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

Existence and Stability of Spatial Plane Waves for the Incompressible Navier–Stokes in... We consider the three-dimensional incompressible Navier–Stokes equation on the whole space. We observe that this system admits a $$L^\infty$$ L ∞ 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 $$L^3(\mathbb {R}^3)$$ L 3 ( R 3 ) and these solutions. Finally, we prove $$L^3$$ L 3 -stability of spatial plane waves, with no condition on their size. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

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

, Volume 20 (1) – Feb 21, 2017
9 pages

/lp/springer_journal/existence-and-stability-of-spatial-plane-waves-for-the-incompressible-8nkNPUJPFj
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0317-6
Publisher site
See Article on Publisher Site

### Abstract

We consider the three-dimensional incompressible Navier–Stokes equation on the whole space. We observe that this system admits a $$L^\infty$$ L ∞ 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 $$L^3(\mathbb {R}^3)$$ L 3 ( R 3 ) and these solutions. Finally, we prove $$L^3$$ L 3 -stability of spatial plane waves, with no condition on their size.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Feb 21, 2017

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