×

On stability of the spatially inhomogeneous Navier-Stokes-Boussinesq system with general nonlinearity. (English) Zbl 1309.35083

Summary: This paper considers \(L^2\)-asymptotic stability of the spatially inhomogeneous Navier-Stokes-Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes-Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal \(L^p\)-in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal \(L^2\)-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal \(L^p\)-in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abels H., Terasawa Y.: On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344, 381-429 (2009) · Zbl 1172.35050 · doi:10.1007/s00208-008-0311-7
[2] Babin A., Mahalov A., Nicolaenko B.: On the regularity of three-dimensional rotating Euler-Boussinesq equations. Math. Models Methods Appl. Sci. 9, 1089-1121 (1999) · Zbl 1035.76055 · doi:10.1142/S021820259900049X
[3] Borchers W., Miyakawa T.: L2 decay for the Navier-Stokes flow in halfspaces. Math. Ann. 282, 139-155 (1988) · Zbl 0627.35076 · doi:10.1007/BF01457017
[4] Borchers W., Miyakawa T.: Algebraic L2 decay for Navier-Stokes flows in exterior domains. Acta Math. 165, 189-227 (1990) · Zbl 0722.35014 · doi:10.1007/BF02391905
[5] Borchers W., Miyakawa T.: L2-decay for Navier-Stokes flows in unbounded domains, with application to exterior stationary flows. Arch. Rational Mech. Anal. 118, 273-295 (1992) · Zbl 0756.76018 · doi:10.1007/BF00387899
[6] Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in Lq-spaces. Math. Z. 196, 415-425 (1987) · Zbl 0636.76027 · doi:10.1007/BF01200362
[7] Bothe, D., Prüss, J.: LP-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39, 379-421 (2007) · Zbl 1172.35052
[8] Brandolese L., Schonbek M.E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364, 5057-5090 (2012) · Zbl 1368.35217 · doi:10.1090/S0002-9947-2012-05432-8
[9] Cholewa, J.W., Dlotko, T.: Global attractors in abstract parabolic problems. London Mathematical Society Lecture Note Series, vol. 278, pp. xii+235. Cambridge University Press, Cambridge (2000) · Zbl 0954.35002
[10] Cosner C.: Pointwise a priori bounds for strongly coupled semilinear systems of parabolic partial differential equations. Indiana Univ. Math. J. 30, 607-620 (1981) · Zbl 0467.35014 · doi:10.1512/iumj.1981.30.30048
[11] de Simon L.: Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. (Italian) Rend. Sem. Mat. Univ. Padova 34, 205-223 (1964) · Zbl 0196.44803
[12] Denk, R., Hieber, M., Prüss, J.: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003) · Zbl 1274.35002
[13] Desch, W., Hieber, M., Prüss, J.: Lp-theory of the Stokes equation in a half space. J. Evol. Equ. 1, 115-142 (2001) · Zbl 0983.35102
[14] Engel, K-J., Nagel, R.: One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, vol. 194, pp. xxii+586. Springer, New York (2000) · Zbl 0952.47036
[15] Farwig R., Sohr H.: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn. 46, 607-643 (1994) · Zbl 0819.35109 · doi:10.2969/jmsj/04640607
[16] Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. 2nd edn. Springer Monographs in Mathematics, pp. xiv+1018. Springer, New York (2011) · Zbl 1245.35002
[17] Gallay T., Wayne C.E.: Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2. Arch. Ration. Mech. Anal. 163, 209-258 (2002) · Zbl 1042.37058 · doi:10.1007/s002050200200
[18] Gallay T., Wayne C.E.: Long-time asymptotics of the Navier-Stokes equation in R2 and R3 [Plenary lecture presented at the 76th Annual GAMM Conference, Luxembourg, 29 March-1 April 2005]. ZAMM Z. Angew. Math. Mech. 86, 256-267 (2006) · Zbl 1094.35089 · doi:10.1002/zamm.200510260
[19] Giga Y.: Domains of fractional powers of the Stokes operator in Lr spaces. Arch. Rational Mech. Anal. 89, 251-265 (1985) · Zbl 0584.76037 · doi:10.1007/BF00276874
[20] Giga Y.: Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62, 186-212 (1986) · Zbl 0577.35058 · doi:10.1016/0022-0396(86)90096-3
[21] Giga Y., Miyakawa T.: Solutions in Lr of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89, 267-281 (1985) · Zbl 0587.35078 · doi:10.1007/BF00276875
[22] Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, vol. 840, pp. iv+348. Springer, Berlin (1981) · Zbl 0456.35001
[23] Hess, M., Hieber, M., Mahalov, A., Saal, J.: Nonlinear stability of Ekman boundary layers. Bull. Lond. Math. Soc. 42, 691-706 (2010) · Zbl 1197.35193
[24] Heywood J.G.: On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61-102 (1976) · Zbl 0347.76016 · doi:10.1007/BF02392043
[25] Heywood J.G.: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639-681 (1980) · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048
[26] Kajikiya R., Miyakawa T.: On L2 decay of weak solutions of the Navier-Stokes equations in Rn. Math. Z. 192, 135-148 (1986) · Zbl 0607.35072 · doi:10.1007/BF01162027
[27] Kato T.: A generalization of the Heinz inequality. Proc. Jpn. Acad. 37, 305-308 (1961) · Zbl 0104.09304 · doi:10.3792/pja/1195523678
[28] Kato T.: Fractional powers of dissipative operators. II. J. Math. Soc. Jpn. 14, 242-248 (1962) · Zbl 0108.11203 · doi:10.2969/jmsj/01420242
[29] Kato, T.: Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions. Math. Z. 187, 471-480 (1984) · Zbl 0545.35073
[30] Kato T., Fujita H.: On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243-260 (1962) · Zbl 0114.05002
[31] Kielhöfer H.: Global solutions of semilinear evolution equations satisfying an energy inequality. J. Differ. Equ. 36, 188-222 (1980) · Zbl 0396.34052 · doi:10.1016/0022-0396(80)90063-7
[32] Koba, H.: Nonlinear stability of ekman boundary layers in rotating stratified fluids. Mem. Am. Math. Soc. 228(1073), viii+127 (2014) · Zbl 1302.35371
[33] Koba, H., Mahalov, A., Yoneda, T.: Global well-posedness for the rotating Navier-Stokes-Boussinesq equations with stratification effects. Adv. Math. Sci. Appl. 22, 61-90 (2012) · Zbl 1283.35086
[34] Kozono, H.: Global Ln-solution and its decay property for the Navier-Stokes equations in half-space \[{\mathbb{R}^n_+}\] R+n. J. Differ. Equ. 79(1), 79-88 (1989) · Zbl 0715.35062
[35] Kozono H., Ogawa T.: Global strong solution and its decay properties for the Navier-Stokes equations in three-dimensional domains with noncompact boundaries. Math. Z. 216, 1-30 (1994) · Zbl 0798.35127 · doi:10.1007/BF02572306
[36] Kunstmann, P.C., Weis, L.: MaximalLp- regularity for parabolic equations, Fourier multiplier theorems and \[{H^\infty}H\]∞- functional calculus. Functional analytic methods for evolution equations, pp. 65-311. Lecture Notes in Mathematics 1855. Springer, Berlin (2004) · Zbl 1097.47041
[37] Lunardi, A.: Interpolation theory. 2nd edn. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] Edizioni della Normale, pp. xiv+191 (2009) · Zbl 0642.35067
[38] Masuda, K.: Weak solutions of Navier-Stokes equations. Tohoku Math. J. (2)36, 623-646 (1984) · Zbl 0568.35077
[39] McCracken, M.: The resolvent problem for the Stokes equations on halfspace in Lp. SIAM J. Math. Anal. 12, 201-228 (1981) · Zbl 0475.35073
[40] Miyakawa, T., Schonbek, M.E.: On optimal decay rates for weak solutions to the Navier-Stokes equations in Rn. In: Proceedings of Partial Differential Equations and Applications (Olomouc, 1999). Math. Bohem. vol. 126, pp. 443-455 (2001) · Zbl 0981.35048
[41] Miyakawa T., Sohr H.: On energy inequality, smoothness and large time behavior in L2 for weak solutions of the Navier-Stokes equations in exterior domains. Math. Z. 199, 455-478 (1988) · Zbl 0642.35067 · doi:10.1007/BF01161636
[42] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol 44, pp. viii+279. Springer, New York (1983) · Zbl 0516.47023
[43] Pedlosky, J.: Geophysical Fluid Dynamics. 2nd edn. Springer, Berlin (1987) · Zbl 0713.76005
[44] Petcu, M., Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. Handbook of numerical analysis. Vol. XIV. Special volume: computational methods for the atmosphere and the oceans, pp. 577-750. Handbook of Numerical Analysis, 14. Elsevier, Amsterdam (2009) · Zbl 0756.76018
[45] Saal, J.: R-Boundedness,\[{H^\infty}H\]∞-calculus, Maximal (Lp-) Regularity and Applications to Parabolic PDE’s. In: Lecture Notes in Mathematical Sciences,The University of Tokyo, Graduate School of Mathematical Sciences (2007) · Zbl 0636.76027
[46] Sattinger, D.H.: The mathematical problem of hydrodynamic stability. J. Math. Mech. 19, 979-817 (1969/1970) · Zbl 0198.30401
[47] Schonbek M.E.: L2 decay for weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 88, 209-222 (1985) · Zbl 0602.76031 · doi:10.1007/BF00752111
[48] Schonbek, M.E.: The Fourier splitting method. Advances in geometric analysis and continuum mechanics (Stanford, CA, 1993), pp. 269-274. Cambridge University Press, Cambridge (1995) · Zbl 0842.35142
[49] Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], pp. x+367. Birkhäuser, Basel (2001) · Zbl 0983.35004
[50] Solonnikov, V.A.: Lp-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. Function theory and partial differential equations. J. Math. Sci. (New York)105, 2448-2484 (2001)
[51] Solonnikov, V.A.: Estimates of the solution of model evolution generalized Stokes problem in weighted Holder spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 336 (2006). Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. vol. 37, pp. 211-238, 277. [Translation in J. Math. Sci. (N.Y.) 143, 2969-2986 (2007)] · Zbl 1127.35051
[52] Tanabe, H.: Equations of evolution. Monographs and Studies in Mathematics, vol. 6, pp. xii+260. Pitman (Advanced Publishing Program), Boston (1979) · Zbl 0417.35003
[53] Temam, R., Ziane, M.: Some mathematical problems in geophysical fluid dynamics. Handbook of mathematical fluid dynamics, vol. III, pp. 535-657. North-Holland, Amsterdam (2004) · Zbl 1222.35145
[54] Triebel, H.: Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library, vol. 18, p. 528. North-Holland, Amsterdam (1978) · Zbl 0387.46032
[55] Wiegner, M.: Decay results for weak solutions of the Navier-Stokes equations on Rn. J. Lond. Math. Soc. (2)35, 303-313 (1987) · Zbl 0652.35095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.