# zbMATH — the first resource for mathematics

On the global well-posedness for Boussinesq system. (English) Zbl 1111.35032
The 2D Cauchy problem to the Boussinesq system is studied in the paper
\begin{aligned} &\frac{\partial v}{\partial t}-\Delta v+(v\cdot\nabla)v+\nabla p=\theta(0,1), \quad x\in \mathbb R^2,\quad t>0,\\ &\frac{\partial \theta}{\partial t}+v\cdot\nabla \theta=0, \quad x\in \mathbb R^2,\quad t>0, \\ &\text{div}\,(v)=0, \quad v| _{t=0}=v_0(x),\quad \theta| _{t=0}=\theta_0(x),\quad x\in \mathbb R^2. \end{aligned}\tag{1} Here $$v=(v_1,v_2)$$ is the velocity, $$p$$ is the pressure and $$\theta$$ is the temperature.
It is proved that problem (1) has a unique global solution $$(v,p,\theta)$$ if $$v_0\in L_2\cap B^{-1}_{\infty,1}$$ and $$\theta_0\in B^{0}_{2,1}$$. The important parts of the proof are estimates to solutions of the Stokes system
\begin{aligned} &\frac{\partial u}{\partial t}-\Delta u+(v\cdot\nabla)u+\nabla p=f, \quad \text{div}\,(v)=0, \quad x\in \mathbb R^2,\quad t>0,\\ &u| _{t=0}=u_0(x) \end{aligned} . and of the transport equation
\begin{aligned} &\frac{\partial a}{\partial t}+(v\cdot\nabla)a=f,\\ &a| _{t=0}=a_0(x). \end{aligned}

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
  Bony, J.-M., Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. école sup., 14, 209-246, (1981) · Zbl 0495.35024  Beale, J.T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for 3D Euler equations, Comm. math. phys., 94, 61-66, (1984) · Zbl 0573.76029  Chae, D., Global regularity for the 2D Boussinesq equations with partial viscous terms, Adv. math., 203, 2, 497-513, (2006) · Zbl 1100.35084  Chemin, J.-Y., Perfect incompressible fluids, Oxford lecture ser. math. appl., vol. 14, (1998), Clarendon New York  Chemin, J.-Y., Théorèmes d’unicité pour le système de navier – stokes tridimensionnel, J. anal. math., 77, 27-50, (1999) · Zbl 0938.35125  Chemin, J.-Y.; Gallagher, I., On the global wellposedness of the 3D navier – stokes equations with large initial data, arXiv:  Córdoba, D.; Fefferman, C.; De La Llave, R., On squirt singularities in hydrodynamics, SIAM J. math. anal., 36, 204-213, (2004) · Zbl 1078.76018  Danchin, R., Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. differential equations, 9, 3-4, 353-386, (2004) · Zbl 1103.35085  Danchin, R., A few remarks on the camassa – holm equation, Differential integral equations, 14, 8, 953-988, (2001) · Zbl 1161.35329  T. Hmidi, S. Keraani, Global well-posedness result for two-dimensional Boussinesq equations, preprint, 2006 · Zbl 1122.35135  Hou, T.Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete contin. dyn. syst., 12, 1-12, (2005) · Zbl 1274.76185  Moffatt, H.K., Some remarks on topological fluid mechanics, (), 3-10 · Zbl 1100.76500  Pedloski, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York  E, W.; Shu, C.-W., Small scale structures in Boussinesq convection, Phys. fluids, 6, 1, 49-58, (1994) · Zbl 0822.76087  Vishik, M., Hydrodynamics in Besov spaces, Arch. ration. mech. anal., 145, 197-214, (1998) · Zbl 0926.35123
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.