On the well-posedness of magnetohydrodynamics system with Hall and ion-slip in critical spaces
- PDF / 497,340 Bytes
- 24 Pages / 547.087 x 737.008 pts Page_size
- 64 Downloads / 184 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
On the well-posedness of magnetohydrodynamics system with Hall and ion-slip in critical spaces Lvqiao Liu Abstract. In this paper, we establish the global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in critical space by constructing the current function J = ∇ × B as an additional unknown. Meanwhile, we consider the long-time behavior of the solutions and get some decay estimates. Mathematics Subject Classification. 35Q35, 76D03. Keywords. Hall-MHD, Global existence, Critical space, Decay estimates, Ion-slip.
1. Introduction and main result In this paper, we consider the following three-dimensional incompressible magnetohydrodynamics equations with Hall and ion-slip effects (Hall-MHDs) in R3 : ⎧ ⎪ ⎨ ut + u · ∇u − μΔu + ∇π = B · ∇B, Bt − ∇ × ((u − ε∇ × B) × B) − νΔB = κ∇ × [B × (B × (∇ × B))], (1.1) ⎪ ⎩ div u = div B = 0, with initial data (u(0, x), B(0, x)) = (u0 (x), B0 (x)) , |B|2 2 ),
(1.2)
u = (u1 , u2 , u3 ) is the fluid velocity field, B = (B1 , B2 , B3 ) is for x ∈ R3 and t ≥ 0, where π = (P + the magnetic field, and P is the scalar pressure of the fluid, respectively. The Hall term ∇ × ((∇ × B) × B) is for the Hall effect, and ∇×[B×(∇×B)] is for the ion-slip effect. The parameters μ, ν, λ and κ denote the viscous coefficient, the resistivity coefficient, the Hall effect coefficient and the ion-slip effect coefficient, respectively. When κ = 0 and ε = 0, the system (1.1) reduces to the Hall-MHD system, which represents the momentum conservation equation for the plasma fluid. Considering its physical applications and mathematical significance, Hall-MHD system has been considered by many researchers. The authors in [1] had derived the Hall-MHD equations from a two-fluid Euler–Maxwell system for electrons and ions by some scaling limit arguments, which also provided a kinetic formulation for the Hall-MHD. Then, the global existence of weak solutions, local classical solutions, blow-up criteria and the global existence of smooth solutions and long-time behavior are obtained in [6,7,23–25]. Later, the authors in [3,23,26] established the well-posedness of strong solutions with improved regularity conditions for initial data in sobolev or Besov spaces. Recently, Danchin and Tan establish the global well-posedness of the Hall-MHD system in critical spaces by a new observation in [14,15]. In addition, if ε = 0 and κ = 0, the system (1.1) reduces to the classical magnetohydrodynamics system [5], which describes the plasma form a nonlinear system that couples Navier–Stokes equations with 0123456789().: V,-vol
134
Page 2 of 24
L. Liu
ZAMP
Maxwell’s equations. The system has caught a great deal of attention by physicists and mathematicians due to its physical importance. In fact, for the classical magnetohydrodynamics system: ⎧ ⎪ ⎨ ut + u · ∇u − μΔu + ∇P = (∇ × B) × B, Bt − ∇ × (u × B) − νδB = 0, (1.3) ⎪ ⎩ div u = 0, which are invariant for all λ > 0 by the rescaling u(t, x) λu(λ2 t, λx),
P (t, x) λ2 P (λ2 t, λx),
B(t, x) B(
Data Loading...
