Optimal three-ball inequalities and quantitative uniqueness for the Stokes system

TL;DR

This paper establishes optimal three-ball inequalities for the Stokes system with singular coefficients, providing a quantitative measure of unique continuation and bounds on the solution's vanishing order using Carleman estimates.

Contribution

It introduces optimal three-ball inequalities for the Stokes system and derives bounds on solution vanishing order, advancing quantitative unique continuation results.

Findings

Derived optimal three-ball inequalities for the Stokes system

Established bounds on the vanishing order of solutions

Provided quantitative unique continuation estimates

Abstract

In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.