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

Previous Article
Robust control of a Cahn-Hilliard-Navier-Stokes model
CPAA Home
This Issue
Next Article
Some properties of positive solutions for an integral system with the double weighted Riesz potentials

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

Nguyen Thieu Huy ^1, , Vu Thi Ngoc Ha ^2, and Pham Truong Xuan ^3,

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

Abstract
Related Papers

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

Mathematics Subject Classification: 35Q30, 35B35, 76D05, 76D0.

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:

[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

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

[1]	Trinh Viet Duoc. Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3387-3405. doi: 10.3934/dcds.2018145
[2]	Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574
[3]	Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[4]	Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[5]	Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[6]	Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[7]	Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[8]	Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[9]	Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[10]	Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[11]	Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635
[12]	Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045
[13]	Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[14]	Satoshi Hashimoto, Mitsuharu Ôtani. Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 323-333. doi: 10.3934/dcds.2007.19.323
[15]	Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[16]	Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065
[17]	Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[18]	Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215
[19]	Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339
[20]	Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences