$ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids

Augusto Visintin

doi:10.3934/dcdss.2022066

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2022, Volume 15, Issue 8: 2331-2343. Doi: 10.3934/dcdss.2022066

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$ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids

Augusto Visintin

Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo di (Trento), Italia

On the occasion of the 60th anniversary of Maurizio Grasselli with friendship

Received: December 31, 2021

Published: August 2022

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Abstract

The flow of a homogeneous, incompressible and heat conducting fluid is here described by coupling a quasilinear Navier-Stokes-type equation with the equation of heat diffusion, convection and buoyancy. This model is formulated variationally as a problem of null-minimization.

First we review how De Giorgi's theory of $ \Gamma $-convergence can be used to prove the compactness and the stability of evolutionary problems under nonparametric perturbations. Then we illustrate how this theory can be applied to the our problem of fluid and heat flow, and to more general coupled flows.

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Mathematics Subject Classification: 35K60, 47H05, 49J40, 58E, 76D03.

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