Infinite Energy Solutions for Damped Navier-Stokes Equations in R2

Abstract

We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field $u_0\in L^{\infty }({\mathbb{R}}^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|\to \infty$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.

The revised version: the error in the interpolation inequality (in Section 5) pointed out by Thierry Gallay is corrected and one more Section with the proof of the inequality is added. The main results remain unchanged

References in corpus (1)

• Inviscid limit for damped and driven incompressible Navier-Stokes equations in ${\mathbb{R}}^2$

Cited by in corpus (10)

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• Global existence of solutions to 2-D Navier-Stokes flow with non-decaying initial data in half-plane
• Infinite energy solutions for Dissipative Euler equations in R^2
• Global well-posedness of the two-dimensional exterior Navier-Stokes equations for non-decaying data
• Using periodic boundary conditions to approximate the Navier-Stokes equations on ${\mathbb{R}}^3$ and the transfer of regularity
• Rigorous results in space-periodic two-dimensional turbulence
• Vanishing viscosity limit for global attractors for the damped Navier--Stokes system with stress free boundary conditions
• No-slip boundary condition for vorticity equation in 2D exterior domains
• Vanishing viscosity limits for axisymmetric flows with boundary
• Hexagonal structures in 2D Navier-Stokes flows