Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1273 - 1290, Volume 28, Issue 3, November 2010      doi:10.3934/dcds.2010.28.1273

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Ching-Lung Lin - Department of Mathematics, NCTS, National Cheng Kung University, Tainan 701, Taiwan (email)
Gunther Uhlmann - Department of Mathematics, University of Washington, Seattle, WA 98195-4350, United States (email)
Jenn-Nan Wang - Department of Mathematics, Taida Institute of Mathematical Sciences, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan (email)

Abstract: 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.

Keywords:  Optimal three-ball inequalities, Carleman estimates, Stokes system.
Mathematics Subject Classification:  Primary: 35A20; Secondary: 76D07.

Received: April 2010;      Available Online: April 2010.