Mathematical Thermodynamics of Viscous Fluids

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2200)

Abstract

This course is a short introduction to the mathematical theory of the motion of viscous fluids. We introduce the concept of weak solution to the Navier-Stokes-Fourier system and discuss its basic properties. In particular, we construct the weak solutions as a suitable limit of a mixed numerical scheme based on a combination of the finite volume and finite elements method. The question of stability and robustness of various classes of solutions is addressed with the help of the relative (modulated) energy functional. Related results concerning weak-strong uniqueness and conditional regularity of weak solutions are presented. Finally, we discuss the asymptotic limit when viscosity of the fluid tends to zero. Several examples of ill- posedness for the limit Euler system are given and an admissibility criterion based on the viscous approximation is proposed.

Notes

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPragueCzech Republic

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