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Mathematical Thermodynamics of Complex Fluids pp 47–100Cite as

Mathematical Thermodynamics of Viscous Fluids

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,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.

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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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  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Eduard Feireisl

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  1. Eduard Feireisl
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  1. Institute of Mathematics AS CR , Prague, Czech Republic

    Eduard Feireisl

  2. Dipartimento di Matematica, University of Pavia, Pavia, Italy

    Elisabetta Rocca

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Feireisl, E. (2017). Mathematical Thermodynamics of Viscous Fluids. In: Feireisl, E., Rocca, E. (eds) Mathematical Thermodynamics of Complex Fluids. Lecture Notes in Mathematics(), vol 2200. Springer, Cham. https://doi.org/10.1007/978-3-319-67600-5_2

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