Global Well-Posedness of 3D Axisymmetric MHD System with Pure Swirl Magnetic Field
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Global Well-Posedness of 3D Axisymmetric MHD System with Pure Swirl Magnetic Field Yanlin Liu1,2
Received: 12 May 2017 / Accepted: 13 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: u0 = ur0 er + uθ0 eθ + uz0 ez , b0 = b0θ eθ . We prove that, this system is globally well-posed provided the scaling-invariant norms ruθ0 L∞ , r −1 b0θ 3 are L2 sufficiently small. Keywords Axisymmetric · MHD · Critical spaces · Mild solutions · Littlewood–Paley Theory Mathematics Subject Classification (2000) 35Q30 · 76D03
1 Introduction In this work, we investigate the global well-posedness of the 3D axisymmetric MHD system. In general, the 3D incompressible MHD system in the Euclidean coordinates reads ⎧ ∂t u + u · ∇u + ∇P = u + b · ∇b, (t, x) ∈ R+ × R3 ⎪ ⎪ ⎨ ∂t b + u · ∇b = b + b · ∇u, (1.1) div u = 0, div b = 0, ⎪ ⎪ ⎩ b|t=0 = b0 . u|t=0 = u0 , where u, P denote the velocity and scalar pressure of the fluid respectively, and b denotes the magnetic field. This system describes the time evolution of viscous electrically-conducting fluids moving through a prevalent magnetic fields, such as plasmas, liquid metals, etc.
B Y. Liu
[email protected]
1
Department of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
2
Academy of Mathematics & Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Y. Liu
Note that when b is identically zero, the system (1.1) reduces to the classical Navier– Stokes equations, hence we can’t expect to have a better theory on MHD than on the Navier– Stokes equations. It is well-known that the global-wellposedness of 3D Navier–Stokes equations is still one of the most challenging open problems in fluid mechanics, thus many efforts are made to study the solutions with some special structures. The geometric structure axisymmetric is such an important case. We call a vector field v is axisymmetric if it can be written as v(t, x) = v r (t, r, z)er + v θ (t, r, z)eθ + v z (t, r, z)ez ,
(1.2)
where(r, θ, z) are the usual cylindrical coordinates in R3 , defined by x = (r cos θ, r sin θ, z), r = x12 + x22 for any x ∈ R3 , and er = (cos θ, sin θ, 0), eθ = (− sin θ, cos θ, 0), ez = (0, 0, 1). v θ is called the swirl component, and we say v is axisymmetric without swirl if v θ = 0. For the axisymmetric without swirl solutions of Navier–Stokes equations, Ladyzhenskaya [8] and independently Ukhovskii and Yudovich [14] proved the existence of weak solutions along with the uniqueness and regularities of such solutions, [11] gave a refined 1 proof. Abidi [1] gave global well-posedness in critical space H˙ 2 . But for the case axisymmetric with non-trivial swirl, the global-wellposedness problem of Navier–Stokes equations is still open, and seems as difficult as for the general case without any geometric structure. The works for this case all need to put some smallness conditions on the initia
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