On the differentiability of weak solutions of a degenerate system of PDE's in fluid mechanics
type: Item
instance of: scholarly article
title: On the differentiability of weak solutions of a degenerate system of PDE's in fluid mechanics
zbMATH Open document ID: 0664.35033
DOI: 10.1007/BF01762796
author: Joachim Naumann
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: Weak solutions, in problems of fluid mechanics, are generalized solutions usually constructed in order to describe the behaviour of phenomena such as discontinuities, for example. In the present paper, the author studies the differentiability of weak solutions for a nonnewtonian fluid (here, a pseudo-plastic fluid). Namely, the author derives a theorem setting that, under some restrictions about the deviatoric stress-tensor, then for any weak solution of the incompressible equations of motion, the velocity u possesses second-order derivatives which satisfy the properties: \[ u_{i,k\ell}\in L^{\sigma}_{loc}(\Omega),\quad (1+\| D(u)\|^ 2)^{(\sigma -2)/4} D_{ij}(u,k)\in L^ 2_{loc}(\Omega), \] \(\sigma\) being a real number related to constitutive law of the fluid. The paper deals only with mathematical results: an application to realistic materials and a physical interpretation of the results would be wellcome.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4086164
zbMATH Keywords: Weak solutions
zbMATH Keywords: fluid mechanics
zbMATH Keywords: generalized solutions
zbMATH Keywords: differentiability
zbMATH Keywords: nonnewtonian fluid
zbMATH Keywords: deviatoric stress-tensor
zbMATH Keywords: incompressible equations of motion
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01762796
OpenAlex ID: W2062732291
description: scientific article; zbMATH DE number 4086164