SPDE in Hilbert Space with Locally Monotone Coefficients

Abstract

In this paper we prove the existence and uniqueness of strong solutions for SPDE in Hilbert space with locally monotone coefficients, which is a generalization of the classical result of Krylov and Rozovskii for monotone coefficients. Our main result can be applied to different types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic $p$-Laplace equation and stochastic porous media equation with some non-monotone perturbations.

20 pages, add Remark 3.1 for stochastic Burgers equation

References in corpus (8)

• Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
• Random attractors for a class of stochastic partial differential equations driven by general additive noise
• Harnack inequality and applications for stochastic generalized porous media equations
• Stochastic Porous Media Equation and Self-Organized Criticality
• Rate of Convergence of Space Time Approximations for stochastic evolution equations
• Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts
• Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations
• Existence and Uniqueness of Solutions to Nonlinear Evolution Equations with Locally Monotone Operators

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