Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes

2025/08/01 by Zachary Bradshaw, Bradshaw, Zachary, Joshua Hudson +1 · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Numerical Methods in Computational Mathematics #Fluid Dynamics and Turbulent Flows

paper · pdf · doi:10.48550/arxiv.2508.00714

Abstract

This paper extends the weak solution theory for the 3D Navier-Stokes equations of Barker, Seregin and Sverak from a critical setting to a supercritical setting making sure to include a useful a priori energy bound as well as a statement about stability under weak-star convergence. Two applications of the a priori bound are then explored. The first provides a spatially local, short-time asymptotic expansion in the time variable starting at t=0 which, as a corollary, provides an upper bound on how fast hypothetical non-unique solutions to the Navier-Stokes equations can separate locally. The second establishes higher-order time regularity at a singular time and at spatial points positioned away from the singularity. This quantifies the degree to which the non-local nature of the pressure allows a far flung singularity to disrupt the time regularity at a regular point.

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Cited by

• Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

Related

• Estimation of non-uniqueness and short-time asymptotic expansions for Navier-Stokes flows
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• The role of the pressure in the regularity theory for the Navier-Stokes equations
• On possible isolated blow-up phenomena of the 3D-Navier-Stokes equation and a regularity criterion in terms of supercritical function space condition and smoothness condition along the streamlines