Evolution problems of Navier–Stokes type with anisotropic diffusion


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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Corresponding author

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

Mathematics Subject Classification

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