Existence and Stability of Spatial Plane Waves for the Incompressible Navier–Stokes in \(\mathbb {R}^3\)

Abstract

We consider the three-dimensional incompressible Navier–Stokes equation on the whole space. We observe that this system admits a \(L^\infty \) family of global spatial plane wave solutions, which are connected with the two-dimensional equation. We then proceed to prove local well-posedness over a space which includes \(L^3(\mathbb {R}^3)\) and these solutions. Finally, we prove \(L^3\)-stability of spatial plane waves, with no condition on their size.

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Correspondence to Simão Correia.

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Communicated by H. Beirão da Veiga.

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Correia, S., Figueira, M. Existence and Stability of Spatial Plane Waves for the Incompressible Navier–Stokes in \(\mathbb {R}^3\) . J. Math. Fluid Mech. 20, 189–197 (2018). https://doi.org/10.1007/s00021-017-0317-6

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Mathematics Subject Classification

  • 35B35
  • 35Q30
  • 76D03

Keywords

  • Incompressible Navier–Stokes
  • local well-posedness
  • stability
  • spatial plane waves