Stochastic two dimensional Euler equations
type: Item
instance of: scholarly article
title: Stochastic two dimensional Euler equations
zbMATH Open document ID: 1032.60055
author: Zdzislaw Brzezniak
author: Szymon Peszat
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The main result of the paper, stated in Section 2, is the following. Consider an open \({\mathcal O}\subset{\mathbb R}^{2}\) satisfying several regularity assumptions (named A), \(q\in [2,\infty)\), the \(W^{1,q}\) spaces of \({\mathcal O}\) and \(H^{1,q}\) the intersection of \(W^{1,q}\) with the closure in \(L^{q}\) of \(C_{\text{sol}}^{\infty }\) defined as the set of all \(v\in C_{0}^{\infty }(\overline {\mathcal O} ,{\mathbb R}^{2})\) with \(\text{div }v=0\), \(\langle v,n\rangle=0\) on \(\partial {\mathcal O}\). Consider, both jointly measurable, an \(F\) which takes \([0,T]\times H^{1,2}\) in \(W^{1,2}\) and \([0,T]\times H^{1,q}\) in \(W^{1,q}\), a \(G\) which takes \([0,T]\times H^{1,2}\) in \(L_{(\text{HS})}({\mathcal H},W^{1,2})\) and \([0,T]\times H^{1,q}\) in \(R({\mathcal H},W^{1,q})\), where \({\mathcal H}\) is a separable Hilbert space, (HS) meaning Hilbert Schmidt operators, \(T\in R\) meaning the convergence in \(L^{2}(P,W^{1,q})\) of \(\sum_{k}\beta_{k}Te_{k}\) for an orthonormal basis \(e_{k}\) and for independent standard Gaussian r.v. \(\beta_{k}\) on the probability space \(P\). Suppose that there exist \(a_{1}\in L^{1}(0,T)\), \(a_{2}\in L^{r}(0,T)\), \(r>2\), such that for \(v\in H^{1,2}\cap H^{1,q}\) the four considered maps satisfy corresponding norm inequalities \(\|F(t,v)\|\leq a_{1}(t)(1+\|\cdot \|)\), \(\|G(t,v)\|\leq a_{2}(t)(1+\|v\|)\). Suppose also that \(\langle F(t,v),z\rangle\) and \(|\langle G^{\ast }(t,v)z\rangle|\) are continuous in \(v\) on H\(_{L^{2}}^{1,2}\), for every \(z\in C_{\text{sol}}^{\infty }\). Then, for every \(u_{0}\in H^{1,2}\cap H^{1,q}\), there exists a martingale \(H^{1,2}\cap H^{1,q}\) valued solution of \[ \partial_{t}u+\langle u,\nabla\rangle u+\nabla p=F(t,u)+G(t,u)\dot W, \] \(\text{div } u=0\), \(\langle u,n\rangle=0\) on \(\partial {\mathcal O}\) (stochastic incompressible Euler equations). This means a filtration on a probability space \(P\), an adapted cylindrical Wiener process \(W(t)\) (i.e. \(W(t)…
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 76M35
zbMATH DE Number: 1906041
zbMATH Keywords: martingale solution
zbMATH Keywords: modified Navier Stokes equations
zbMATH Keywords: cylindrical Wiener process
zbMATH Keywords: Wiener random field
zbMATH Keywords: radonifying mapping
zbMATH Keywords: stochastic integral
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1015345773
OpenAlex ID: W2079505476
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DOI: 10.1214/AOP/1015345773
description: scientific article; zbMATH DE number 1906041