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Mathematical Theory of Compressible Viscous Fluids pp 49–53Cite as

Birkhäuser

Weak Formulation Revisited

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Part of the book series: Advances in Mathematical Fluid Mechanics ((LNMFM))

Abstract

In Chap. 3, we have introduced the weak formulation of both the equation of continuity (3.5) and the momentum balance (3.11). On the other hand, we have seen in Chap. 4 that regular solutions of the Navier–Stokes system satisfy also the renormalized equation of continuity (4.8), together with the total energy balance (4.10). Under the general hypothesis considered in this book, the piece of information encoded in (4.8) and (4.10) cannot be obtained directly from the weak formulation (3.5) and (3.11). Accordingly, it seems convenient to include both (4.8) and (4.10) in the definition of weak solutions to the problem (2.7), (2.8), (2.13)–(2.15) as a kind of additional admissibility criteria.

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Authors and Affiliations

  1. Institute of Mathematics CAS, Praha, Czech Republic

    Eduard Feireisl

  2. Department of Mathematical Sciences, Norwegian University of Science & Tech., Trondheim, Norway

    Trygve G. Karper

  3. Charles University, Faculty Mathematics and Physics Charles University, Praha, Czech Republic

    Milan Pokorný

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  1. Eduard Feireisl
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Feireisl, E., Karper, T.G., Pokorný, M. (2016). Weak Formulation Revisited. In: Mathematical Theory of Compressible Viscous Fluids. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-44835-0_5

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