Blow-up of Critical Besov Norms at a Potential Navier–Stokes Singularity

By Isabelle Gallagher, G. Koch, F. Planchon. 2014

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We prove that if an initial datum to the incompressible Navier–Stokes equations in any critical Besov space $${\dot B^{-1+\frac 3p}_{p,q}({\mathbb {R}}^{3})}$$B˙p,q-1+3p(R3), with $${3 < p, q < \infty}$$3 0}$$T>0, then the norm of the solution in that Besov space becomes unbounded at time T. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) concerning suitable weak solutions blowing up in $${L^{3}({\mathbb R}^{3})}$$L3(R3). Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527–1559, 2013), which provided an alternative proof of the $${L^{3}({\mathbb R}^{3})}$$L3(R3) result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the $${L^{3}({\mathbb R}^{3})}$$L3(R3) setting.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e4e4f94ec0d7ea08e14c79715f8163ad0386489c