Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

Hao Jia; Vladim\'ir \v{S}ver\'ak

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

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arxiv 1204.0529 v1 pith:24JLRFGL submitted 2012-04-02 math.AP

Hao Jia , Vladimír Šverák This is my paper · ORCID

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keywords initialequationsestimateslocal-in-spacenavier-stokesnearsolutionstime

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abstract

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.

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Constructs exact finite-time self-similar blowup solutions for the 1D weak-advection Hou-Li model via fixed-point near origin and ODE extension, for 2/3<a<1 periodic and 0<a≤1 whole-space with Neumann condition.

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