November  2016, 15(6): 2103-2116. doi: 10.3934/cpaa.2016029

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, Vietnam

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, Vietnam

Received  November 2015 Revised  January 2016 Published  September 2016

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.
Citation: Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
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show all references

References:
[1]

Springer, Berlin-Heidelberg-NewYork, 1976.  Google Scholar

[2]

Acta Math., 174 (1995), 311-382. doi: 10.1007/BF02392469.  Google Scholar

[3]

Math. Z., 211 (1992), 409-447. doi: 10.1007/BF02571437.  Google Scholar

[4]

Funkcial. Ekvac., 50 (2007), 371-403. doi: 10.1619/fesi.50.371.  Google Scholar

[5]

Springer Tracts in Natural Philosophy, 38, Springer, Berlin, Heidelberg, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[6]

Arch. Ration. Mech. Anal., 138 (1997), 307-318. doi: 10.1007/s002050050043.  Google Scholar

[7]

Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9.  Google Scholar

[8]

J. Reine Angew. Math., 596 (2006), 45-62. doi: 10.1515/CRELLE.2006.051.  Google Scholar

[9]

J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[10]

Arch. Rational Mech. Anal., 193 (2009), 339-421. doi: 10.1007/s00205-008-0130-8.  Google Scholar

[11]

Journal of Differential Equations, 246 (2009), 1820-1844. doi: 10.1016/j.jde.2008.10.010.  Google Scholar

[12]

Arch. Ration. Mech. Anal., 213 (2014), 689-703. doi: 10.1007/s00205-014-0744-y.  Google Scholar

[13]

Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[14]

Math. Ann., 310 (1998), 1-45. doi: 10.1007/s002080050134.  Google Scholar

[15]

Tôhoku Math. J., 33 (1981), 383-393. doi: 10.2748/tmj/1178229401.  Google Scholar

[16]

Gordon & Breach, New York, 1969.  Google Scholar

[17]

Nonlinearity, 4 (1991), 503-529.  Google Scholar

[18]

Ric. Mat., 40 (1991), 81-135.  Google Scholar

[19]

J. Math. Sci., (New York), 93 (1999), 719-746. doi: 10.1007/BF02366850.  Google Scholar

[20]

Hiroshima Math. J., 12 (1982), 115-140.  Google Scholar

[21]

Math. Z., 32 (1930), 329-375. doi: 10.1007/BF01194638.  Google Scholar

[22]

North-Holland, Amsterdam, New York, Oxford, 1978.  Google Scholar

[23]

Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar

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