Existence and regularity of the pressure for the stochastic Navier-Stokes equations
type: Item
instance of: scholarly article
title: Existence and regularity of the pressure for the stochastic Navier-Stokes equations
zbMATH Open document ID: 1049.60058
author: José Real
author: J. A. Langa
author: Jacques Simon
published in: Applied Mathematics and Optimization
publication date: 2004-05-27T00:00:00Z
review text: The authors study the stochastic Navier-Stokes equation with domain \(D \times (0,T)\) where \(D\) is a connected and bounded open subset of \(\mathbb R^d\) with \(d=\)2, 3 or 4. The existence of a velocity \(u\) and a pressure \(p\) solution to this problem is proved when the body force \(F\) is a measurable sublinear mapping from \((0,T] \times (L^2(D))^d\) into \((H^{-1}(D))^d\). The solution is obtained in two steps: first the authors consider a velocity \(u\) satisfying the ``variational N-S equation'' (in which the pressure is eliminated) and then they associate a pressure \(p\) to such a \(u\) by using a generalization of de Rham's theorem to processes. Finally, they prove that with a body force \(F\) with values in the dual \(V'\) of the space \(V=\{v \in (H_0^1(D))^d: \nabla \cdot v=0 \}\), in general it is not possible to solve the stochastic Navier-Stokes equation.
reviewed by: Carles Rovira
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76M35
zbMATH DE Number: 2067291
zbMATH Keywords: stochastic Navier-Stokes equation
MaRDI profile type: Publication
full work available at URL: s00245-003-0773-7
OpenAlex ID: W2115473533
DOI: 10.1007/S00245-003-0773-7
description: scientific article; zbMATH DE number 2067291