Global Well-Posedness of an Inviscid Three-Dimensional Pseudo-Hasegawa-Mima Model

Abstract

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-Bénard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations.MSC Subject Classifications: 35Q35, 76B03, 86A10.

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

“…Similar inequalities have been established in [4] and [7] for the 2D case. We follow here the ideas of the proof presented in [4]. for any R, λ > 0, and for some constant C N,p,λ > 0.…”

Global strong solutions for the three-dimensional Hasegawa-Mima model with partial dissipation

“…Now we are ready to state the main result of the paper: the global existence, uniqueness, and continuous dependence on initial data of strong solutions for our model (1.7)-(1.9). 3 , then system (1.7)-(1.9) admits a unique strong solution (u, w) tr on [0, T ] in the sense of Definition 1.1 satisfying the initial condition (u(0), w(0)) tr = (u 0 , w 0 ) tr . Moreover, the energy equality is valid for every t ∈ [0, T ]:…”

“…The three-dimensional inviscid Hasegawa-Mima equations can be written as (cf. [2,3,10,11,17,22]): where J(f, g) = ∂f ∂x ∂g ∂y − ∂f ∂y ∂g ∂x is the Jacobian and ∆ h = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is the horizontal Laplacian. System (1.1)-(1.2) describes the coupling of the drift modes to the ionacoustic waves that propagate along the magnetic field.…”

“…Nevertheless, by adding the full viscosity to (1.1)-(1.2), Zhang and Guo [22] proved the global regularity and the existence of global attractors for a viscous and forced 3D Hasegawa-Mima model using standard tools from the theory of Navier-Stokes equations. On the other hand, Cao, Farhat and Titi [3] proposed and studied an inviscid three-dimensional modified version of (1.1)-(1.2), the pseudo-Hasegawa-Mima equations:…”

Recent Advances Concerning Certain Class of Geophysical Flows

“…Thanks to the above proposition, we can control the main part of the quantity v 2 ∞ . To this end, we use a logarithmic Sobolev limiting inequality, generalizing the classical Brézis-Gallout-Wainger inequality [7,8] (see also [9]), stated in the next proposition. This logarithmic inequality serves as a bridge between the L q norms and the L ∞ norm.…”