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The paper deals with the Navier-Stokes equations in a strip in the class of spatially non-decaing (infinite-energy) solutions belonging to the properly chosen uniformly local Sobolev spaces. The global well-posedness and dissipativity of the Navier-Stokes equations in a strip in such spaces has been first established in [S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip. Glasg. Math. J., 49 (2007), no. 3, 525--588]. However, the proof given there contains rather essential error and the aim of the present paper is to correct this error and to show that the main results of that paper remain true.
In this paper we give optimal lower bounds for the blow-up rate of the $dot{H}^{s}left(mathbb{T}^3right)$-norm, $frac{1}{2}<s<frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof for a l
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hol
In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $lim_{tto T}sqrt{T-t}||u(t)||_{BMO}<in