Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

Abstract

<p style='text-indent:20px;'>We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.</p>

Global well-posedness of a Prandtl model from MHD in Gevrey function spaces

“…We also mention some other related works [13,39,41]. For the Prandtl system with a suitable background magnetic field, the local solutions in Sobolev or Gevrey function spaces were obtained in [19,21,23,25], and the global analytic solution was established recently in [15,26]. Note that all of these global-in-time existence results are in 2D setting under some suitable structural conditions on the initial data.…”

3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space

“…On the other hand, in the fully nonlinear regime, Prandtl type system can be derived from the MHD system. In this regime, the tangential magnetic field has stabilizing effort as shown in the 2D case by Liu et al [34] (see also [28,31] for the further generalization), where the Sobolev well-posedness theory was established without Oleinik's monotonicity condition on the velocity field provided the tangential magnetic field dominates. Without any structural assumption, the Gevrey well-posedness was studied in [30] with Gevrey index less or equal to 3 2 that is not known to be optimal.…”

3D hyperbolic Navier-Stokes equations in a thin strip: global well-posedness and hydrostatic limit in Gevrey space

“…On the other hand, in the fully nonlinear regime, Prandtl type system can be derived from the MHD system. In this regime, the tangential magnetic field has stabilizing effort as shown in the 2D case by Liu-Xie-Yang [34](see also [28,31] for the further generalization), where the Sobolev well-posedness theory was established without Oleinik's monotonicity condition on the velocity field provided the tangential magnetic field dominates. Without any structural assumption, the Gevrey well-posedness was studied in [30] with Gevrey index less or equal to 3/2 that is not known to be optimal.…”