Regularization by noise and stochastic Burgers equations

Gubinelli, Massimiliano; Jara, Milton

Gubinelli, Massimiliano; Jara, Milton (2013), Regularization by noise and stochastic Burgers equations, Stochastic Partial Differential Equations: Analysis and Computations, 1, 2, p. 325-350. http://dx.doi.org/10.1007/s40072-013-0011-5

Type

Article accepté pour publication ou publié

External document link

http://arxiv.org/abs/1208.6551v2

Date

2013

Journal name

Stochastic Partial Differential Equations: Analysis and Computations

Volume

Number

Publisher

Springer

Pages

325-350

Abstract (EN)

We study a generalized 1d periodic SPDE of Burgers type: ∂tu=−Aθu+∂xu2+Aθ/2ξ where θ>1/2 , −A is the 1d Laplacian, ξ is a space–time white noise and the initial condition u0 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 θ>1/2 . When θ>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.

Subjects / Keywords

Noise regularization; SPDEs; Kardar–Parisi–Zhang equation