Global regularity for the 2D Boussinesq equations with partial viscosity terms.(English)Zbl 1100.35084
Summary: We prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed byH. K. Moffatt [R. L. Ricca (ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 3–10 (2001;Zbl 1100.76001)].
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D09 | Viscous-inviscid interaction |
Keywords:
Boussinesq equations;vanishing viscosity limit;vanishing diffusivity limit;global regularityCitations:
Zbl 1100.76001Cite
Full Text:DOI
References:
| [1] | Brezis, H.; Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5, 7, 773-789 (1980) ·Zbl 0437.35071 |
| [2] | J.R. Cannon, E. DiBenedetto, The initial problem for the Boussinesq equations with data in \(L^p\); J.R. Cannon, E. DiBenedetto, The initial problem for the Boussinesq equations with data in \(L^p\) ·Zbl 0429.35059 |
| [3] | Chae, D.; Nam, H.-S., Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh, Sect. A, 127, 5, 935-946 (1997) ·Zbl 0882.35096 |
| [4] | 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 |
| [5] | Córdoba, D.; Fefferman, C.; De La LLave, R., On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36, 1, 204-213 (2004) ·Zbl 1078.76018 |
| [6] | W.E., C. Shu, Small scale structures un Boussinesq convection, Phys. Fluids 6 (1994) 48-54.; W.E., C. Shu, Small scale structures un Boussinesq convection, Phys. Fluids 6 (1994) 48-54. |
| [7] | Engler, H., An alternative proof of the Brezis-Wainger inequality, Commun. Partial Differential Equations, 14, 4, 541-544 (1989) ·Zbl 0688.46016 |
| [8] | Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Continuous Dynamical System, 12, 1, 1-12 (2005) ·Zbl 1274.76185 |
| [9] | A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.; A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003. ·Zbl 1278.76004 |
| [10] | 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: Kluwer Academic Publishers Dordrecht, The Netherlands), 3-10 ·Zbl 1100.76500 |
| [11] | Y. Taniuchi, A note on the blow-up criterion for the inviscid 2-D Boussinesq equations, in: R. Salvi (Ed.), The Navier-Stokes Equations: Theory and Numerical Methods, Lecture Notes in Pure and Applied Mathematics, vol. 223, 2002, pp. 131-140.; Y. Taniuchi, A note on the blow-up criterion for the inviscid 2-D Boussinesq equations, in: R. Salvi (Ed.), The Navier-Stokes Equations: Theory and Numerical Methods, Lecture Notes in Pure and Applied Mathematics, vol. 223, 2002, pp. 131-140. ·Zbl 0991.35070 |
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