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A profile decomposition approach to the \(L^\infty _t(L^{3}_x)\) Navier-Stokes regularity criterion. (English) Zbl 1291.35180

The authors prove that strong solutions of the incompressible Navier-Stokes equations in \({\mathbb{R}}^3\) which are bounded in \(L^2({\mathbb{R}}^3)\) do not become singular in finite time. This result was established by L. Escauriaza et al. [Russ. Math. Surv. 58, No. 2, 211–250 (2003); translation from Usp. Mat. Nauk 58, No. 2, 3–44 (2003; Zbl 1064.35134)] for weak solutions. Here, the method of “critical elements” (or minimal blow-up solutions) is used, as developed recently by C. E. Kenig and F. Merle. The main tool is a “profile decomposition” for the Navier-Stokes equations in critical Besov spaces.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
35D35 Strong solutions to PDEs
35B44 Blow-up in context of PDEs

Citations:

Zbl 1064.35134
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References:

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