A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

TL;DR

This paper investigates the initial-time behavior of suitable weak solutions to the Navier-Stokes equations, showing it mirrors the resolvent operator of the Stokes operator, and uses this to study spatial decay properties.

Contribution

It establishes the time behavior of the local $L^ abla_{loc}$-norm near zero for solutions with certain initial data, extending partial regularity results.

Findings

The $L^ abla_{loc}$-norm behaves like the Stokes resolvent near t=0.

Provides a key tool for analyzing spatial decay of solutions.

Builds on and extends classical partial regularity results.

Abstract

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg, as a corollary, under suitable assumptions of local character on the initial data, we prove a behavior in time of the $L^\infty_{loc}$-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator. Besides its own interest, the result is a main tool to study the spatial decay estimates of a suitable weak solution, performed in paper F. Crispo and P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem (submitted).