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Journal of Mathematical Fluid Mechanics

, Volume 15, Issue 4, pp 717–745 | Cite as

Infinite Energy Solutions for Damped Navier–Stokes Equations in \({\mathbb{R}^2}\)

  • Sergey ZelikEmail author
Article

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 \({\mathbb{R}^2}\), we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.

Mathematics Subject Classification (2010)

35Q30 35Q35 

Keywords

Navier–Stokes equations infinite-energy solutions weighted energy estimates unbounded domains 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK

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