Evolution problems of Navier–Stokes type with anisotropic diffusion

Abstract

In this work, we consider the evolutive problem for the incompressible Navier–Stokes equations with a general diffusion which can be fully anisotropic. The existence of weak solutions is proved for the associated initial problem supplemented with no-slip boundary conditions. We prove also the properties of extinction in a finite time, exponential time decay and power time decay. With this respect, we consider the important case of a forces fields with possible different behavior in distinct directions. Perturbations of the asymptotically stable equilibrium are established as well.

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Correspondence to H. B. de Oliveira.

Additional information

S. N. Antontsev and H. B. de Oliveira were partially supported by the Portuguese Foundation for Science and Technology (FCT) through PEstOE/MAT/UI 0209/2014. S. N. Antontsev was also partially supported by the Research Project Grant No. 15-11-20019 of the Russian Science Foundation.

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Antontsev, S.N., de Oliveira, H.B. Evolution problems of Navier–Stokes type with anisotropic diffusion. RACSAM 110, 729–754 (2016). https://doi.org/10.1007/s13398-015-0262-2

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Keywords

  • Anisotropic diffusion
  • Navier–Stokes
  • Existence
  • Finite time extinction
  • Exponential time decay
  • Power time decay

Mathematics Subject Classification

  • 35Q35
  • 76D05
  • 35Q30
  • 76D03
  • 35D30
  • 35B40