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Generalized 2D Euler-Boussinesq equations with a singular velocity. (English) Zbl 1291.35221
Summary: This paper studies the global (in time) regularity problem concerning a system of equations generalizing the two-dimensional incompressible Boussinesq equations. The velocity here is determined by the vorticity through a more singular relation than the standard Biot-Savart law and involves a Fourier multiplier operator. The temperature equation has a dissipative term given by the fractional Laplacian operator \(\sqrt{-{\Delta}}\). We establish the global existence and uniqueness of solutions to the initial-value problem of this generalized Boussinesq equations when the velocity is “double logarithmically” more singular than the one given by the Biot-Savart law. This global regularity result goes beyond the critical case. In addition, we recover a result of D. Chae et al. [Arch. Ration. Mech. Anal. 202, No. 1, 35–62 (2011; Zbl 1266.76010)] when the initial temperature is set to zero.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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[1] Adhikari, D.; Cao, C.; Wu, J., The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differential Equations, 249, 1078-1088, (2010) · Zbl 1193.35144
[2] Adhikari, D.; Cao, C.; Wu, J., Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations, 251, 1637-1655, (2011) · Zbl 1232.35111
[3] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier analysis and nonlinear partial differential equations, (2011), Springer-Verlag · Zbl 1227.35004
[4] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0344.46071
[5] Cannon, J. R.; DiBenedetto, E., The initial value problem for the Boussinesq equations with data in \(L^p\), (Lecture Notes in Math., vol. 771, (1980), Springer-Verlag Berlin), 129-144
[6] Cao, C.; Wu, J., Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 985-1004, (2013) · Zbl 1284.35140
[7] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[8] Chae, D.; Constantin, P.; Wu, J., Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202, 35-62, (2011) · Zbl 1266.76010
[9] Chae, D.; Wu, J., The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230, 1618-1645, (2012) · Zbl 1248.35156
[10] Chemin, J.-Y., Fluides parfaits incompressibles, Astérisque, 230, (1995) · Zbl 0829.76003
[11] 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
[12] Constantin, P.; Doering, C. R., Infinite Prandtl number convection, J. Stat. Phys., 94, 159-172, (1999) · Zbl 0935.76083
[13] Constantin, P.; Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 1289-1321, (2012) · Zbl 1256.35078
[14] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 511-528, (2004) · Zbl 1309.76026
[15] 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
[16] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 421-457, (2011) · Zbl 1223.35249
[17] E, W.; Shu, C., Small-scale structures in Boussinesq convection, Phys. Fluids, 6, 49-58, (1994) · Zbl 0822.76087
[18] Gill, A. E., Atmosphere-Ocean dynamics, (1982), Academic Press London
[19] Hmidi, T., On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4, 247-284, (2011) · Zbl 1264.35173
[20] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249, 2147-2174, (2010) · Zbl 1200.35228
[21] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36, 420-445, (2011) · Zbl 1284.76089
[22] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185
[23] Larios, A.; Lunasin, E.; Titi, E. S., Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid Voigt-a regularization, (25 Oct. 2010)
[24] Majda, A. J., Introduction to PDEs and waves for the atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, (2003), AMS/CIMS · Zbl 1278.76004
[25] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, (2001), Cambridge University Press
[26] Miao, C.; Wu, J.; Zhang, Z., Littlewood-Paley theory and its applications in partial differential equations of fluid dynamics, (2012), Science Press Beijing, China, (in Chinese)
[27] 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
[28] Miao, C.; Zheng, X., On the global well-posedness for Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321, 33-67, (2013) · Zbl 1307.35233
[29] Miao, C.; Zheng, X., Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl., (2013)
[30] Moffatt, H. K., Some remarks on topological fluid mechanics, (Ricca, R. L., An Introduction to the Geometry and Topology of Fluid Flows, (2001), Kluwer Academic Publishers Dordrecht, The Netherlands), 3-10 · Zbl 1100.76500
[31] Ohkitani, K., Comparison between the Boussinesq and coupled Euler equations in two dimensions, Tosio Kato’s Method and Principle for Evolution Equations in Mathematical Physics, Sapporo, 2001, Surikaisekikenkyusho Kokyuroku No. 1234, 127-145, (2001)
[32] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
[33] Runst, T.; Sickel, W., Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations, (1996), Walter de Gruyter Berlin, New York · Zbl 0873.35001
[34] Stein, E., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501
[35] Triebel, H., Theory of function spaces II, (1992), Birkhäuser Verlag · Zbl 0763.46025
[36] Wu, J., Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263, 803-831, (2006) · Zbl 1104.35037
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