# zbMATH — the first resource for mathematics

On the global well-posedness of the Euler-Boussinesq system with fractional dissipation. (English) Zbl 1194.35329
Summary: We study the global well-posedness of the Euler-Boussinesq system with the term dissipation $$|D|^{\alpha }$$ on the temperature equation. We prove that for $$\alpha >1$$ the coupled system has a global unique solution for initial data with critical regularities.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
Full Text:
##### References:
 [1] Clavin, P., Instabilities and nonlinear patterns of overdriven detonations in gases, (), 49-97 · Zbl 1271.76102 [2] Woyczyński, W.A., Lévy processes in the physical sciences, Lévy processes, (2001), Birkhäuser Boston Boston, MA, pp. 241-266 · Zbl 0982.60043 [3] Droniou, J.; Imbert, C., Fractal first-order partial differential equations, Arch. ration. mech. anal., 182, 299-331, (2006) · Zbl 1111.35144 [4] Karch, G.; Woyczyński, W.A., Fractal Hamilton-Jacobi-KPZ equations, Trans. amer. math. soc., 360, 2423-2442, (2008) · Zbl 1136.35012 [5] () [6] Chae, D., Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. in math., 203, 2, 497-513, (2006) · Zbl 1100.35084 [7] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with yudovich’s type data, Comm. math. phys., 290, 1, 1-14, (2009) · Zbl 1186.35157 [8] R. Danchin, M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, arXiv:0809.4984v1 [math.AP] 29 Sep (2008) [9] Hmidi, T.; Keraani, S., On the global well-posedness of the Boussinesq system with zero viscosity, Adv. differential equations, 12, 4, 461-480, (2007) · Zbl 1154.35073 [10] Kiselev, A.; Nasarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. math., 167, 3, 445-453, (2007) · Zbl 1121.35115 [11] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (in press) · Zbl 1204.35063 [12] Vishik, M., Hydrodynamics in Besov spaces, Arch. ration. mech. anal., 145, 197-214, (1998) · Zbl 0926.35123 [13] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press [14] Bony, J.-M., Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. de l’ecole norm. sup., 14, 209-246, (1981) · Zbl 0495.35024 [15] Triebel, H., Theory of function spaces, (1983), Leipzig · Zbl 0546.46028 [16] Hmidi, T.; Keraani, S., On the global well-posedness of the Boussinesq system with zero viscosity, Indiana univ. math. J., 58, 4, 1591-1618, (2009) · Zbl 1178.35303 [17] Hmidi, T.; Keraani, S., Incompressible viscous flows in borderline Besov spaces, Arch. rational. mech. anal., 189, 283-300, (2008) · Zbl 1147.76014 [18] Chae, D., Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. anal., 38, 3-4, 339-358, (2004) · Zbl 1068.35097 [19] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. math. phys., 249, 511-528, (2004) · Zbl 1309.76026 [20] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein inequality and the 2D dissipative quasigeostrophic equation, Comm. math. phys., 271, 821-838, (2007) · Zbl 1142.35069 [21] Abidi, H.; Hmidi, T., On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. math. anal., 40, 1, 167-185, (2008) · Zbl 1157.76054 [22] Hmidi, T., Régularité höldérienne des poches de tourbillon visqueuses, J. math. pures appl. (9), 84, 11, 1455-1495, (2005) · Zbl 1095.35024 [23] Chemin, J.-Y., Théorèmes d’unicité pour le système de navier – stokes tridimensionnel, J. anal. math., 77, 27-50, (1999) · Zbl 0938.35125 [24] Hmidi, T.; Keraani, S., Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces, Adv. math., 214, 618-638, (2007) · Zbl 1119.76070 [25] Oliver, M., The Lagrangian averaged Euler equations as the short-time inviscid limit of the navier – stokes equations with Besov class data in $$\mathbb{R}^2$$, Commun. pure appl. anal., 1, 2, 221-235, (2002) · Zbl 1014.35076
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.