GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

doi:10.1007/s10473-022-0609-7

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (6): 2343-2366.doi: 10.1007/s10473-022-0609-7

GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

Weixi LI^1,2, Rui XU¹, Tong YANG³

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China;
3. Department of Mathematics, City University of Hong Kong, Hong Kong, China

收稿日期:2022-07-07 出版日期:2022-12-25 发布日期:2022-12-16
通讯作者: Tong YANG, E-mail: matyang@cityu.edu.hk E-mail:matyang@cityu.edu.hk
基金资助:
W.-X. Li’s research was supported by NSF of China (11871054, 11961160716, 12131017) and the Natural Science Foundation of Hubei Province (2019CFA007). T. Yang’s research was supported by the General Research Fund of Hong Kong CityU (11304419).

GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

Weixi LI^1,2, Rui XU¹, Tong YANG³

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China;
3. Department of Mathematics, City University of Hong Kong, Hong Kong, China

Received:2022-07-07 Online:2022-12-25 Published:2022-12-16
Contact: Tong YANG, E-mail: matyang@cityu.edu.hk E-mail:matyang@cityu.edu.hk
Supported by:
W.-X. Li’s research was supported by NSF of China (11871054, 11961160716, 12131017) and the Natural Science Foundation of Hubei Province (2019CFA007). T. Yang’s research was supported by the General Research Fund of Hong Kong CityU (11304419).

摘要/Abstract

摘要： We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

关键词: magnetic Prandtl equation, Gevrey function space, global well-posedness, auxiliary functions, loss of derivative

Abstract: We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

Key words: magnetic Prandtl equation, Gevrey function space, global well-posedness, auxiliary functions, loss of derivative

中图分类号:

76W05

Weixi LI, Rui XU, Tong YANG. GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES[J]. 数学物理学报(英文版), 2022, 42(6): 2343-2366.

Weixi LI, Rui XU, Tong YANG. GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES[J]. Acta mathematica scientia,Series B, 2022, 42(6): 2343-2366.

参考文献

[1] Aarach N. Global well-posedness of 2D Hyperbolic perturbation of the Navier-Stokes system in a thin strip. arXiv e-prints, arXiv:2111.13052
[2] Alexandre R, Wang Y -G, Xu C -J, Yang T. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28(3): 745-784
[3] Chen D, Ren S, Wang Y, Zhang Z. Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Asymptot Anal, 2020, 120(3/4): 373-393
[4] Chen D, Wang Y, Zhang Z. Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow. Ann Inst H Poincaré Anal Non Linéaire, 2018, 35(4): 1119-1142
[5] Dietert H, Gérard-Varet D. Well-posedness of the Prandtl equations without any structural assumption. Ann PDE, 2019, 5(1): Art 8
[6] E W, Engquist B. Blowup of solutions of the unsteady Prandtl’s equation. Comm Pure Appl Math, 1997, 50(12): 1287-1293
[7] Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equation. J Amer Math Soc, 2010, 23(2): 591-609
[8] Gérard-Varet D, Iyer S, Maekawa Y. Improved well-posedness for the Triple-Deck and related models via concavity Preprint. arXiv:2205.15829
[9] Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann Sci Ec Norm Supér (4), 2015, 48(6): 1273-1325
[10] Gérard-Varet D, Masmoudi N, Vicol V. Well-posedness of the hydrostatic Navier-Stokes equations. Anal PDE, 2020, 13(5): 1417-1455
[11] Gérard-Varet D, Prestipino M. Formal derivation and stability analysis of boundary layer models in MHD. Z Angew Math Phys, 2017, 68(3): Art 76
[12] Guo Y, Nguyen T. A note on Prandtl boundary layers. Comm Pure Appl Math, 2011, 64(10): 1416-1438
[13] Ignatova M, Vicol V. Almost global existence for the Prandtl boundary layer equations. Arch Ration Mech Anal, 2016, 220(2): 809-848
[14] Kukavica I, Masmoudi N, Vicol V, Wong T K. On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J Math Anal, 2014, 46(6): 3865-3890
[15] S. Li and F. Xie. Global solvability of 2D MHD boundary layer equations in analytic function spaces. J Differential Equations, 299:362-401, 2021.
[16] Li W -X, Masmoudi N, Yang T. Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption. Comm Pure Appl Math, 2022, 75(8): 1755-1797
[17] Li W -X, Wu D, Xu C -J. Gevrey class smoothing effect for the Prandtl equation. SIAM J Math Anal, 2016, 48(3): 1672-1726
[18] Li W -X, Xu R. Gevrey well-posedness of the hyperbolic Prandtl equations. Commun Math Res, doi: 10.4208/cmr.2021-0104, 2022
[19] Li W -X, Xu R. Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity. Electron Res Arch, 2021, 29(6): 4243-4255
[20] Li W -X, Yang T. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J Eur Math Soc, 2020, 22(3): 717-775
[21] Li W -X, Yang T. Well-posedness of the MHD boundary layer system in Gevrey function space without structural assumption. SIAM J Math Anal, 2021, 53(3): 3236-3264
[22] Li W -X, Yang T. 3D hyperbolic Navier-Stokes equations in a thin strip: global well-posedness and hydrostatic limit in Gevrey space. Commun Math Anal Appl, doi:10.4208/cmaa.2022-0007, 2022
[23] Liu C -J, Wang D, Xie F, Yang T. Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces. J Funct Anal, 2020, 279(7): 108637
[24] Liu C -J, Wang Y -G, Yang T. A well-posedness theory for the Prandtl equations in three space variables. Adv Math, 2017, 308: 1074-1126
[25] Liu C -J, Xie F, Yang T. MHD boundary layers theory in Sobolev spaces without monotonicity I: Wellposedness theory. Comm Pure Appl Math, 2019, 72(1): 63-121
[26] Liu N, Zhang P. Global small analytic solutions of MHD boundary layer equations. J Differential Equations, 2021, 281: 199-257
[27] Masmoudi N, Wong T K. On the H^s theory of hydrostatic Euler equations. Arch Ration Mech Anal, 2012, 204(1): 231-271
[28] Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68(10): 1683-1741
[29] Paicu M, Zhang P. Global existence and the decay of solutions to the Prandtl system with small analytic data. Arch Ration Mech Anal, 2021, 241(1): 403-446
[30] Paicu M, Zhang P. Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data. Sci China Math, 2022, 65(6): 1109-1146
[31] Paicu M, Zhang P, Zhang Z. On the hydrostatic approximation of the Navier-Stokes equations in a thin strip. Adv Math, 2020, 372: 107293
[32] Renardy M. Ill-posedness of the hydrostatic Euler and Navier-Stokes equations. Arch Ration Mech Anal, 2009, 194(3): 877-886
[33] Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm Math Phys, 1998, 192(2): 433-461
[34] Tao T. Nonlinear Dispersive Equations: Local and Global Analysis. Volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; Providence, RI: American Mathematical Society, 2006
[35] Wang C, Wang Y -X. Optimal Gevrey stability of hydrostatic approximation for the Navier-Stokes equations in a thin domain. arXiv e-prints, page arXiv:2206.03873v2
[36] Wang C, Wang Y, Zhang P. On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class. arXiv e-prints, page arXiv:2103.00681
[37] Xie F, Yang T. Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime. SIAM J Math Anal, 2018, 50(6): 5749-5760
[38] Xin Z, Zhang L. On the global existence of solutions to the Prandtl’s system. Adv Math, 2004, 181(1): 88-133
[39] Xu C -J, Zhang X. Long time well-posedness of Prandtl equations in Sobolev space. J Differential Equations, 2017, 263(12): 8749-8803
[40] Yang T. Vector fields of cancellation for the Prandtl operators. Commun Math Anal Appl, 2022, 1(2): 345-354
[41] Zhang P, Zhang Z. Long time well-posedness of Prandtl system with small and analytic initial data. J Funct Anal, 2016, 270(7): 2591-2615

GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 13

Metrics

本文评价

推荐阅读 0

[1]	Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J]. 数学物理学报(英文版), 2022, 42(6): 2230-2256.
[2]	Jinlu LI, Zhaoyang YIN, Xiaoping ZHAI. GLOBAL WELL-POSEDNESS FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(5): 2131-2148.
[3]	晋雪婷, 肖跃龙, 于幻. GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION[J]. 数学物理学报(英文版), 2022, 42(4): 1293-1309.
[4]	陈停停, 屈爱芳, 王振. EXISTENCE AND UNIQUENESS OF THE GLOBAL L₁ SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS[J]. 数学物理学报(英文版), 2021, 41(3): 941-958.
[5]	Muhammad Zainul ABIDIN, 陈杰诚. GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 164-176.
[6]	苏仕斌, 赵小奎. GLOBAL WELLPOSEDNESS OF MAGNETOHYDRODYNAMICS SYSTEM WITH TEMPERATURE-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2018, 38(3): 898-914.
[7]	张再云, 黄建华, 孙明保. ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²[J]. 数学物理学报(英文版), 2017, 37(2): 385-394.
[8]	霍朝辉. GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION[J]. 数学物理学报(英文版), 2016, 36(4): 1117-1152.
[9]	夏红强. GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS[J]. 数学物理学报(英文版), 2015, 35(1): 255-274.
[10]	陈明涛. GLOBAL WELL-POSEDNESS OF THE 2D INCOMPRESSIBLE MICROPOLAR FLUID FLOWS WITH PARTIAL VISCOSITY AND ANGULAR VISCOSITY[J]. 数学物理学报(英文版), 2013, 33(4): 929-935.
[11]	蒲学科, 郭柏灵. GLOBAL WELL-POSEDNESS OF THE STOCHASTIC 2D BOUSSINESQ EQUATIONS WITH PARTIAL VISCOSITY[J]. 数学物理学报(英文版), 2011, 31(5): 1968-1984.
[12]	李用声, 杨兴雨. GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE[J]. 数学物理学报(英文版), 2011, 31(4): 1303-1317.
[13]	郭柏灵, 黄代文. ON THE 3D VISCOUS PRIMITIVE EQUATIONS OF THE LARGE-SCALE ATMOSPHERE[J]. 数学物理学报(英文版), 2009, 29(4): 846-866.