An $\epsilon$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data

Abstract

We prove an $\epsilon$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.

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Cited by in corpus (4)

• Spatial decay of discretely self-similar solutions to the Navier-Stokes equations
• Remarks on sparseness and regularity of Navier-Stokes solutions
• The importance of Luis Caffarelli's work in the study of fluids
• Local regularity criteria in terms of one velocity component for the Navier-Stokes equations