Lemmas on compensated compactness in elliptic and parabolic equations
type: Item
instance of: scholarly article
title: Lemmas on compensated compactness in elliptic and parabolic equations
zbMATH Open document ID: 1216.35045
DOI: 10.1134/S0081543810030089
author: Svetlana E. Pastukhova
author: V. V. Zhikov
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2011-03-17T00:00:00Z
review text: The authors study the solvability of parabolic and elliptic equations of monotone type with nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a solution, they consider the approximating equations \[ \begin{alignedat}{2} \text{div} (A_n(x,\nabla u_n)+g)&=0,&\quad x&\in \Omega\subset\mathbb R^d,\\ \frac{\partial}{\partial t}u_n -\text{div} (A_n(x,t,\nabla u_n)+g)&=0,&\quad (x,t)&\in Q_T =\Omega\times [0,T]\subset\mathbb R^{d+1}, \end{alignedat} \] where \(n=1,2,\dots\) and \(A_n(\cdot,\xi)\to A(\cdot,\xi)\) as \(n\to\infty\) pointwise. A key element of passing to the limit in the above approximation scheme is a generalized lemma on compensated compactness. The parabolic version of this lemma is proved for the first time in the present paper. The new technique applies to stationary and nonstationary problems of fast diffusion in an incompressible flow, to a parabolic equation with a \(p(x,t)\)-Laplacian, and its generalization, and to a nonstationary thermistor system.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5866958
zbMATH Keywords: elliptic and parabolic equations of monotone type
zbMATH Keywords: compensated compactness
zbMATH Keywords: weak convergence
reviewed by: Lubomira Softova
MaRDI profile type: Publication
Wikidata QID: Q124791911
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full work available at URL: s0081543810030089
OpenAlex ID: W1999697749
description: scientific article; zbMATH DE number 5866958