American Institute of Mathematical Sciences

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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