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.
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Communicated by A. V. Fursikov
The author would like to thank Gregory Seregin and Thierry Gallay for the fruitful discussions.
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Zelik, S. Infinite Energy Solutions for Damped Navier–Stokes Equations in \({\mathbb{R}^2}\) . J. Math. Fluid Mech. 15, 717–745 (2013). https://doi.org/10.1007/s00021-013-0144-3
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DOI: https://doi.org/10.1007/s00021-013-0144-3