Infinite Energy Solutions for Damped Navier-Stokes Equations in R2

TL;DR

This paper establishes global well-posedness, dissipativity, and regularity of solutions to the damped Navier-Stokes equations in 2D space, allowing for initial data with no decay assumptions and showing polynomial growth bounds over time.

Contribution

It develops a weighted energy theory for Navier-Stokes type problems and applies it to demonstrate solution properties without decay constraints in R2.

Findings

Weak solutions exist globally and are regular in uniformly-local spaces.

Solutions can grow at most polynomially (quintic) over time.

The theory applies to classical Navier-Stokes equations in R2, extending understanding of solution behavior.

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.