TY - JOUR
T1 - Optimal three-ball inequalities and quantitative uniqueness for the Stokes system
AU - Lin, Ching Lung
AU - Uhlmann, Gunther
AU - Wang, Jenn Nan
N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2010/11
Y1 - 2010/11
N2 - We study the local behavior of a solution to the Stokes system with singular coefficients in Rn 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.
AB - We study the local behavior of a solution to the Stokes system with singular coefficients in Rn 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.
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U2 - 10.3934/dcds.2010.28.1273
DO - 10.3934/dcds.2010.28.1273
M3 - Article
AN - SCOPUS:77955334340
VL - 28
SP - 1273
EP - 1290
JO - Discrete and Continuous Dynamical Systems
JF - Discrete and Continuous Dynamical Systems
SN - 1078-0947
IS - 3
ER -