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arXiv · 1407.1712

Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations

Jacek Cyranka·Piotr Zgliczyński
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Abstract

We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish existence of a locally attracting solution.

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Jacek Cyranka, Piotr Zgliczyński. 2014-07-07. Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations. https://arxiv.org/abs/1407.1712

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