Regularity and attractors for the three-dimensional generalized Boussinesq system. (English) Zbl 1538.35077
Summary: We study the three-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties.
{© 2023 John Wiley & Sons, Ltd.}
{© 2023 John Wiley & Sons, Ltd.}
MSC:
| 35B41 | Attractors |
| 35Q35 | PDEs in connection with fluid mechanics |
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76F20 | Dynamical systems approach to turbulence |
| 76F65 | Direct numerical and large eddy simulation of turbulence |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
attractors; evolutionary system; fractional Laplacian; regularity; three-dimensional Boussinesq system; weak solutionReferences:
| [1] | P.Constantin and C. R.Doering, Infinite Prandtl number convection, J. Stat. Phys.94 (1999), 159-172. · Zbl 0935.76083 |
| [2] | C.Doering and J.Gibbon, Applied analysis of the Navier‐Stokes equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995. · Zbl 0838.76016 |
| [3] | A. E.Gill, Atmosphere‐ocean dynamics, Academic Press, London, 1982. |
| [4] | A. J.Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, Vol. 9, AMS/CIMS, 2003. · Zbl 1278.76004 |
| [5] | J.Pedlosky, Geophysical fluid dynamics, Springer‐Verlag, New York, 1987. · Zbl 0713.76005 |
| [6] | B.Birnir and N.Svanstedt, Existence theory and strong attractors for the Rayleigh‐Bénard problem with a large aspect ratio, Discrete Contin. Dyn Syst.10 (2004), 53-74. · Zbl 1174.76308 |
| [7] | R.Temam, Infinite dimensional dynamical systems in mechanics and physics, 2nd ed., Springer‐Verlag, Berlin, 1997. · Zbl 0871.35001 |
| [8] | M.Cabral, R.Rosa, and R.Temam, Existence and dimension of the attractor for the Bénard problem on channel‐like domains, Discrete Contin. Dyn. Syst.10 (2004), no. 1‐2, 89-116. · Zbl 1049.37046 |
| [9] | C.Foias, O.Manley, and R.Temam, Attractors for the Bénard problem, Nonlinear Anal.11 (1987), 939-967. · Zbl 0646.76098 |
| [10] | O.Manley, M.Marion, and R.Temam, Equations of combustion in the presence of complex chemistry, Indiana Univ. Math. J.42 (1993), no. 3, 941-967. · Zbl 0796.35164 |
| [11] | H.Abidi and T.Hmidi, On the global well‐posedness for Boussinesq system, J. Differ. Equ.233 (2007), no. 1, 199-220. · Zbl 1111.35032 |
| [12] | F.Cheng and C. J.Xu, Analytical smoothing effect of solution for the Boussinesq equations, Acta Math. Sci.39 (2019), no. 1, 165-179. · Zbl 1499.35495 |
| [13] | J.Fan, G.Nakamura, and H.Wang, Blow‐up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain, Nonlinear Anal.75 (2012), 3436-3442. · Zbl 1243.35137 |
| [14] | S.Gala and M. A.Ragusa, A regularity criterion of weak solutions to the 3D Boussinesq equations, Bull. Braz. Math. Soc.51 (2020), no. 2, 513-525. · Zbl 1437.35574 |
| [15] | O. V.Kapustyan, V. S.Melnik, and J.Valero, A weak attractors and properties of solutions for the three‐dimensional Bénard problem, Discrete Contin. Dyn Syst.18 (2007), 449-481. · Zbl 1143.35013 |
| [16] | O. V.Kapustyan, A. V.Pankov, and J.Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system, Set‐Valued Var Anal.20 (2012), 445-465. · Zbl 1259.35042 |
| [17] | O. V.Kapustyan, A. V.Pankov, and J.Valero, On the existence and connectedness of a global attractor for solutions of the three‐dimensional Bénard system that satisfy a system of energy inequalities, J. Math. Sci.191 (2013), no. 3, 388-402. · Zbl 1291.35222 |
| [18] | O. V.Kapustyan and A. V.Pankov, Global [( \varphi \]\)‐attractor for a modified 3D Bénard system on channel‐like domains, Nonauton. Dyn. Syst.1 (2014), no. 1, 1-9. · Zbl 1288.35081 |
| [19] | J.Li, X.Wu, and W.Zhu, Global smooth solutions of the damped Boussinesq equations with a class of large initial data. arXiv:1909.08360. |
| [20] | C.Miao and X.Zheng, On the global well‐posedness for the Boussinesq system with horizontal dissipation, Commun. Math. Phys.321 (2013), 33-67. · Zbl 1307.35233 |
| [21] | H.Qiu, Y.Du, and Z.Yao, Serrin‐type blow‐up criteria for 3D Boussinesq equations, Appl. Anal.89 (2010), 1603-1613. · Zbl 1387.35501 |
| [22] | H.Qiu, Y.Du, and Z.Yao, A blow‐up criterion for 3D Boussinesq equations in Besov spaces, Nonlinear Anal.73 (2010), 806-815. · Zbl 1193.76025 |
| [23] | X. M.Wang, Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number, Commun. Pure Appl. Math.60 (2007), no. 9, 1293-1318. · Zbl 1143.35087 |
| [24] | Z.Xiang, The regularity criterion of the weak solution to the 3D viscous Boussinesq equations in Besov spaces, Math. Methods Appl. Sci.34 (2011), 360-372. · Zbl 1205.35235 |
| [25] | H.Bessaih and B.Ferrario, The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion, J. Differ. Equ.262 (2017), no. 3, 1822-1849. · Zbl 1354.35087 |
| [26] | NBoardman, R.Ji, H.Qiu, and J.Wu, Uniqueness of weak solutions to the Boussinesq equations without thermal diffusion, Commun. Math. Sci.17 (2019), no. 6, 1595-1624. · Zbl 1433.35260 |
| [27] | T.Hmidi, On a maximum principle and its application to logarithmically critical Boussinesq system, Anal. PDE4 (2011), 247-284. · Zbl 1264.35173 |
| [28] | Q.Jiu and H.Yu, Global well‐posedness for 3D generalized Navier‐Stokes‐Boussinesq equations, Acta Math. Appl. Sin. Engl. Ser.32 (2016), no. 1, 1-16. · Zbl 1334.76033 |
| [29] | M.Kaya and A. O.Çelebi, Global attractor for the regularized Bénard problem, Appl. Anal.93 (2014), no. 9, 1989-2001. · Zbl 1302.35063 |
| [30] | H.Qiu, Y.Du, and Z.Yao, Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces, Math Methods Appl. Sci.36 (2013), no. 1, 86-98. · Zbl 1257.35153 |
| [31] | Z.Xiang and W.Yan, Global regularity of solutions to the Boussinesq equations with fractional diffusion, Adv. Differ. Equ.18 (2013), no. 11‐12, 1105-1128. · Zbl 1280.35117 |
| [32] | K.Yamazaki, On the global regularity of [( N \]\)‐dimensional generalized Boussinesq system, Appl. Math.60 (2015), no. 2, 109-133. · Zbl 1363.35065 |
| [33] | W.Yang, Q.Jiu, and J.Wu, The 3D incompressible Boussinesq equation with fractional partial dissipation, Commun. Math. Sci.16 (2018), no. 3, 617-633. · Zbl 1404.35373 |
| [34] | Z.Ye, A note on global well‐posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci. Ser. B Engl. Ed.35 (2015), no. 1, 112-120. · Zbl 1340.35277 |
| [35] | J. D.Gibbon, The three‐dimensional Euler equations: where do we stand?Phys. D237 (2008), no. 14‐17, 1894-1904. · Zbl 1143.76389 |
| [36] | A.Ilyin, A.Kostianko, and S.Zelik, Sharp upper and lower bounds of the attractor dimension for 3D damped Euler‐Bardina equations. arXiv:2106.09077. · Zbl 1490.35273 |
| [37] | J. C.Robinson, J. L.Rodrigo, and W.Sadowski, The three‐dimensional Navier‐Stokes equations: classical theory, Cambridge University Press, Cambridge, 2016. · Zbl 1358.35002 |
| [38] | R.Temam, Navier‐Stokes equations, theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2, North‐Holland Publishing Co., Amsterdam, 1979. · Zbl 0426.35003 |
| [39] | C.Gal and Y.Guo, Inertial manifolds for the hyperviscous Navier‐Stokes equations, J. Differ. Equ.265 (2018), 4335-4374. · Zbl 1397.35041 |
| [40] | C. T.Anh, L. T.Thuy, and L. T.Tinh, Long‐time behavior of a family of incompressible three‐dimensional Leray‐ [( \alpha \]\)‐like models, Bull. Korean Math. Soc.58 (2021), no. 5, 1109-1127. · Zbl 1477.35152 |
| [41] | D.Barbato, F.Morandin, and M.Romito, Global regularity for a slightly supercritical hyperdissipative Navier‐Stokes system, Anal. PDE7 (2014), no. 8, 2009-2027. · Zbl 1309.76053 |
| [42] | J.Fan, F.Li, and G.Nakamura, Global solutions to the Navier‐Stokes‐ [( \omega \]\) and related models with rough initial data, Z. Angew. Math. Phys.65 (2014), 301-314. · Zbl 1293.35203 |
| [43] | M.Holst, E.Lunasin, and G.Tsogtgerel, Analysis of a general family of regularized Navier‐Stokes and MHD models, J. Nonlinear Sci.20 (2010), 523-567. · Zbl 1207.35241 |
| [44] | J. L.Lions, Quelques Méthodes de Resolution des Problèmes aux Limites Non Linéaires, Vol. 1, Dunod, Paris, 1969. · Zbl 0189.40603 |
| [45] | E.Olson and E. S.Titi, Viscosity versus vorticity stretching: global well‐posedness for a family of Navier‐Stokes‐alpha‐like models, Nonlinear Anal.66 (2007), 2427-2458. · Zbl 1110.76011 |
| [46] | T.Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier‐Stokes equation, Anal. PDE2 (2009), 361-366. · Zbl 1190.35177 |
| [47] | C.Zhu, Global attractor for the time discretized modified three‐dimensional Bénard systems, Miskolc. Math. Notes22 (2021), no. 1, 457-483. · Zbl 1474.35131 |
| [48] | V. V.Chepyzhov and M. I.Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris Sér. I Math.321 (1995), no. 10, 1309-1314. · Zbl 0843.35038 |
| [49] | V. V.Chepyzhov and M. I.Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl.76 (1997), no. 10, 913-964. · Zbl 0896.35032 |
| [50] | A.Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differ. Equ.21 (2009), no. 2, 249-268. · Zbl 1176.35035 |
| [51] | O. V.Kapustyan and J.Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat, J. Bifur. Chaos Appl. Sci Engrg.20 (2010), no. 9, 2723-2734. · Zbl 1202.37023 |
| [52] | S.Lu, Strongly compact strong trajectory attractors for evolutionary systems and their applications, Asymptot. Anal.2022 (2022), 1-63, DOI 10.3233/ASY‐221805. |
| [53] | A.Cheskidov and C.Foias, On global attractors of the 3D Navier‐Stokes equations, J. Differ. Equ.231 (2006), 714-754. · Zbl 1113.35140 |
| [54] | A.Cheskidov and S.Lu, The existence and the structure of uniform global attractors for nonautonomous reaction‐diffusion systems without uniqueness, Discrete Contin. Dyn. Syst. Ser. S2 (2009), 55-66. · Zbl 1201.35051 |
| [55] | A.Cheskidov and S.Lu, Uniform global attractor of the 3D Navier‐Stokes equations, Adv. Math.267 (2014), 277-306. · Zbl 1304.35113 |
| [56] | T.Kato and G.Ponce, Commutator estimates and the Euler and Navier‐Stokes equations, Comm. Pure Appl. Math.41 (1988), 891-907. · Zbl 0671.35066 |
| [57] | P.Constantin and C.Foias, Navier‐Stokes equations, University of Chicago Press, 1988. · Zbl 0687.35071 |
| [58] | A.Behzadan and M.Holst, Multiplication in Sobolev spaces, revisited, Ark. Mat.59 (2021), no. 2, 275-306. · Zbl 1490.46027 |
| [59] | J. L.Lions and E.Magenes, Problèmes aux limites non homogènes et Applications, Vol. 1, Dunod, Paris, 1968. · Zbl 0165.10801 |
| [60] | Y.Dai, W.Hu, J.Wu, and B.Xiao, The Littlewood‐Paley decomposition for periodic functions and applications to the Boussinesq equations, Anal. Appl.18 (2020), no. 4, 639-682. · Zbl 1444.42023 |
| [61] | T.Kato and G.Ponce, Well posedness of the Euler and Navier‐Stokes equations in the Lebesgue spaces [( {L}_s^p\left({\mathbb{R}}^2\right) \]\), Rev. Mat. Iberoam.2 (1986), no. 1‐2, 73-88. · Zbl 0615.35078 |
| [62] | V.Chepyzhov and M.Vishik, Attractors for equations of mathematical physics, Vol. 49, American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. · Zbl 0986.35001 |
| [63] | S.Lu, Attractors for nonautonomous reaction‐diffusion systems with symbols without strong translation compactness, Asymptot. Anal.54 (2007), 197-210. · Zbl 1139.35028 |
| [64] | R.Rosa, The global attractor for the 2D Navier‐Stokes flow on some unbounded domains, Nonlinear Anal.32 (1998), 71-85. · Zbl 0901.35070 |
| [65] | S.Lu, H.Wu, and C.Zhong, Attractors for nonautonomous 2D Navier‐Stokes equations with normal external forces, Discrete Contin. Dyn. Syst.13 (2005), 701-719. · Zbl 1083.35094 |
| [66] | S.Lu, Attractors for nonautonomous 2D Navier‐Stokes equations with less regular normal forces, J. Differ. Equ.230 (2006), 196-212. · Zbl 1112.35029 |
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