Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids
type: Item
instance of: scholarly article
title: Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids
zbMATH Open document ID: 0949.35115
DOI: 10.1007/BF02433444
author: T. N. Shilkin
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-08-24T00:00:00Z
review text: The author studies the boundary value problem that describes a~slow stationary flow of a~generalized Newtonian liquid \[ \begin{gathered} -{\text{div}}{\partial D\over \partial \tau}\bigl(\varepsilon(u)\bigr) =f-\nabla p\quad \text{in } \Omega\subset \mathbb R^n, \\ \text{div }u=0\quad\text{in } \Omega, \\ u=0\quad\text{on } \partial\Omega. \end{gathered} \] Here \(u\) is the unknown velocity field, \(p\) is the unknown pressure field, \(\varepsilon(u)={1\over 2}\bigl(\nabla u+(\nabla u)^T\bigr)\) is the~strain rate tensor, \(D\) is a~dissipative potential on the space of real symmetric matrices of order~\(n\). For a~generalized solution to this problem, the author proves that \(u\in W^2_2(\Omega)\) and \(p\in W^1_2(\Omega)\) and the Hölder continuity of the strain rate tensor \(\varepsilon(u)\) on some set ~ \(\Omega_0\subset\overline{\Omega}\). In two dimensions, the Hölder continuity of \(\varepsilon(u)\) is proven everywhere on the set ~\(\overline{\Omega}\). Finally, \(\varepsilon(u)\) belongs to \(L^\infty\) for a~special class of potentials (\(u(x)\) stands for a~generalized solution).
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76A05
zbMATH DE Number: 1495305
zbMATH Keywords: stationary flow of a generalized Newtonian liquid
zbMATH Keywords: regularity of the strain velocity tensor
MaRDI profile type: Publication
cites work: Comments on the validity of a common category of constitutive equations
cites work: Some Remarks on Non-Newtonian Fluids Including Nonconvex Perturbations of the Bingham and Powell–Eyring Model for Viscoplastic Fluids
cites work: Regularity results for the quasi-static Bingham variational inequality in dimensions two and three
cites work: On differential properties of extremals of variational problems of the mechanics of viscoplastic media
cites work: Q4949495
cites work: Q5663431
cites work: Q4115554
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Regularity for minimizers of some variational problems in plasticity theory
cites work: Q3364034
full work available at URL: https://doi.org/10.1007/bf02433444
OpenAlex ID: W2087141599
description: scientific article; zbMATH DE number 1495305