Regularity of Solutions to the Navier-Stokes Equations Evolving from Small Data in BMO−1

By Pierre Germain, Natavsa Pavlovi'c, Gigliola Staffilani. 2006

101 citations recorded by the source corpus.

In 2001, H. Koch and D. Tataru proved the existence of global in time solutions to the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$ for initial data small enough in $BMO^{-1}$. We show in this article that the Koch and Tataru solution has higher regularity. As a consequence, we get a decay estimate in time for any space derivative, and space analyticity of the solution. Also as an application of our regularity theorem, we prove a regularity result for self-similar solutions.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3609b4b3dd62fbc88bc653925a053a63c22c85a2

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In Applied Mathematics

In Computational Mechanics

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Citations in the corpus

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions (2012, background): “…initial value problem (1.1), (1.2) has a unique local-in-time “mild solution” defined on some time interval (0, T ), which is smooth in R3 × (0, T ) and has in many respects the same regularity as the solution of the heat equation in R3 × [0, T ) for times close to t = 0, see for example [6, 10].”
Liouville theorems for the Navier–Stokes equations and applications (2007, unknown): “Taking difference quotients, we see that, for u0 = 0, we have similar estimates for spatial derivatives: ||∇kxu||Cαpar(Q(z0,R)) ≤ C(α, R)||∇kxf ||L∞x, t(Rn×(0,T )) and (3.12) ||∇k+1x u||Lpx, t(Q(z0,R)) ≤ C(p, R)||∇ k xf ||L∞x, t(Rn×(0,T )) (3.13) Moreover, a routine inspection of representation…”