Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics

Nguyen Thieu Huy; Vu Thi Ngoc  Ha; Pham Truong  Xuan

doi:10.3934/cpaa.2016029

Article Contents

2016, Volume 15, Issue 6: 2103-2116. Doi: 10.3934/cpaa.2016029

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Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics

1.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi
2.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology
3.
Faculty of Information Technology, Department of Mathematics, Water Resources University, Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Hanoi

Received: October 31, 2015

Revised: December 31, 2015

Published: October 2016

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Abstract

For an exterior domain $\Omega\subset R^d$ with smooth boundary, we study the existence and stability of bounded mild solutions in time $t$ to the abstract semi-linear evolution equation $u_t + Au = Pdiv (G(u)+F(t))$ where $-A$ generates a $C_0$-semigroup on the solenoidal space $L^d_{\sigma,w}(\Omega)$ (known as weak-$L^d$), $P$ is Helmholtz projection; $G$ is a nonlinear operator acting from $L^d_{\sigma,w}(\Omega)$ into $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$, and $F(t)$ is a second-order tensor in $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$. Our obtained abstract results can be applied not only to reestablish the known results on Navier-Stokes flows on exterior domains and/or around rotating obstacles, but also to obtain a new result on existence and polynomial stability of bounded solutions to Navier-Stokes-Oseen equations on exterior domains.

Keywords:

Semi-linear evolution equations,
boundedness and polynomial stability of solutions,
Navier-Stokes-Oseen equations,
fluid dynamics,
exterior domains.

Mathematics Subject Classification: 35Q30, 35B35, 76D05, 76D07.

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