On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

Abstract

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u 0 and magnetic field B 0 in critical regularity spaces.In the case where u 0 , B 0 and the current J 0 := ∇ × B 0 belong to the homo-and are small enough, we establish a global result and the conservation of higher regularity. If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided u 0 , B 0 and J 0 are small enough in the larger Besov spaceḂ 1 2 2,r , r ≥ 1. If r = 1, then we also establish the local existence for large data, and exhibit continuation criteria for solutions with critical regularity.Our results rely on an extended formulation of the Hall-MHD system, that has some similarities with the incompressible Navier-Stokes equations.

Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

“…) and H s (R 3 ) are not embedded into each other when s > 0, Theorem 1.2 is different from the corresponding local well-posedness result (for large initial data) in [11].…”

“…)) was proved in [11]. Although their initial data is less regular than that in Theorem 1.1, their proof relies on (1.13) and thus cannot be applied to the general case.…”

“…Therefore, it seems that σ in Theorem 1.1 cannot be reduced to 0. Recently, Danchin and Tan in [11] observed that if µ = ν, regarding v := u−κ∇×B as a new unknown, the system (1.1) can be rewritten as (one can see [11, P36-37] for derivation):…”

“…Fan, Huang and Nakamura [12] established the global well-posedness of (1.1) with axisymmetric initial data. The authors in [11,16] proved the global well-posedness of (1.1) for small initial data in critical Besov spaces by means of some new observations. Regarding the large time behavior of solutions to (1.1), Chae and Schonbek [7] obtained by using the Fourier splitting techniques that if in addition (u 0 , B 0 ) ∈ L 1 (R 3 ), the global small solutions of (1.1) in H m (R 3 ) with m ≥ 3 satisfy…”

“…

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system with initial data (uIn particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce σ to 0 with the aid of the new formulation of the Hall-MHD system observed by [11]. Moreover, we establish the global well-posedness in H 1 2 +σ (R 3 ) × H 3 2 (R 3 ) with σ ∈ (0, 2) for small initial data and get the optimal time-decay rates of solutions.

…”