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Mathematical Analysis of Fluids in Motion

  • Eduard Feireisl
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 153)

Abstract

Continuum fluid mechanics is a phenomenological theory based on macroscopic observable state variables, the time evolution of which is described by means of systems of partial differential equations. The resulting mathematical problems are highly non-linear and rather complex, even in the simplest physically relevant situations. We discuss several recent results and newly developed methods based on the concept of weak solution. The class of weak solutions is happily large enough in order to guarantee the existence of global-in-time solutions without any essential restrictions on the size of the relevant data. On the other hand, the underlying structural hypotheses impose quite severe restrictions on the specific form of constitutive relations. The best known open problems - hypothetical presence of vacuum zones, propagation of density oscillations, sequential stability of the temperature field, among others - are discussed. The final part of the study addresses the simplified problems, in particular, the incompressible Navier-Stokes system.

Keywords

Weak Solution Entropy Production Rate Incompressible Limit Renormalize Solution Total Energy Balance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Eduard Feireisl
    • 1
  1. 1.Mathematical Institute of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic

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