On compactness of solutions to the Navier-Stokes equations of compressible flow
type: Item
instance of: scholarly article
title: On compactness of solutions to the Navier-Stokes equations of compressible flow
zbMATH Open document ID: 0952.35091
author: Eduard Feireisl
author: Hana Petzeltová
published in: Journal of Differential Equations
publication date: 2001-01-04T00:00:00Z
review text: The papers deals with the system describing the motion of a compressible flow in the following form: \[ \rho_t+\text{ div}(\rho u)=0,\quad (\rho u)_t+\text{ div}(\rho u\otimes u)+\nabla P=\text{ div} T(u)+\rho f \] for the velocity of the fluid \(u=(u_1,u_2,u_3)\) and for the mass density \(\rho\). Here, the pressure \(P\) and the density \(\rho\) are functionally dependent: \(P=p(\rho)\) for \(p\) nondecreasing, and locally Lipschitz continuous. This system is supplemented with periodic boundary conditions. The main goal of this paper is to show that any sequence of weak solutions satisfying the natural a priori estimates contains a subsequence converging (not necessarily strongly) to another weak solution of the same problem. Such a result is crucial for proving the existence of solutions for the Cauchy problem, the existence of time periodic solutions, or when the long-time behavior of solutions is examined. Assumptions imposed in this paper are weaker than those in Section 1 of the book of \textit{P. L. Lions} [Mathematical topics in fluid mechanics. Vol. 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications. 10. Oxford: Clarendon Press. xiv, 348 p. (1998; Zbl 0908.76004).].
reviewed by: Grzegorz Karch
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N10
zbMATH DE Number: 1454726
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: compressible flow
zbMATH Keywords: weak solutions
zbMATH Keywords: compactness
zbMATH Keywords: a priori estimates
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3720
OpenAlex ID: W2086576393
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DOI: 10.1006/JDEQ.1999.3720
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description: scientific article; zbMATH DE number 1454726