Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global well-posedness of strong solutions to a tropical climate model

Pages: 4495 - 4516, Volume 36, Issue 8, August 2016      doi:10.3934/dcds.2016.36.4495

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Jinkai Li - Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel (email)
Edriss Titi - Department of Mathematics, Texas A&M University, 3368-TAMU, College Station, TX 77843-3368, United States (email)

Abstract: In this paper, we consider the Cauchy problem to the TROPICAL CLIMATE MODEL derived by Frierson--Majda--Pauluis in [15], which is a coupled system of the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature. The system considered in this paper has viscosities in the momentum equations, but no diffusivity in the temperature equation. We establish here the global well-posedness of strong solutions to this model. In proving the global existence of strong solutions, to overcome the difficulty caused by the absence of the diffusivity in the temperature equation, we introduce a new velocity $w$ (called the pseudo baroclinic velocity), which has more regularities than the original baroclinic mode of the velocity. An auxiliary function $\phi$, which looks like the effective viscous flux for the compressible Navier-Stokes equations, is also introduced to obtain the $L^\infty$ bound of the temperature. Regarding the uniqueness, we use the idea of performing suitable energy estimates at level one order lower than the natural basic energy estimates for the system.

Keywords:  Tropical atmospheric dynamics, primitive equations, global well-posedness, strong solutions, logarithmic Gronwall inequality.
Mathematics Subject Classification:  Primary: 35D35, 76D03; Secondary: 86A10.

Received: April 2015;      Revised: October 2015;      Available Online: March 2016.