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March  2020, 40(3): 1555-1593. doi: 10.3934/dcds.2020086

## Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation

 1 Department of Mathematical Science, Tsinghua University, Beijing 100084, China 2 Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100190, China 3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Peng-Fei Yao

Received  February 2019 Revised  September 2019 Published  December 2019

Fund Project: The first and third author are supported by the National Science Foundation of China, grants no. 61473126 and no. 61573342, and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011. The second author is supported by the National Science Foundation of China, grants no. 11771235.

We consider a nonlinear, free boundary fluid-structure interaction model in a bounded domain. The viscous incompressible fluid interacts with a nonlinear elastic body on the common boundary via the velocity and stress matching conditions. The motion of the fluid is governed by incompressible Navier-Stokes equations while the displacement of elastic structure is determined by a nonlinear elastodynamic system with boundary dissipation. The boundary dissipation is inserted in the velocity matching condition. We prove the global existence of the smooth solutions for small initial data and obtain the exponential decay of the energy of this system as well.

Citation: Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086
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