Quantitative uniqueness estimates for the generalized non-stationary Stokes system

* Dedicated to Professor Sergio Vessella on the occasion of his 65th birthday.

ABSTRACT

We study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in $R^n$ with $n\ge 2$. One of our main results is a bound on the vanishing order of a non-trivial solution u satisfying the generalized non-stationary Stokes system, which is a quantitative version of the (strong) unique continuation property for u. Different from the previously known results, our 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-cylinder inequalities for u.

Keywords:

• Stokes system
• quantitative uniqueness estimates
• Carleman estimates

2010 Mathematics Subject Classification:

• 35B60

Acknowledgments

Ching-Lung Lin was supported in part by the MOST. Jenn-Nan Wang was partially supported by MOST 108-2115-M-002-002-MY3.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

* Dedicated to Professor Sergio Vessella on the occasion of his 65th birthday.

Additional information

Funding

Ching-Lung Lin and Jenn-Nan Wang were supported in part by the MOST – Ministry of Science and Technology, Taiwan.