On the backward uniqueness of the stochastic primitive equations with additive noise

Boling Guo; Guoli Zhou

doi:10.3934/dcdsb.2018305

Article Contents

2019, Volume 24, Issue 7: 3157-3174. Doi: 10.3934/dcdsb.2018305

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On the backward uniqueness of the stochastic primitive equations with additive noise

Boling Guo^1, and
Guoli Zhou^2, ,

1.
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
2.
School of Mathematics and Statistics, Chongqing University, Chongqing city 401331, China

* Corresponding author: Guoli Zhou
* Corresponding author: Guoli Zhou

Received: April 30, 2018

Revised: June 30, 2018

Early access: October 2018

Published: May 2019

The second author is supported by NSF NNSF of China(Grant No. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central Universities(Grant No. 2018CDXYST0024, ) and China Scholarship Council (Grant No.201506055003).

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Abstract

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

Keywords:

Stochastic primitive equations,
backward uniqueness.

Mathematics Subject Classification: Primary: 35Q35, 60H15; Secondary: 86A10.

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