Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation

Bahouri, 2011, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 Brandolese, 2012, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364, 5057, 10.1090/S0002-9947-2012-05432-8 Cao, 2013, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 985, 10.1007/s00205-013-0610-3 Chandrasekhar, 1981 Chae, 2006, Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math., 203, 497, 10.1016/j.aim.2005.05.001 Chae, 1999, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155, 55, 10.1017/S0027763000006991 Chen, 2007, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271, 821, 10.1007/s00220-007-0193-7 Constantin, 2008, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1103, 10.1016/j.anihpc.2007.10.001 Constantin, 2009, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 26, 159, 10.1016/j.anihpc.2007.10.002 Córdoba, 2004, A maximum principle applied to the quasi-geostrophic equations, Comm. Math. Phys., 249, 511, 10.1007/s00220-004-1055-1 Danchin, 2013, Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Amer. Math. Soc., 141, 1979, 10.1090/S0002-9939-2012-11591-6 Danchin, 2008, Le théorème de Leray et le théorème de Fujita–Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 261, 10.24033/bsmf.2557 Danchin, 2009, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290, 1, 10.1007/s00220-009-0821-5 Danchin, 2011, Global existence results for the anisotropic Boussinesq system in dimensional two, Math. Models Methods Appl. Sci., 21, 421, 10.1142/S0218202511005106 Hmidi, 2011, On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4, 247, 10.2140/apde.2011.4.247 Hmidi, 2009, On the global well-posedness of the two-dimensional Boussinesq system with a zero viscosity, Indiana Univ. Math. J., 58, 1591, 10.1512/iumj.2009.58.3590 Hmidi, 2011, Global well-posedness for Euler–Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36, 420, 10.1080/03605302.2010.518657 Hmidi, 2010, On the global well-posedness of the Euler–Boussinesq system with fractional dissipation, Phys. D, 239, 1387, 10.1016/j.physd.2009.12.009 Hou, 2005, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. Ser. A, 12, 1, 10.3934/dcds.2005.12.1 Jiu Ju, 2005, The maximal principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255, 161, 10.1007/s00220-004-1256-7 Larios Liu, 2010, Local well-posedness and blow-up criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12, 280, 10.1007/s00021-008-0286-x Majda, 2002 Miao, 2011, On the global well-posedness of a class of Boussinesq–Navier–Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18, 707, 10.1007/s00030-011-0114-5 Pedlosky, 1987 Vishik, 1998, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145, 197, 10.1007/s002050050128 Wang, 2011, A frequency localized maximum principle applied to the 2D quasi-geostrophic equation, Comm. Math. Phys., 301, 105, 10.1007/s00220-010-1144-2 J. Wu, X. Xu, Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, preprint. Wu, 2012, Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich's type data, J. Differential Equations, 253, 100, 10.1016/j.jde.2012.02.025 Xu, 2010, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677, 10.1016/j.na.2009.07.008 Yudovich, 1963, Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3, 1032