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Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation. (English) Zbl 1452.76030
Summary: In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of \({{\theta}_0}\). We also give a refined blowup criterion in the supercritical case.

MSC:
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
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[1] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wiss., vol. 343, (2011), Springer · Zbl 1227.35004
[2] Brandolese, L.; Schonbek, M., Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364, 10, 5057-5090, (2012) · Zbl 1368.35217
[3] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 3, 985-1004, (2013) · Zbl 1284.35140
[4] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (1981), Dover Publications, Inc. · Zbl 0142.44103
[5] Chae, D., Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math., 203, 2, 497-513, (2006) · Zbl 1100.35084
[6] Chae, D.; Kim, S.-K.; Nam, H.-S., Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155, 55-80, (1999) · Zbl 0939.35150
[7] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271, 821-838, (2007) · Zbl 1142.35069
[8] Constantin, P.; Wu, J., Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 6, 1103-1110, (2008) · Zbl 1149.76052
[9] Constantin, P.; Wu, J., Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 26, 1, 159-180, (2009) · Zbl 1163.76010
[10] Córdoba, A.; Córdoba, D., A maximum principle applied to the quasi-geostrophic equations, Comm. Math. Phys., 249, 511-528, (2004) · Zbl 1309.76026
[11] Danchin, R., Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Amer. Math. Soc., 141, 1979-1993, (2013) · Zbl 1283.35080
[12] Danchin, R.; Paicu, M., Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 2, 261-309, (2008) · Zbl 1162.35063
[13] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. Math. Phys., 290, 1-14, (2009) · Zbl 1186.35157
[14] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimensional two, Math. Models Methods Appl. Sci., 21, 3, 421-457, (2011) · Zbl 1223.35249
[15] Hmidi, T., On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4, 2, 247-284, (2011) · Zbl 1264.35173
[16] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero viscosity, Indiana Univ. Math. J., 58, 4, 1591-1618, (2009) · Zbl 1178.35303
[17] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36, 3, 420-445, (2011) · Zbl 1284.76089
[18] Hmidi, T.; Zerguine, M., On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239, 15, 1387-1401, (2010) · Zbl 1194.35329
[19] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. Ser. A, 12, 1, 1-12, (2005) · Zbl 1274.76185
[20] Jiu, Q.; Miao, C.; Wu, J.; Zhang, Z., The 2D incompressible Boussinesq equations with general critical dissipation
[21] Ju, N., The maximal principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255, 161-181, (2005) · Zbl 1088.37049
[22] Larios, A.; Lunasin, E.; Titi, E., Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid Voigt-α regularization
[23] Liu, X.; Wang, M.; Zhang, Z., Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12, 280-292, (2010) · Zbl 1195.76136
[24] Majda, A.; Bertozzi, A., Vorticity and incompressible flows, (2002), Cambridge University Press
[25] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18, 707-735, (2011) · Zbl 1235.76020
[26] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer New York · Zbl 0713.76005
[27] Vishik, M., Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145, 197-214, (1998) · Zbl 0926.35123
[28] Wang, H.; Zhang, Z., A frequency localized maximum principle applied to the 2D quasi-geostrophic equation, Comm. Math. Phys., 301, 105-129, (2011) · Zbl 1248.35211
[29] J. Wu, X. Xu, Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, preprint. · Zbl 1301.35115
[30] Wu, G.; Xue, L., Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich’s type data, J. Differential Equations, 253, 100-125, (2012) · Zbl 1305.35119
[31] Xu, X., Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677-681, (2010) · Zbl 1177.76024
[32] Yudovich, V., Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3, 1032-1066, (1963)
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