Strong solutions for stochastic partial differential equations of gradient type

Gess, Benjamin

doi:10.1016/j.jfa.2012.07.001

Cited by 73 publications

(61 citation statements)

References 21 publications

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“…Applied to driving signals given by independent Brownian motions this implies a new regularity result for the variational stochastic solution X corresponding to (1.2), namely P-a.s. local equicontinuity on O T . This complements the regularity results given in [26], where it is shown that |X| m sgn(X t ) ∈ L 2 ([0, T ] × Ω; H 1 0 (O)) and X ∈ L ∞ ([0, T ]; L m+1 (Ω × O)) if the initial condition is regular enough.…”

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Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Gess

2012

Comptes Rendus. Mathématique

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Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in L 1 (O) on bounded domains O. The generation of a continuous, order-preserving random dynamical system on L 1 (O) and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in L ∞ (O) norm. Uniform L ∞ bounds and uniform spacetime continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong-Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space-time rough signals, existence of solutions is proven for initial data in L m+1 (O).1. Introduction. The qualitative study of stochastic dynamics induced by stochastic partial differential equations (SPDE) especially in the case of non-Markovian noise is based on the theory of random dynamical systems (RDS); cf., for example, [2]. Since the foundational work [17,18,43] the long-time behavior of several quasilinear SPDE has been investigated by means of the existence of random attractors. However, all these results are restricted to simple models of the noise (e.g., additive or real multiplicative 2 ) not including the important case of linear multiplicative space-time

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confidence: 86%

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Gess

2012

Comptes Rendus. Mathématique

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“…When applied to the Allen-Cahn equation, see [13,Rem. 4.9], the regularity properties are slightly weaker than the ones in Theorem 2.2.…”

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confidence: 99%

Stochastic Allen–Cahn equation with mobility

Bertini

Buttà

Pisante

2017

Nonlinear Differ. Equ. Appl.

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Abstract. We introduce a class of stochastic Allen-Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the d-dimensional torus in the time interval [0, T ]. Assuming that d ≤ 3 and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in L 2 with continuous paths, satisfying almost surely the regularity properties u ∈ C([0, T ]; H 1 ) and u ∈ L 2 ([0, T ]; H 2 ).

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“…The question of existence of a unique (variational) solution to equations of the form (1.1) is well-understood: first results were established in [15,14], for an overview in the above stated generality and further references we refer the reader to [16]. Existence of a strong solution under various assumptions appeared in [3,10] and numerical approximations were studied in [11,12]. In the case of linear operator A which generates a strongly continuous semigroup, more is known concerning regularity and maximal regularity (see e.g.…”

Section: Time Regularitymentioning

confidence: 99%

On time regularity of stochastic evolution equations with monotone coefficients

Breit

Hofmanová

2015

Comptes Rendus. Mathématique

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We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity we obtain a fractional Sobolev type time regularity of order up to 1 2 for a certain functional G(u) of the solution. Namely, G(u) = ∇u in the case of the heat equation and G(u) = |∇u| p−2 2 ∇u for the p-Laplacian. The motivation is twofold. On the one hand, it turns out that this is the natural time regularity result that allows to establish the optimal rates of convergence for numerical schemes based on a time discretization. On the other hand, in the linear case, i.e. where the solution is given by a stochastic convolution, our result complements the known stochastic maximal space-time regularity results for the borderline case not covered by other methods.

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Strong solutions for stochastic partial differential equations of gradient type

Cited by 73 publications

References 21 publications

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Stochastic Allen–Cahn equation with mobility

On time regularity of stochastic evolution equations with monotone coefficients

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