Optimal three-ball inequalities and quantitative uniqueness for the Stokes system
Resource
Discrete and Continuous Dynamical Systems - Series A, 28(3), 1273-1290
Journal
Discrete and Continuous Dynamical Systems
Journal Volume
28
Journal Issue
3
Pages
1273-1290
Date Issued
2010
Author(s)
Abstract
We study the local behavior of a solution to theStokes system with singular coefficients in $R^n$ with $n=2,3$. Oneof our main results is a bound on the vanishing order of anontrivial solution $u$ satisfying the Stokes system, which is aquantitative version of the strong unique continuation property for$u$. Different from the previous known results, our strong uniquecontinuation result only involves the velocity field $u$. Our proofrelies on some delicate Carleman-type estimates. We first use theseestimates to derive crucial optimal three-ball inequalitiesfor $u$. Taking advantage of the optimality, we then derive an upperbound on the vanishing order of any nontrivial solution $u$ to theStokes system from those three-ball inequalities. As an application,we derive a minimal decaying rate at infinity of any nontrivial $u$satisfying the Stokes equation under some a priori assumptions.
SDGs
Type
journal article