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Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs · arXivDesk

0806.0616Jun 3, 200831 pages;Major Changes: New theorem which covers the case of SPDEs with quadratic nonlinearity (such as stochastic Navier-Stokes equation with multiplicative noise) is added

Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs

Z. Brzeźniak, M. Neklyudov

Analysis of PDEs Mathematical Physics Probability

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Abstract

The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in [Ghidaglia-1986] for deterministic PDEs.

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The paper addresses: Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs.
The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type.

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Author comment: 31 pages;Major Changes: New theorem which covers the case of SPDEs with quadratic nonlinearity (such as stochastic Navier-Stokes equation with multiplicative noise) is added