ∇u n ) B in L p (D) d for n → ∞. Using these convergence results and (2.7), we get and since A τ is pseudomonotone, (2.9) , a priori for a subsequence. Since u is unique, it follows that the whole sequence (u n ) converges to u weakly in W 1,p 0 (D) for n → ∞ and A −1 τ is demi-continuous.

Estimates
Lemma 2.2. For u 0 ∈ L 2 (D), and k = 0, . . . , N − 1, let u k+1 be the solution to (2.1). Then, Proof. Taking u k+1 as a test function in (2.1), we get where , and, using (A2), From Gauss-Green's theorem it follows that and we still have to estimate Combining (2.11) with the above estimates and taking expectation we arrive at Using Hölder's and Young's inequalities it follows that for any α > 0 and therefore (2.10) holds.
Definition 2.1. For N ∈ N * , τ > 0, we introduce the right-continuous step function and the piecewise affine functions There exist a generic constants K i 0, i = 1, . . . , 6 not depending on the discretization parameters such that Proof. We fix n ∈ {1, . . . , N}, take the sum over 0, . . . , n − 1 in (2.10) to get From (2.17) and lemma A.1 (see appendix) it follows that and we get (2.13), (2.14) and the first inequality of (2.16). Now, from (2.14) it follows that E we have shown both inequalities of (2.15). Thanks to lemma A.1 it follows that and this yields the second inequality of (2.16).
where L > 0 is the Lipschitz constant of F . Thanks to Young's and Poincaré's inequalities, a positive constant C 1 exists such that with continuous injection. Therefore, using Hölder's, Sobolev's and Young's inequalities, we get for a constant C 2 0. Using (2.24), (2.25), (2.26) and (2.27) and taking expectation it follows that for a constant C 3 0. Now, the assertion follows from (2.16) of lemma 2.3.
Lemma 2.6. There exists a constant K 0 such that where C 0 is a constant that may change from line to line. Now, from [7], [14, example 2.4.1] for any q 1 and α > 1/q, it follows that where U is defined in (1.2) and X := Taking expectation in (2.33) and using (2.34) and (2.35), (2.30) holds true.
As a consequence, we can estimate the difference between the piecewise affine and the time-continuous approximations of the stochastic integral: and the assertion follows from lemma 2.7.

Tightness
In the multiplicative noise case, the stochastic integral inherits the spatial Sobolev regularity from the solution. In the additive noise case, this is not possible and the compactness argument for processes with values in L 2 (D) is more subtle.
Proof. From Burkholder-Davies-Gundy's inequality it follows that )), the right-hand side of (2.37) converges to 0 when N → ∞. Now, the convergence result for ( M N ) is a direct consequence of lemma 2.8. Lemma 2.10. For N ∈ N * , let μ N be the law of Then, (μ N ) N ∈N * is a uniformly tight sequence.
Proposition 2.11. The sequence (ν N ) of laws induced by where U is defined in (1.2), is tight and therefore, passing to a not relabelled subsequence if necessary, there exist probability measures ν 1 on L 2 (Q T ), ν 2 on C([0, T ]; W −1,p (D)) and a probability measure Proof. Note that for all N ∈ N * , ν N = (L(u r N − u N ), μ N ) and the tightness of (μ N ) follows from lemma 2.10. The tightness of (L(u r N − u N )) is a direct consequence of lemma 2.6 and the converse part of Prokhorov's theorem.

2.3.2.
Compactness Now, we are in position to apply the theorem of Skorokhod: by Appendix A.3, there exists a new probability space (Ω , F , P ) 1 such that, passing to a subsequence if necessary, (Y N ) converges almost surely in X . Without changing the notation of random variables with the same law in order not to overload the presentation, the semi-implicit Euler scheme (2.1) is satisfied on Ω . 2 Moreover, remark A.1 and in particular the a priori estimates developed in lemmas 2.2 to 2.6 hold true on (Ω , F , P ). Thus, on (Ω , F , P ), • there exists a L 2 (Q T )-valued random variable u ∞ such that L(u ∞ ) = L(ν 1 ) and lim N →∞ u N = u ∞ in L 2 (Q T ) a.s. in Ω . Then, lemma 2.4 and Vitali's theorem yield the convergence of ( u N ) to u ∞ in L (Ω ; L 2 (Q T )) for all 1 < 2.
• (u r N − u N ) converges a.s. to 0 in L 2 (Q T ) and one proves similarly, or by using (2.29), that lim N →∞ u r N = u ∞ in L 2 (Q T ) a.s. in Ω and in L (Ω ; L 2 (Q T )) for all 1 < 2.

First identifications at the limits Note that
With similar arguments, one can show that W N t = tQ for all t ∈ [0, T ] and all N ∈ N * , where Q is the covariance operator of W . From Levy's theorem (see [3, proposition 3.11, p. 75]) it follows that W N is a cylindrical Wiener process with values in L 2 (D) with respect to (F N t ). In particular (Ω , F , (F N t ), P , W N ) is a stochastic basis in the sense of [4, lemma 2.1, p. 1126]. We recall that we recovered the time discretization scheme in Ω and therefore the following approximate equations hold true: for all N ∈ N * , we have Now we are ready to pass to the limit in the approximate equations: Remark 2.1. Using lemma 2.4, one gets that ( u N ) is bounded in L 2 (Ω ; L ∞ (0, T ; L 2 (D))), thus in L 2 w (Ω ; L ∞ (0, T ; L 2 (D))) where w stands for the weak- * measurability. Since [5, theorem 8.20.3, p. 606] L 2 w (Ω ; L ∞ (0, T ; L 2 (D))) L 2 (Ω ; L 1 (0, T ; L 2 (D))) * , the theorem of Banach-Alaoglu yields u ∞ ∈ L 2 w (Ω ; L ∞ (0, T ; L 2 (D))). On the other hand, since u N converges to u ∞ in L (Ω ; C([0, T ]; W −1,p (D))), it follows that u ∞ is a random variable with values in the space of weakly continuous functions C w ([0, T ]; L 2 (D)) [15, lemma 1.4, p. 263]. In particular, u ∞ (t) ∈ L 2 (D) a.s. in Ω for all t ∈ [0, T ]. Therefore, (2.39) holds in L 2 (D), a.s. in Ω , for all t ∈ [0, T ].

Martingale identification argument
We denote the augmentation of the filtration σ(W ∞ (s), Φ ∞ (s), u ∞ (s)) 0 s t , t ∈ [0, T ] by (F ∞ t ). In the following two lemmas, we will show that W ∞ is a cylindrical Wiener process with values in L 2 (D) with respect to (F ∞ t ). To this end, we first show that W ∞ is a (F ∞ t )-martingale. Since u ∞ is in C([0, T ]; W −1,p (D)) a.s. in Ω , it is a stochastic process and therefore (F ∞ t ) is well defined.
since Φ N and u r N are (F t )-adapted processes for all N ∈ N * and W is a (F t )-martingale. Lemma 2.15. W ∞ is a cylindrical Wiener process with values in L 2 (D) with respect to (F ∞ t ).
Proof. Since we already know that W ∞ is a (F ∞ t )-martingale with W ∞ (0) = 0, according to [3, theorem 4.4, p. 89], it is left to show that (2.46) where Q is the covariance operator of W . Recall that W N t = tQ for all t ∈ [0, T ] and all N ∈ N * . Let U be the Hilbert space defined in (1.2) and (g l ) be an orthonormal basis of U . For all t ∈ [0, T ], 0 s t, ψ ∈ C b (C([0, s]; U ) × L 2 (0, s; HS(L 2 (D))) × L 2 ((0, s) × D) and n, m ∈ N * , combining the convergence results from the previous section with the dominated convergence theorem of Lebesgue, it follows that where (W, g n , g m )(r) := (W (r), g n ) U (W (r), g m ) U for W (r) ∈ U , r ∈ [0, T ], thus (2.46) holds true. In particular, (Ω , F , (F ∞ t ), P , W ∞ ) is a stochastic basis in the sense of [4, lemma 2.1, p. 1126].
From § A.3 of the appendix it follows that Φ N is a (F N t )-predictable process with values in HS(L 2 (D)) and that M N (t) = t 0 Φ N dW N (s). In order to pass to the limit, we recall the following lemma: Lemma 2.16. [4, lemma 2.1, p. 1126] Let (Ω, F, P ) be a fixed probability space, W a cylindrical Wiener process on (Ω, F, P ) with values in L 2 (D) with respect to a given filtration (F t ) and X a separable Hilbert space. Consider a sequence of stochastic bases S n = (Ω, F, (F n t ), P, W n ), that is a sequence so that each W n is a cylindrical Wiener process with values in L 2 (D) with respect to (F n t ). Assume that (G n ) is a collection of HS(L 2 (D); X)-valued, (F n t )-predictable processes such that G n ∈ L 2 (0, T ; HS(L 2 (D); X)) a.s. in Ω and G ∈ L 2 (0, T ; HS(L 2 (D); X)) is (F t )predictable. If W n → W in probability in C([0, T ]; U ), where U is given in (1.2) and G n → G in probability in L 2 (0, T ; HS(L 2 (D); X)), then in probability in L 2 (0, T ; X). (2.47) in W −1,p (D), for all t ∈ [0, T ], a.s. in Ω . By lemma 2.4 and Itô's formula (see, e.g. [10, theorem 4.2.5]), it follows that u ∞ is a square-integrable, (F ∞ t )-adapted, continuous process with values in L 2 (D), and (2.47) holds in L 2 (D) a.s. in Ω .
for any t ∈ [0, T ] a.s. in Ω. For L 0 being the Lipschitz constant of F we have and it follows, by arguments similar to the ones on p. 6, that lim δ→0 + I 3 = {u1=u2} |∇(u 1 − u 2 )| dx ds = 0 a.s. in Ω and the result holds true.
In particular, proposition 3.1 implies that whenever a solution (in the sense of definition 1.1) to (1.1) exists, it is unique. The following lemma (see, e.g. [8, lemma 1.1, pp. 144-145]) contains a suitable necessary and sufficient condition for strong convergence: Lemma 3.2. Let V be a Polish space equipped with the Borel σ-algebra. A sequence of V -valued random variables (X n ) converges in probability if and only if for every pair of subsequences (X l ) and (X m ) there exists a joint subsequence (X l k , X m k ) which converges for k → ∞ in law to a probability measure μ such that μ({(w, z) ∈ V × V | w = z}) = 1. Proof. We apply lemma 3.2 to prove the assertion. Let ( u K ), and ( u L ) be a pair of subsequences of ( u N ). Then, repeating the arguments of § 2.3, it follows that the random vector (Y K,L ) defined by is tight on the appropriate product space, and therefore relatively compact; thus we can extract a joint subsequence (Y Kj ,Lj ) which converges in law. Thus, in particular, there exist probability measures μ 1 , μ 2 such that (μ 1 , μ 2 ) = lim j→∞ L( u Kj , u Lj ). Now we continue as in § § 2.3, 2.4 and 2.5: passing to a new probability space (Ω , F , P ) and not changing notation for random variables with the same law, we find random variables u 1 ∞ , u 2 ∞ such that L(u 1 ∞ ) = μ 1 , L(u 2 ∞ ) = μ 2 and, since we have the strong convergences of M N , M N and Φ N , it implies that M Kj , M Kj , M Lj , M Lj and Φ Kj , Φ Lj are converging to the same limits and and therefore (u τ ) converges to v = u 0 in the sense of distributions and weakly in L 2 (D). As a consequence, the whole sequence (u τ ) converges to u 0 weakly in L 2 (D) and strongly in L 2 (D) thanks to the uniform convexity property and the above inequality. Therefore, u τ 0 := u τ for all τ > 0.

A.3. From Ω to Ω
In this section, we are interested in proving that the time discretization scheme is preserved by changing the probability space. Following the notations of § § 2.3.1 and 2.3.2, by Skorokhod representation, we consider a vector of random elements with the same law ν N as Y N on X . Let us notice that thanks to Appendix A.2.1, Y N and Y N have both the same law on [L 2 (Q T ) ∩ L p (0, T ; W 1,p 0 (D))] 2 × W 1,q,p (0, T ; W 1,q 0 (D), W −1,p (D)) × [C([0, T ]; L 2 (D))] 2 × L 2 (0, T ; HS(L 2 (D))) × C ([0, T ]; U ) denoted X where q = min (2, p). Note that the semi-implicit Euler-Maruyama scheme (2.1) means, for any k, t k div(a(x, u r N , ∇u r N ) + F (u r N )) dt = 0, and since for any ϕ ∈ C ∞ c (D), the following mapping Ψ defined by is a positive Borel function on X , one gets that t k div(a(x, u r N , ∇u r N ) + F (u r N )) dt = 0.
For a fixed N ∈ N * , u r N is a W 1,p 0 (D)-valued, right continuous step function on the fixed time discretization (t k ) N k=0 : it can be identified with an element of W 1,p 0 (D) N by the Borel measurable mapping N k=1 u k 1 [t k−1 ,t k ) → (u 1 , . . . , u N ). By Appendix A.2.1, one concludes that u r N follows the same structure: N k=1 u k 1 [t k−1 ,t k ) and t k+1 t k div(a(x, u r N , ∇u r N ) + F (u r N )) dt = τ div(a(x, u k+1 , ∇u k+1 ) + F (u k+1 )).
By a similar reasoning, Φ N , u N and M N are respectively right continuous step function and piecewise affine continuous functions for the two last ones. Taking into account those information, one gets that the discretization is conserved by changing the probability space, i.e. u k+1 − u k − τ div(a(x, u k+1 , ∇u k+1 ) + F (u k+1 )) = Φ k Δ k+1 W N .
Then, by Appendix A.2.2, since Φ k is F t k measurable where (F t ) is the filtration generated by W , then, Φ k will be F t k measurable where (F t ) is the filtration generated by W N . Therefore, the time discretization (2.1) is totally recovered and by the uniqueness of lemma 2.1, u k is F t k measurable for any k.