M. Gubinelli and M. Jara
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.
M. Gubinelli and M. Jara. Regularization by noise and stochastic Burgers equations. Papers of Massimiliano Gubinelli, 2012, version 2. https://mgubi.org/papers/paper/burgers-regularisation-by-noise/
Licence: arXiv.org perpetual, non-exclusive license to distribute; all other rights reserved to the authors
Pin in the collection (SHA-512 of the inventory it fixes):
734ea2ba5a2b3bc54167eb603c81154523c8af197be6706fe3468d32ac1cf1c1c98847632284d482a7efa81477fb9984a012575491a440234daaf336ee67efd4
Papers of Massimiliano Gubinelli: an OpenSheaf archive (format 0.1, a draft); this page is generated by opensheaf-py from the archive.