Anisotropic Inequalities Research Articles

Stability and exponential decay of the magnetic Bénard system with horizontal dissipation

This article is concerned with the global stability problem for the two-dimensional magnetic Bénard system. The focus here involves only horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity. The lack of vertical dissipation makes it difficult to study the stability and long-time behavior of the magnetic Bénard system. To handle these, we use a new method to decompose the velocity u, magnetic field b, and temperature Θ into the horizontal average (u‾,b‾,Θ‾) and the oscillation part (u˜,b˜,Θ˜). Then we establish the global stability by using several anisotropic inequalities when the initial data are small in H2(Ω). Moreover, we mathematically show that the H1-norm of the oscillation part eventually decays exponentially to zero. Furthermore, the solution (u,b,Θ) approaches the horizontal average (u‾,b‾,Θ‾) asymptotically.

Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is \({\mathbb {R}}^2\), the stability problem in a Sobolev setting remains open. When the spatial domain is \({\mathbb {T}}\times {\mathbb {R}}\), this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity u and temperature \(\theta \) into the horizontal average \(({\bar{u}}, {\bar{\theta }})\) and the corresponding oscillation \(({\widetilde{u}}, {\widetilde{\theta }})\), and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space \(H^2\). In addition, we prove that the oscillation \(({\widetilde{u}}, {\widetilde{\theta }})\) decays exponentially to zero in \(H^1\) and \((u, \theta )\) converges to \(({\bar{u}}, {\bar{\theta }})\). This result reflects the stratification phenomenon of buoyancy-driven fluids.

On two-dimensional large-scale primitive equations in oceanic dynamics (II)

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially. Here the depth of the ocean is positive but not always a constant. By Faedo-Galerkin method and anisotropic inequalities, the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained. Moreover, by studying the asymptotic behavior of solutions for the above problem, the energy is exponential decay with time is proved.