带真空无磁扩散不可压磁流体方程柯西问题的局部适定性

Abstract

Abstract:

In this paper, we investigate the Cauchy problem of the nonhomogeneous incompressible and non-resistive MHD on ${\Bbb R}$² with vacuum as far field density and prove that the 2D Cauchy problem has a unique local strong solution provided that the initial density and magnetic field decay not too slow at infinity. Furthermore, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one. Moreover, we also establish a blowup criterion which depends only on the Magnetic fields.

Key words: 2D non-resistive MHD equations, Vacuum, Classical solutions, Blowup criterion

CLC Number:

O175.2

Mingtao Chen,Wenhuo Su,Aibin Zang. Local well-Posedness for the Cauchy Problem of 2D Nonhomogeneous Incompressible and Non-Resistive MHD Equations with Vacuum[J].Acta mathematica scientia,Series A, 2021, 41(1): 100-125.

Trendmd

References 34

1	Antontsev S N, Kazhikov A V, Monakhov V N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Amsterdam: North-Holland, 1990
2	Bian D , Guo B . Well-posedness in critical spaces for the full compressible MHD equations. Acta Math Sci, 2013, 33B (4): 273- 296
3	Cabannes H . Theoretical Magnetofluiddynamics. New York: Academic Press, 1970
4	Chandrasekhar S . Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press, 1961
5	Chemin J Y , McCormick D S , Robinson J C , Rodrigo J L . Local existence for the non-resistive MHD equations in Besov spaces. Adv Math, 2016, 286: 1- 31 doi: 10.1016/j.aim.2015.09.004
6	Chen M , Zang A . On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum. Nonlinearity, 2017, 30: 3637- 3675 doi: 10.1088/1361-6544/aa7e97
7	Chen Q , Tan Z , Wang Y . Strong solutions to the incompressible magnetohydrodynamic equations. Math Meth Appl Sci, 2011, 34: 94- 107 doi: 10.1002/mma.1338
8	Cho Y , Kim H . Unique solvability for the density-dependent Navier-Stokes equations. Nonlinear Anal, 2004, 59: 465- 489 doi: 10.1016/j.na.2004.07.020
9	Choe H , Kim H . Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm Partial Differential Equations, 2003, 28: 1183- 1201 doi: 10.1081/PDE-120021191
10	Craig W , Huang X , Wang Y . Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations. J Math Fluid Mech, 2013, 15: 747- 758 doi: 10.1007/s00021-013-0133-6
11	Stein Elias M . Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993
12	Fefferman C L , McCormick D S , Robinson J C , Rodrigo J L . Higher order commutator estimates and local existence for the non-resistive MHD equations and relatedmodels. Journal of Functional Analysis, 2014, 267: 1035- 1056 doi: 10.1016/j.jfa.2014.03.021
13	Fefferman C L , McCormick D S , Robinson J C , Rodrigo J L . Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch Rational Mech Anal, 2017, 223: 677- 691 doi: 10.1007/s00205-016-1042-7
14	Freidberg J P . Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev Modern Phys, 1982, 54: 801- 902 doi: 10.1103/RevModPhys.54.801
15	Galdi G P . An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problem. New York: Springer, 2011
16	Huang X D, Li J. Global well-posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data. 2012, https://arxiv.org/abs/1207.3746
17	Huang X , Wang Y . Global strong solution to the 2D nonhomogeneous incompressible MHD system. J Differential Equations, 2013, 254: 511- 517 doi: 10.1016/j.jde.2012.08.029
18	Huang X , Wang Y . Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity. J Differential Equations, 2015, 259: 1606- 1627 doi: 10.1016/j.jde.2015.03.008
19	Jiu Q , Niu D . Mathematical results related to a two-dimensional magnetohydrodynamic equations. Acta Math Sci, 2006, 26B (4): 744- 756
20	Kazhikov A V . Resolution of boundary value problems for nonhomogeneous viscous fluids. Dokl Akad Nauk, 1974, 216: 1008- 1010
21	Kim H . A blow-up criterion for the nonhomogeneous nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37: 1417- 1434 doi: 10.1137/S0036141004442197
22	Kulikovskiy A G , Lyubimov G A . Magnetohydrodynamics. Reading, MA: Addison-Wesley, 1965
23	Li J , Liang Z . On local classical solutions to the cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum. J Math Pures Appl, 2014, 102: 640- 671 doi: 10.1016/j.matpur.2014.02.001
24	Liang Z . Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids. J Differential Equations, 2015, 7: 2633- 2654
25	Lin F , Xu L , Zhang P . Global small solutions to 2D incompresible MHD system. J Differ Equa, 2015, 259: 5440- 5485 doi: 10.1016/j.jde.2015.06.034
26	Lions P L . Mathematical Topics in Fluid Mechanics, Vol I: Incompressible Models. Oxford: Oxford University Press, 1996
27	Lü B, Xu Z, Zhong X. On local strong solutions to the Cauchy problem of two-dimensional density-dependent Magnetohydrodynamic equations with vacuum. 2015, https://arxiv.org/abs/1506.02156
28	Lü B , Xu Z , Zhong X . Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Magnetohydrodynamic equations with vacuum. J Math Pures Appl, 2017, 108 (1): 41- 62 doi: 10.1016/j.matpur.2016.10.009
29	Lü B, Shi X, Zhong X. Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum. 2015, https://arxiv.org/abs/1506.03143
30	Pan R, Zhou Y, Zhu Y. Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes. 2015, https://arxiv.org/abs/1503.05644
31	Ren X , Wu J , Xiang Z , Zhang Z . Global existence and decay of smooth solution for the 2D MHD equations without magnetic diffusion. J Funct Anal, 2014, 267: 503- 541 doi: 10.1016/j.jfa.2014.04.020
32	Si X, Ye X. Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients. Z Angew Math Phys, 2016, 67, Article number: 126
33	Zhang J . Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient. J Differ Equa, 2015, 259: 1722- 1742 doi: 10.1016/j.jde.2015.03.011
34	Zhou Y , Fan J . A regularity criterion for the 2D MHD system with zero magnetic diffusivity. J Math Anal Appl, 2011, 378: 169- 172 doi: 10.1016/j.jmaa.2011.01.014

[1]	Qingling Zhang,Ying Ba. The Perturbed Riemann Problem for the Pressureless Euler Equations with a Flocking Dissipation [J]. Acta mathematica scientia,Series A, 2020, 40(1): 49-62.
[2]	LIU Jian, LIAN Ru-Xu, QIAN Mao-Fu. Global Existence of Solution to Bipolar Navier-Stokes-Poisson System [J]. Acta mathematica scientia,Series A, 2014, 34(4): 960-976.
[3]	YE Yu-Lin, DOU Chang-Sheng, JIU Quan-Sen. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY [J]. Acta mathematica scientia,Series A, 2014, 34(3): 851-871.
[4]	ZHANG Ting-Ting. The Stability of the Riemann Solutions for Aw-Rascle Model [J]. Acta mathematica scientia,Series A, 2013, 33(2): 230-241.
[5]	Zhang Ting. Discontinuous Solutions of the Navier-Stokes Equations for Compressible Flow with Density-Dependent Viscosity [J]. Acta mathematica scientia,Series A, 2008, 28(2): 214-221.
[6]	Zhu Qingguo. The Concept of Group and Local Invariant Solution of Vacuum Einstein Equation [J]. Acta mathematica scientia,Series A, 1998, 18(S1): 35-43.

Local well-Posedness for the Cauchy Problem of 2D Nonhomogeneous Incompressible and Non-Resistive MHD Equations with Vacuum

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 34

Related Articles 6

Metrics

Comments

Recommended 10