Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

Abstract

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces B s p,r (R d), d ≥ 2, with 1 < p < +∞ and under the Lipschitz condition s > 1 + d/p and r ∈ [1, +∞], or s = 1 + d/p and r = 1. A key ingredient is the reformulation of the system via the so-called Elsässer variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when d = 2, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients.

Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

“…It is worth to notice that, adapting the relative entropy arguments presented in Subsection 4.3 of [11], we can replace (in the statement above) the C 1 T (L 2 ) requirement for δ̺ ε and δu ε with the C 0 T (L 2 ) regularity. However, one needs to pay an additional L 2 assumption on the densities.…”

“…Actually, equations (2.9) are locally well-posedness in all B s p,r Besov spaces, under the condition (1.4). We refer to [11] where the authors apply the standard Littlewood-Paley machinery to the quasi-homogeneous ideal MHD system to recover local in time well-posedness in spaces B s p,r for any 1 < p < +∞. The case p = +∞ was reached in [12] with a different approach based on the vorticity formulation of the momentum equation.…”

“…We have also to mention [11], where the authors rigorously prove the convergence of the ideal magnetohydrodynamics (MHD) equations towards a quasi-homogeneous type system (see also [10] in this respect). Their method relies on a relative entropy inequality for the primitive system that allows to treat also the inviscid limit but requires well-prepared initial data.…”

“…The strategy consists in making use of the algebraic structure hidden behind the system (recasted as a wave system) to reveal strong convergence properties for special quantities: in our case, γ ε := curl (̺ ε u ε ) (see Section 4.2 below). We refer also to [11], for a different approach based on relative entropy inequalities for the primitive equations to prove the convergence towards the limit system. Now, once the limit system is rigorously depicted, one could address its well-posedness issue: it is worth noticing that system (1.6) is not globally well-posed even in two dimensions.…”

“…In Section 4, we study the singular perturbation problem, establishing constraints that the limit points have to satisfy and proving the convergence to the quasi-homogeneous Euler system thanks to a compensated compactness technique. In Section 5 we review, for the limit system (1.6), the results presented in [11] and [12], and we explicitly derive the lifespan of solutions to equations (1.6) (see relation (2.10)).…”