不可压缩磁流体方程组在Besov空间中的爆破准则

Abstract

Abstract:

In this paper, we study the blow-up criterion of smooth solutions to the threedimensional incompressible magnetohydrodynamic(MHD) equations in Besov space of negative regular index. We show that a smooth solution breaks down if and only if the certain norm of the velocity u blows up at the same time. Here the norm‖·‖_{V_Θ} is defined in nonhomogenous Besov space and it is weaker than‖·‖_{B_{∞, ∞}^α-1}-norm, for 0 < α < 1.

Key words: Incompressible MHD equations, Blow-up criterion, Besov space

CLC Number:

O175.2

Zhaoyang Shang. Blow-Up Criterion for Incompressible Magnetohydrodynamics Equations in Besov Space[J].Acta mathematica scientia,Series A, 2019, 39(1): 67-80.

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References 47

1	Alfvén H . Existence of electromagnetic-hydrodynamic waves. Nature, 1942, 150: 405- 406
2	Brezis H , Mironescu P . Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J Evol Equa, 2001, 1 (4): 387- 404 doi: 10.1007/PL00001378
3	Bahouri H , Chemin J Y , Danchin R . Fourier Analysis and Nonlinear Partial Differential Equations. Heidelberg: Springer, 2011
4	Beale J T , Kato T , Majda A . Remarks on the breakdown of smooth solutions for the 3D Euler equations. Comm Math Phys, 1984, 94: 61- 66 doi: 10.1007/BF01212349
5	Cannone M , Chen Q , Miao C . A losing estimate for the Ideal MHD equations with application to Blow-up criterion. SIAM J Math Anal, 2007, 38: 1847- 1859 doi: 10.1137/060652002
6	Chen Q , Miao C , Zhang Z . The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations. Comm Math Phys, 2007, 275: 861- 872 doi: 10.1007/s00220-007-0319-y
7	Chen Q , Miao C , Zhang Z . On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Comm Math Phys, 2008, 284: 919- 930 doi: 10.1007/s00220-008-0545-y
8	Chen Q , Miao C , Zhang Z . On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces. Arch Ration Mech Anal, 2010, 195 (2): 561- 578 doi: 10.1007/s00205-008-0213-6
9	Caflisch R E , Klapper I , Steele G . Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun Math Phys, 1997, 184: 443- 455 doi: 10.1007/s002200050067
10	Duvaut G , Lions J L . Inéquations en thermoélasticité et magnéto-hydrodynamique. Arch Rational Mech Anal, 1972, 46: 241- 279 doi: 10.1007/BF00250512
11	Furioli G , Lemarié-Rieusset P G , Terraneo E . Sur l'unicité dans ${L^3}({{mathbb{R}}^3})$ des solutions "mild" des équations de Navier-Stokes[On the uniqueness in ${L^3}({{mathbb{R}}^3})$ of mild solutions of the Navier-Stokes equations]. C R Acad Sci Paris Sér I Math, 1997, 325: 1253- 1256 doi: 10.1016/S0764-4442(97)82348-8
12	Hasegawa A . Self-organization processes in continuous media. Adv Phys, 1985, 34: 1- 42 doi: 10.1080/00018738500101721
13	He C , Xin Z . On the regularity of weak solutions to the magnetohydrodynamic equations. J Differential Equations, 2005, 213: 235- 254 doi: 10.1016/j.jde.2004.07.002
14	He C , Xin Z . Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J Funct Anal, 2005, 227: 113- 152 doi: 10.1016/j.jfa.2005.06.009
15	Hajaiej H, Molinet L, Ozawa T, Wang B. Necessary and sufficient conditions for the fractional GagliardoNirenberg inequalities and applications to Navier-Stokes and generalized boson equations. 2011, arXiv: 1004.4287v3
16	Kozono H , Ogawa T , Taniuchi Y . The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math Z, 2002, 242: 251- 278 doi: 10.1007/s002090100332
17	Kozono H , Taniuchi Y . Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Comm Math Phys, 2000, 214: 191- 200 doi: 10.1007/s002200000267
18	Kato T , Ponce G . Commutator estimates and Euler and Navier-Stokes equations. Comm Pure Appl Math, 1988, 41: 891- 907 doi: 10.1002/(ISSN)1097-0312
19	Lifschitz A . Magnetohydro-dynamics and Spectral Theory, Developments in Electromagnetic Theory and Applications. Dordrecht: Kluwer Academic Publishers Group, 1989
20	Lions P L , Masmoudi N . Uniqueness of mild solutions of the Navier-Stokes system in L^N. Comm Partial Differential Equations, 2001, 26: 2211- 2226 doi: 10.1081/PDE-100107819
21	Lei Z , Zhou Y . BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst, 2009, 25: 575- 583 doi: 10.3934/dcdsa
22	Majda A J . Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer-Verlag, 1984
23	Miao C , Yuan B . On the well-posedness of the Cauchy problem for an MHD system in Besov spaces. Math Methods Appl Sci, 2009, 32 (1): 53- 76 doi: 10.1002/mma.v32:1
24	Ogawa T , Taniuchi Y . On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain. J Differential Equations, 2003, 190: 39- 63 doi: 10.1016/S0022-0396(03)00013-5
25	Prodi G . Un teorema di unicitá per le equazioni di Navier-Stokes[A uniqueness theorem for the NavierStokes equations]. Ann Mat Pura Appl, 1959, 48: 173- 182 doi: 10.1007/BF02410664
26	Politano H , Pouquet A , Sulem P L . Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys Plasmas, 1995, 2: 2931- 2939 doi: 10.1063/1.871473
27	Ren W . On the blow-up criterion for the 3D Boussinesq system with zero viscosity constant. Appl Anal, 2015, 94 (48): 56- 862
28	Serrin J. The Initial Value Problem for the Navier-Stokes Equations//Langer R E. Proceedings of the Symposium on Nonlinear Problems. Madison: Univ of Wisconsin Press, 1963: 69-98
29	Sermange M , Temam R . Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36 (5): 635- 664 doi: 10.1002/(ISSN)1097-0312
30	Shang Z . Osgood type blow-up criterion for the 3D Boussinesq equations with partial viscosity. AIMS Mathematics, 2018, 3 (1): 1- 11 doi: 10.3934/Math.2018.1.1
31	Wen H , Zhu C . Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data. SIAM J Math Anal, 49 (1): 162- 221 doi: 10.1137/16M1055414
32	Wu J . Bounds and new approaches for the 3D MHD equations. J Nonlinear Sci, 2002, 12: 395- 413 doi: 10.1007/s00332-002-0486-0
33	Wu J . Regularity results for weak solutions of the 3D MHD equations. Discrete Contin Dynam Syst, 2004, 10: 543- 556
34	Wu J . Regularity criteria for the generalized MHD equations. Commun Partial Differ Equa, 2008, 33 (2): 285- 306
35	Wu J . Global regularity for a class of generalized magnetohydrodynamic equations. J Math Fluid Mech, 2011, 13 (2): 295- 305
36	Xu X , Ye Z , Zhang Z . Remark on an improved regularity criterion for the 3D MHD equations. Appl Math Lett, 2015, 42: 41- 46 doi: 10.1016/j.aml.2014.11.004
37	Xu X , Ye Z . Note on global regularity of 3D generalized magnetohydrodynamicmodel with zero diffusivity. Commun Pure Appl Anal, 2015, 14 (2): 585- 595
38	Ye Z . Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations. Ann Mat Pura Appl, 2016, 4 (4): 1111- 1121
39	Ye Z , Xu X . Global regularity of 3D generalized incompressible magnetohydrodynamic model. Appl Math Lett, 2014, 35: 1- 6 doi: 10.1016/j.aml.2014.03.018
40	Ye Z , Xu X . Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system. Nonlinear Anal, 2014, 100: 86- 96 doi: 10.1016/j.na.2014.01.012
41	Zhang Q . Refined blow-up criterion for the 3D magnetohydrodynamics equations. Appl Anal, 2013, 92 (12): 2590- 2599 doi: 10.1080/00036811.2012.751589
42	Zhou Y . Remarks on regularities for the 3D MHD equations. Discrete Contin Dynam Systems, 2005, 12: 881- 886 doi: 10.3934/dcdsa
43	Zhang Z , Liu X . On the blow-up criterion of smooth solutions to the 3D Idea lMHD equations. Acta Math Appl Sinica E, 2004, 20: 695- 700
44	Zhang Z , Tang T , Liu L . An Osgood type regularity criterion for the liquid crystal flows. Nonlinear Differ Equa Appl, 2014, 21: 253- 262 doi: 10.1007/s00030-013-0245-y
45	Zhang Z , Sadek G . Osgood type regularity criterion for the 3D Newton-Boussinesq equation. Electron J Dierential Equations, 2013, 223: 1- 6
46	Zhang Z , Yang X . Navier-Stokes equations with vorticity in Besov spaces of negative regular indices. J Math Anal Appl, 2016, 440: 415- 419 doi: 10.1016/j.jmaa.2016.03.037
47	Zhang Z . A logarithmically improved regularity criterion for the 3D MHD system involving the velocity field in homogeneous Besov spaces. Ann Polon Math, 2016, 18 (1): 51- 57

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Blow-Up Criterion for Incompressible Magnetohydrodynamics Equations in Besov Space

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