Strong solutions for stochastic partial differential equations of gradient type

被引:57
作者
Gess, Benjamin [1 ]
机构
[1] Univ Bielefeld, Fac Math, Bielefeld, Germany
关键词
Stochastic partial differential equations; Strong solutions; Regularity; Subdifferential; Stochastic porous medium equation; Stochastic reaction-diffusion equation; Stochastic p-Laplace equation; GENERALIZED POROUS-MEDIA; EVOLUTION EQUATIONS; REGULARITY; EXISTENCE;
D O I
10.1016/j.jfa.2012.07.001
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the "distance" defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained. (C) 2012 Elsevier Inc. All rights reserved.
引用
收藏
页码:2355 / 2383
页数:29
相关论文
共 29 条
[1]  
Adams R.A., 1975, Sobolev Spaces. Adams. Pure and applied mathematics
[2]  
[Anonymous], 1992, ENCY MATH ITS APPL, DOI DOI 10.1017/CBO9780511666223
[3]  
[Anonymous], 1994, DIRICHLET FORMS SYMM
[4]  
Barbu V., 1993, Analysis and control of nonlinear infinite dimensional systems
[5]  
Barbu V, 2008, INDIANA U MATH J, V57, P187
[6]   On a random scaled porous media equation [J].
Barbu, Viorel ;
Roeckner, Michael .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2011, 251 (09) :2494-2514
[7]   Stochastic Porous Media Equations and Self-Organized Criticality [J].
Barbu, Viorel ;
Da Prato, Giuseppe ;
Roeckner, Michael .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2009, 285 (03) :901-923
[8]   The Global Random Attractor for a Class of Stochastic Porous Media Equations [J].
Beyn, Wolf-Juergen ;
Gess, Benjamin ;
Lescot, Paul ;
Roeckner, Michael .
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2011, 36 (03) :446-469
[9]   Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation [J].
Brzezniak, Z. ;
van Neerven, J. M. A. M. ;
Veraar, M. C. ;
Weis, L. .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2008, 245 (01) :30-58
[10]   Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise [J].
Brzezniak, Z ;
van Neerven, J .
JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY, 2003, 43 (02) :261-303