Improved regularity for the stochastic fast diffusion equation

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space $W^{1,m+1}_0$ for initial data in $L^{2}$.


Introduction
In this work, we establish higher order regularity of the strong solutions to the stochastic fast diffusion equation perturbed by linear multiplicative Wiener noise.The equations are set on a bounded domain O ⊂ R d with sufficiently smooth boundary, and formulated with zero Dirichlet boundary conditions.Our approach is independent of the space dimension.
The singular-degenerate stochastic fast diffusion equation, m ∈ (0, 1), until the timehorizon T > 0, is given by (1.1) where we employ the notation x [m] := |x| m−1 x, x ∈ R, m ∈ (0, 1).The stochastic driving term is given by an independent family of standard one-dimensional Brownian motions {β k (t)} t≥0 , k ∈ N supported by a filtered probability space (Ω, F, {F t } t≥0 , P) satisfying the usual assumptions of completeness and right-continuity.The noise coefficients g k , k ∈ N, are assumed to satisfy Denote H := H −1 0 (O), that is, the topological dual space of H 1 0 (O) = W 1,2 0 (O).Furthermore, denote the L 2 (O)-norm by • 2 and the H −1 0 (O)-norm by • H .For v ∈ H, we introduce the following notation for the noise coefficient, where e k ∈ H, k ∈ N are the elements of an orthonormal basis of H. Section 3].Here, L 2 (H, H) denotes the space of linear Hilbert-Schmidt operators from H to H. We also obtain The stochastic fast diffusion equation is closely related to the stochastic porous medium equation, see [7] and the references therein.Several properties of the solutions to stochastic fast diffusion equations have been studied, for instance, finite time extinction [6,17], random attractors [16], invariance of subspaces [24], ergodicity and uniqueness of invariant measures [4,21,25,27], convergence of solutions [11,22], under general pseudodifferential operators [29], and regularity [19].The limiting case m = 0 exhibits two particular frameworks, depending on how one interprets the passage to the limit for m → 0. The multivalued case with a step-function nonlinearity is related to models of self-organized criticality and has been first studied in [3,5,8,17] and is still an active topic of research [1,26].The logarithmic diffusion case has been studied in [2,10].The case m ∈ (−1, 0) is treated in [9].For an initial datum u 0 ∈ L 2 (Ω, F 0 , P; H) and all spatial dimensions d ∈ N, it is known that there exists a unique solution {u(t)} t∈[0,T ] in the sense of stochastic variational inequalities (SVI) [19,Definition 2.1] to (1.1) in the space L 2 (Ω; C([0, T ], H)) by [19,Theorem 2.3], which is also a unique generalized strong solution in the sense of [19,Definition A.1] by [19,Theorem 3.1].At the same place, for initial data u 0 ∈ L m+1 (Ω, F 0 , P; L m+1 (O)) ∩ L 2 (Ω, F 0 , P; H), the authors obtain that u is in fact a unique pathwise strong solution in the sense of [19,Definition A.1], such that Stronger notions of solutions and non-negativity of solutions are discussed in [7, Section 3.6], where the authors obtain u Our main result is given as follows.
Theorem 1.1.Assume that (1.2) holds.Then the unique strong solution u to equation Note that also by the results of Gess and Röckner [19].
Our main idea is based on the observation that formally, so the nonlinear drift is of the form u → div(A(u)∇u), that is, a divergence-form quasilinear partial differential operator.The structure of the drift operator resembles the quasi-linear operators in [14,28], however, we would like to point out that their result requires strong ellipticity of the nonlinear coefficient A, whereas A(u) = m|u| m−1 becomes singular for u = 0 in our case.We will justify the formal chain rule (1.4) by a choice of suitable approximations for the nonlinearity u [m] .
For the degenerate drift case, there are several strong regularity results for the stochastic porous medium equation, that is (1.1) with m > 1, as in this case one can treat the second order terms occurring in Itô's formula more directly, see [15].Recently, optimal regularity for the stochastic porous medium equation in one spatial dimension with multiplicative space-time white noise was obtained using the so-called Stroock-Varopoulos inequality by Dareiotis, Gerencsér and Gess [13], see [18,20,23] for further results proving improved regularity for porous media equations.We note that the application of the Stroock-Varopoulos inequality requires m > 1 and cannot be applied in our case.
Furthermore, we would like to point out that our upper estimate contains a factor m − m+1 2 , so our argument does not to carry over to the singular multi-valued case m = 0. Looking closer at our proof below, one observes that for m = 0, we may obtain an upper bound containing a term 1 δ u 3 L 3 (O) , with δ → 0, so even the improved integrability results from [5] in spatial dimensions d = 1, 2, 3 cannot resolve this issue.A result of improved regularity in this limiting case remains an open problem.

Proof of the main result
Let us introduce a regularization for the nonlinearity r → r [m] , for δ ≥ 0, let Now, we need to regularize the original equation (1.1) with parameter ε > 0, δ ≥ 0 (2.1) Here, {W (t)} t≥0 denotes the cylindrical Wiener process in H on (Ω, F, {F t } t≥0 , P), constructed with respect to the {β k } k∈N introduced in the description of the equation (1.1) and the orthonormal basis {e k } k∈N of H.By [19, Proof of Theorem 3.1], see also [22,Theorem 6.4], we get that there exist unique solutions u ε,δ to (2.1) for any ε > 0, δ ≥ 0, and we obtain the following weak convergences (weak * convergence, respectively) for a subsequence {δ n } n∈N , lim n→∞ δ n = 0, (2.2) where u is the unique solution to (1.1) in the sense of [19,Definition A.1].On the other hand, Proof of Theorem 1.1.We recall that we have assumed (1.2) to hold.Note that by the chain rule for Sobolev functions, as φ δ ∈ C 1 (R) for δ > 0 (composing φ δ with a smooth cut-off function if necessary), we get that for all v ∈ H 1 0 (O), ∇(φ δ (v)) = φ ′ δ (v)∇v.In the sequel, let us fix t ∈ [0, T ].By Itô's formula [12,Theorem 4.32] for the functional for some K ≥ 0, and by integration by parts in O, we get that For notation purposes, we have dropped the dependency on the time s ∈ [0, t] and the space variable ξ ∈ O for the functions u ε,δ under the integrals.This convention will be kept throughout the arguments below.Choosing K = C g , we obtain for ε, δ ∈ (0, 1], We shall use this estimate to get the regularity of the solution as follows.We first rewrite for β > 0, and use Hölder's inequality for p = 2 m + 1 and q = 2 1 − m .
We obtain .