Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force
type: Item
instance of: scholarly article
title: Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force
zbMATH Open document ID: 0931.35124
DOI: 10.1023/A:1021937715194
author: Franco Flandoli
author: Björn Schmalfuss
published in: Journal of Dynamics and Differential Equations
publication date: 2000-02-23T00:00:00Z
review text: The authors consider a viscous incompressible Newtonian fluid in a bounded open domain \(D\subset\mathbb{R}^3\) with smooth boundary \(\partial D\), described by the classical Navier-Stokes equations \[ {\partial u\over\partial t}+(u\cdot\nabla)u+\nabla p=\nu\Delta u+f+{\partial g\over \partial t}\text{ in }[0,\infty)\times D, \] \[ \begin{aligned} \text{div} & u=0\text{ in }[0, \infty)\times D,\\ & u=0\text{ in }[0,\infty)\times\partial D,\\ & u(0,x)=u_0(x),\;x\in D,\end{aligned} \] where \(u\) is the velocity field, \(p\) the pressure field, \(\nu>0\) the kinematic viscosity, and \(f+(\partial g/\partial t)\) the body force. Let \(X=L^2_{\text{loc}}[0,\infty;H)\), and \(W(g)\subset X\) be the space of generalized weak solutions. Let \(G\) be the space of all \(g\in C_0(\mathbb{R};V)\) that have polynomial growth at \(-\infty\) and satisfy the property \[ \liminf_{\alpha\to\infty}\limsup_{t\to\infty}{1\over t}\int^0_{-t} \bigl|z_\alpha(s,g)\bigr|^8_{L^4}ds=0, \] where \(z_\alpha(t,g)=g(t)-\int^t_{-\infty}(A+\alpha)e^{-(t-s)(A+\alpha)}g(s)ds\). Let \[ B(M_0,g)= \left\{ u\in W(g): \int^1_0\bigl|u(s)\bigr|^2 ds\leq M_0\right\} \] be defined for all constants \(M_0>0\) and all \(g\in G\). Let \(M(\cdot)\) be a function, defined on the backward orbit of \(g\) (the set \(\{\theta_{-\tau}g;\tau\geq 0\})\) with values in \((0,\infty)\), so that the subexponential growth condition is fulfilled, \[ \lim_{\tau\to+\infty}{\log^+M(\theta-\tau g)\over\tau}=0.\tag{1} \] For \(g\in G\) and a function \(M(\cdot)\) satisfying the assumption (1) let us define the set \[ {\mathcal A}\bigl(g,M(\cdot)\bigr)=\bigcap_{\tau\geq 0}\overline {\bigcup_{t\geq\tau}\Phi(t,\theta_{-t}g)B\bigl(M(\theta_{-t}g), \theta_{-t}g \bigr)} \] and let for \(g\in G\) \[ {\mathcal A}(g)=\overline{\bigcup_{M(\cdot)}{\mathcal A}(g,M(\cdot))}.\tag{2} \] Let \({\mathcal D}\) be the set of all measurable multifunctions contained in a ball \(B(M(g),g)\), so that (1) is fulfilled with \(P_Q\)-probability 1. There ex…
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 1333046
zbMATH Keywords: viscous incompressible Newtonian fluid
zbMATH Keywords: smooth boundary
zbMATH Keywords: body force
zbMATH Keywords: generalized weak solutions
zbMATH Keywords: measurable multifunctions
zbMATH Keywords: attractor
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
full work available at URL: a:1021937715194
OpenAlex ID: W152863877
description: scientific article; zbMATH DE number 1333046