Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

Abstract

Abstract. We establish global existence and uniqueness theorems for the twodimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary "stream-function" associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.

On the attractor for the semi-dissipative Boussinesq equations

“…The requirements on the initial data were weakened in [9], where the following theorem was proven (see also [26,36] for related results and additional discussion).…”

“…This means that the usual notion of global attractors no longer applies. In this work, we propose a new type of global attractor-which we call a weak sigma-attractor-that captures the large-time dynamics of a well-known semi-dissipative model for ocean flows known as the semi-dissipative 2D Boussinesq system [36]. The notion of the weak sigma-attractor, which in some respects is a hybrid of the attractors of the 2D and 3D Navier-Stokes equations [23,25], exploits the fact that while only part of the system is dissipative, the other part has a hyperbolic structure that gives rise to certain conserved quantities which in turn correspond to invariant sets in the phase space.…”

Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

“…The case of partial anisotropic dissipation has been considered in several settings (see for instance [1,8,16,29,28]). For the global smooth solutions to the damped Boussinesq equations with small initial datum, we refer the readers to the recent works [2,42].…”

“…It is worth remarking that there are several works concerning the global regularity for the 2D Boussinesq equations with logarithmical dissipation (see, e.g., [20,12,26]). Many other interesting recent results on the Boussinesq equations can be found, with no intention to be complete (see, e.g., [11,14,24,27,28,31,43,46] and the references therein).…”

Existence and Uniqueness of Solutions to Heat Equations with Hysteresis Coupled with Navier–Stokes Equations in 2D and 3D

“…Mathematically, the problem (P) is the Boussinesq system with hysteresis formulated in a quasivariational inequality, which represents the phenomenon by thermostat devices. Boussinesq systems are dealt with in many works such as Morimoto [28], Fukao and Kenmochi [9], Kubo [23], Fukao and Kubo [11,12], Sobajima et al [31], Larios et al [24], Li and Xu [25], Miao and Zheng [26], Fukao and Kenmochi [10] and the author [35]. In particular, the mathematical theory of 2D and 3D Navier-Stokes equations is discussed in detail in Temam [33,34] and Constantin and Foias [5].…”