Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation
type: Item
instance of: scholarly article
title: Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation
zbMATH Open document ID: 0859.35011
DOI: 10.1215/S0012-7094-95-08110-1
author: Michael Loss
author: Eric Carlen
published in: Duke Mathematical Journal
publication date: 1997-04-09T00:00:00Z
review text: The authors study a generalization of the vorticity formulation of the 2D-Navier-Stokes equations, that is \(w_t-\Delta w=\text{div}(\underline f(x,w)w)\) with suitable assumptions on the vector field \(\underline f\). They optimize the constants in the \(L_p-L_q\)-estimate \[ |w(.,t)|_q\leq C\cdot t^{-{n\over 2}({1\over p}-{1\over q})}|w(.,0)|_p,\quad 1\leq p\leq q, \] and give even decay results for \(w\) in \(L_1\), as well as spatial estimates. The results are sharp to some extent, yet the authors missed some earlier papers, where qualitatively similar results can be found, e.g. \textit{M. E. Schonbek} [Nonlinear Anal., Theory Methods, Appl. 10, 943-956 (1986; Zbl 0617.35060)] and \textit{Y. Giga}, \textit{T. Miyakawa} and \textit{H. Osada} [Arch. Ration. Mech. Anal. 104, No. 3, 223-250 (1988; Zbl 0666.76052)].
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 880074
zbMATH Keywords: vorticity
zbMATH Keywords: \(L_ p-L_ q\)-estimate
reviewed by: Michael Wiegner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 880074