Generalized solutions to models of compressible viscous fluids

Anna Abbatiello; Eduard Feireisl; Antoní Novotný

doi:10.3934/dcds.2020345

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2021, Volume 41, Issue 1: 1-28. Doi: 10.3934/dcds.2020345

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Generalized solutions to models of compressible viscous fluids

1.
Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
2.
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic
3.
Université de Toulon, IMATH, EA 2134, BP 20132, 83957 La Garde, France

^* Corresponding author: Eduard Feireisl
^* Corresponding author: Eduard Feireisl

Received: November 30, 2019

Revised: July 31, 2020

Early access: October 2020

Published: January 2021

The research of A.A. is supported by Einstein Foundation, Berlin. The research of E.F. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18-12719S. The work of A.N. was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea(NRF-2019H1D3A2A01101128)

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Abstract

We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non–linear viscosity tensor and non–homogeneous boundary conditions, for which the problem of existence of global–in–time weak/strong solutions is largely open.

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Mathematics Subject Classification: Primary: 35A01, 35D30; Secondary: 76N10.

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