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

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.

[1]  Zhouping Xin,et al.  On the global existence of solutions to the Prandtl's system , 2004 .

[2]  Tong Yang,et al.  Justification of Prandtl Ansatz for MHD Boundary Layer , 2017, SIAM J. Math. Anal..

[3]  Di Wu,et al.  Gevrey Class Smoothing Effect for the Prandtl Equation , 2015, SIAM J. Math. Anal..

[4]  Nader Masmoudi,et al.  Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods , 2012, 1206.3629.

[5]  D. Gérard-Varet,et al.  Well-Posedness of the Prandtl Equations Without Any Structural Assumption , 2018, Annals of PDE.

[6]  D. Gérard-Varet,et al.  Formal derivation and stability analysis of boundary layer models in MHD , 2016, 1612.02641.

[7]  Chao-Jiang Xu,et al.  Long time well-posdness of the Prandtl equations in Sobolev space , 2015, 1511.04850.

[8]  Emmanuel Dormy,et al.  On the ill-posedness of the Prandtl equation , 2009, 0904.0434.

[9]  Wei-Xi Li,et al.  Well-posedness of the MHD Boundary Layer System in Gevrey Function Space without Structural Assumption , 2020, SIAM J. Math. Anal..

[10]  Tong Yang,et al.  Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points , 2019, Journal of the European Mathematical Society.

[11]  Michael Ruzhansky,et al.  On the Fourier analysis of operators on the torus , 2006, math/0612575.

[12]  V. Vicol,et al.  Almost Global Existence for the Prandtl Boundary Layer Equations , 2015, 1502.04319.

[13]  Zhifei Zhang,et al.  Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow , 2016, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[14]  Chao-Jiang Xu,et al.  Gevrey Hypoellipticity for a Class of Kinetic Equations , 2009, 1102.5432.

[15]  V. N. Samokhin,et al.  Mathematical Models in Boundary Layer Theory , 1999 .

[16]  Radjesvarane Alexandre,et al.  Well-posedness of the Prandtl equation in Sobolev spaces , 2012, 1203.5991.

[17]  Tong Yang,et al.  Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption , 2020, Communications on Pure and Applied Mathematics.

[18]  E Weinan,et al.  BLOWUP OF SOLUTIONS OF THE UNSTEADY PRANDTL'S EQUATION , 1997 .

[19]  Ping Zhang,et al.  Long time well-posdness of Prandtl system with small and analytic initial data , 2014, 1409.1648.

[20]  Tong Yang,et al.  Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces , 2020, 2002.11888.

[21]  Nader Masmoudi,et al.  Well-posedness for the Prandtl system without analyticity or monotonicity , 2013 .

[22]  Igor Kukavica,et al.  On the Local Well-posedness of the Prandtl and Hydrostatic Euler Equations with Multiple Monotonicity Regions , 2014, SIAM J. Math. Anal..

[23]  Yan Guo,et al.  A note on Prandtl boundary layers , 2010, 1011.0130.

[24]  Tong Yang,et al.  MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory , 2016, Communications on Pure and Applied Mathematics.