July  2010, 28(3): 1273-1290. doi: 10.3934/dcds.2010.28.1273

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

1. 

Department of Mathematics, NCTS, National Cheng Kung University, Tainan 701, Taiwan

2. 

Department of Mathematics, University of Washington, Seattle, WA 98195-4350

3. 

Department of Mathematics, Taida Institute of Mathematical Sciences, NCTS (Taipei), National Taiwan University, Taipei 106

Received  April 2010 Published  April 2010

We study the local behavior of a solution to the Stokes system with singular coefficients in $R^n$ with $n=2,3$. One of our main results is a bound on the vanishing order of a nontrivial solution $u$ satisfying the Stokes system, which is a quantitative version of the strong unique continuation property for $u$. Different from the previous known results, our strong unique continuation result only involves the velocity field $u$. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequalities for $u$. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution $u$ to the Stokes 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.
Citation: Ching-Lung Lin, Gunther Uhlmann, Jenn-Nan Wang. Optimal three-ball inequalities and quantitative uniqueness for the Stokes system. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1273-1290. doi: 10.3934/dcds.2010.28.1273
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