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
References:
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J. Bergh and J. Löfström, Interpolation Spaces,, Springer, (1976). Google Scholar

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W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, \emph{Acta Math.}, 174 (1995), 311. doi: 10.1007/BF02392469. Google Scholar

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R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, \emph{Math. Z.}, 211 (1992), 409. doi: 10.1007/BF02571437. Google Scholar

[4]

R. Farwig and T. Hishida, Stationary Navier-Stokes flows around a rotating obstacle,, \emph{Funkcial. Ekvac.}, 50 (2007), 371. doi: 10.1619/fesi.50.371. Google Scholar

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G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems,, Springer Tracts in Natural Philosophy, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

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G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flows past a body,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 363. doi: 10.1007/s00205-004-0306-9. Google Scholar

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M. Geissert, H. Heck and M. Hieber, $L_p$-Theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle,, \emph{J. Reine Angew. Math.}, 596 (2006), 45. doi: 10.1515/CRELLE.2006.051. Google Scholar

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Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system,, \emph{J. Differential Equations}, 61 (1986), 186. doi: 10.1016/0022-0396(86)90096-3. Google Scholar

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T. Hishida and Y. Shibata, $L_p-L_q$ estimate of the stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle,, \emph{Arch. Rational Mech. Anal.}, 193 (2009), 339. doi: 10.1007/s00205-008-0130-8. Google Scholar

[11]

Nguyen Thieu Huy, Invariant manifolds of admissible classes for semi-linear evolution equations,, \emph{Journal of Differential Equations}, 246 (2009), 1820. doi: 10.1016/j.jde.2008.10.010. Google Scholar

[12]

Nguyen Thieu Huy, Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle,, \emph{Arch. Ration. Mech. Anal.}, 213 (2014), 689. doi: 10.1007/s00205-014-0744-y. Google Scholar

[13]

T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb R^n$ with applications to weak solutions,, \emph{Math. Z.}, 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar

[14]

T. Kobayashi and Y. Shibata, On the Oseen equation in the three dimensional exterior domains,, \emph{Math. Ann.}, 310 (1998), 1. doi: 10.1007/s002080050134. Google Scholar

[15]

H. Komatsu, A general interpolation theorem of Marcinkiewicz type,, \emph{T\^ohoku Math. J.}, 33 (1981), 383. doi: 10.2748/tmj/1178229401. Google Scholar

[16]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon & Breach, (1969). Google Scholar

[17]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space,, \emph{Nonlinearity}, 4 (1991), 503. Google Scholar

[18]

P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space,, \emph{Ric. Mat.}, 40 (1991), 81. Google Scholar

[19]

P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains,, \emph{J. Math. Sci.}, 93 (1999), 719. doi: 10.1007/BF02366850. Google Scholar

[20]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, \emph{Hiroshima Math. J.}, 12 (1982), 115. Google Scholar

[21]

F. K. G. Odqvist, Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten,, \emph{Math. Z.}, 32 (1930), 329. doi: 10.1007/BF01194638. Google Scholar

[22]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar

[23]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force,, \emph{Math. Ann.}, 317 (2000), 635. doi: 10.1007/PL00004418. Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces,, Springer, (1976). Google Scholar

[2]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows,, \emph{Acta Math.}, 174 (1995), 311. doi: 10.1007/BF02392469. Google Scholar

[3]

R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, \emph{Math. Z.}, 211 (1992), 409. doi: 10.1007/BF02571437. Google Scholar

[4]

R. Farwig and T. Hishida, Stationary Navier-Stokes flows around a rotating obstacle,, \emph{Funkcial. Ekvac.}, 50 (2007), 371. doi: 10.1619/fesi.50.371. Google Scholar

[5]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems,, Springer Tracts in Natural Philosophy, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar

[6]

G. P. Galdi, J. G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest,, \emph{Arch. Ration. Mech. Anal.}, 138 (1997), 307. doi: 10.1007/s002050050043. Google Scholar

[7]

G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flows past a body,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 363. doi: 10.1007/s00205-004-0306-9. Google Scholar

[8]

M. Geissert, H. Heck and M. Hieber, $L_p$-Theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle,, \emph{J. Reine Angew. Math.}, 596 (2006), 45. doi: 10.1515/CRELLE.2006.051. Google Scholar

[9]

Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system,, \emph{J. Differential Equations}, 61 (1986), 186. doi: 10.1016/0022-0396(86)90096-3. Google Scholar

[10]

T. Hishida and Y. Shibata, $L_p-L_q$ estimate of the stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle,, \emph{Arch. Rational Mech. Anal.}, 193 (2009), 339. doi: 10.1007/s00205-008-0130-8. Google Scholar

[11]

Nguyen Thieu Huy, Invariant manifolds of admissible classes for semi-linear evolution equations,, \emph{Journal of Differential Equations}, 246 (2009), 1820. doi: 10.1016/j.jde.2008.10.010. Google Scholar

[12]

Nguyen Thieu Huy, Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle,, \emph{Arch. Ration. Mech. Anal.}, 213 (2014), 689. doi: 10.1007/s00205-014-0744-y. Google Scholar

[13]

T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb R^n$ with applications to weak solutions,, \emph{Math. Z.}, 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar

[14]

T. Kobayashi and Y. Shibata, On the Oseen equation in the three dimensional exterior domains,, \emph{Math. Ann.}, 310 (1998), 1. doi: 10.1007/s002080050134. Google Scholar

[15]

H. Komatsu, A general interpolation theorem of Marcinkiewicz type,, \emph{T\^ohoku Math. J.}, 33 (1981), 383. doi: 10.2748/tmj/1178229401. Google Scholar

[16]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Gordon & Breach, (1969). Google Scholar

[17]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space,, \emph{Nonlinearity}, 4 (1991), 503. Google Scholar

[18]

P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space,, \emph{Ric. Mat.}, 40 (1991), 81. Google Scholar

[19]

P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains,, \emph{J. Math. Sci.}, 93 (1999), 719. doi: 10.1007/BF02366850. Google Scholar

[20]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, \emph{Hiroshima Math. J.}, 12 (1982), 115. Google Scholar

[21]

F. K. G. Odqvist, Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten,, \emph{Math. Z.}, 32 (1930), 329. doi: 10.1007/BF01194638. Google Scholar

[22]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar

[23]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force,, \emph{Math. Ann.}, 317 (2000), 635. doi: 10.1007/PL00004418. Google Scholar

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