On solutions for a generalized Navier–Stokes–Fourier system fulfilling the entropy equality. (14th November 2022)

• Record Type: Journal Article
• Authors: Abbatiello, Anna; Bulíček, Miroslav; Kaplický, Petr
• Is Part Of: Philosophical transactions. Volume 380:Number 2236(2022)
• Publication Date: 2022-11-14
• Subjects: incompressible fluid -- heat conducting fluid -- stability -- inhomogeneous temperature -- non-Newtonian fluid -- entropy equality -- renormalized solution; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rsta.2021.0351
• Languages: English
• ISSNs: 1364-503X

Abstract

Abstract : We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to 11 / 5, we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna–Lions theory only for p > 5 / 2 . The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality. This article is part of the theme issue 'Non-smooth variational problems and applications'.