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Article Dans Une Revue Stochastic partial differential equations: analysis and computations Année : 2013

Regularization by noise and stochastic Burgers equations

Résumé

We study a generalized 1d periodic SPDE of Burgers type: $$ \partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi $$ where $\theta > 1/2$, $-A$ is the 1d Laplacian, $\xi$ is a space-time white noise and the initial condition $u_0$ is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when $\theta>1/2$. When $\theta>5/4$ we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of the same equation and for a model of stationary 2d stochastic Navier-Stokes evolution.

Dates et versions

hal-00858787 , version 1 (06-09-2013)

Identifiants

Citer

Massimiliano Gubinelli, M. Jara. Regularization by noise and stochastic Burgers equations. Stochastic partial differential equations: analysis and computations, 2013, 1 (2), pp.325-350. ⟨10.1007/s40072-013-0011-5⟩. ⟨hal-00858787⟩
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