Lower bounds on blowing-up solutions of the three-dimensional Navier–Stokes equations in H˙^{3/2}, H˙^{5/2}, and B˙^{5/2}_{2,1}

McCormick, David S, Olson, Eric J, Robinson, James C, Rodrigo, Jose L, Vidal-López, Alejandro and Zhou, Yi (2016) Lower bounds on blowing-up solutions of the three-dimensional Navier–Stokes equations in H˙^{3/2}, H˙^{5/2}, and B˙^{5/2}_{2,1}. SIAM Journal on Mathematical Analysis (SIMA), 48 (3). pp. 2119-2132. ISSN 0036-1410

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Abstract

If u is a smooth solution of the Navier–Stokes equations on R^ 3 with first blowup time T, we prove lower bounds for u in the Sobolev spaces H˙^(3/2) , H˙^( 5/2) , and the Besov space B˙^(5/2)_( 2,1 ), with optimal rates of blowup: we prove the strong lower bounds ||u(t)||_(H˙^(3/2))≥ c(T − t) ^(−1/2) and ||u(t)||_(B˙^(5/2)_( 2,1))≥ c(T − t) −1 , but in H˙^(5/2) we only obtain the weaker result lim supt→T − (T −t)||u(t)||_(H˙^(5/2)) ≥ c. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of u.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Numerical Analysis and Scientific Computing Research Group
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: David McCormick
Date Deposited: 01 Dec 2017 10:17
Last Modified: 01 Dec 2017 13:29
URI: http://sro.sussex.ac.uk/id/eprint/66125

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