Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equation in \(\mathbb R^{3}\). (English) Zbl 1267.76130
Summary: We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, and then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony’s paraproduct decomposition, it is proved that the strong solution \((u,b)\) can be extended after \(t=T\) if either \(u\in L^q_T(\dot B^0_{p,\infty})\) with \(\frac 1q+\frac 3p\leq 1\) and \(b\in L^1_T(\dot B^0_{\infty,\infty})\) or \((\omega,J)\in L^q_T(\dot B^0_{p,\infty})\) with \(\frac 2q+\frac 3p\leq 2\), where \(\omega(t)= \nabla\times u\) denotes the vorticity of the velocity and \(J=\nabla\times b\) stands for the current density.
MSC:
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35B65 | Smoothness and regularity of solutions to PDEs |
References:
| [1] | Beale, J. T.; Kato, T.; Majda, A. J., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94, 61-66 (1984) · Zbl 0573.76029 |
| [2] | Beirao da Veiga, H., A new regularity class for the Navier-Stokes equations in \(R^n\), Chinese Ann. Math. Ser. B, 16, 407-412 (1995) · Zbl 0837.35111 |
| [3] | Bergh, J.; Löfstrom, J., Interpolation Spaces, An Introduction (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0344.46071 |
| [4] | Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14, 209-246 (1981) · Zbl 0495.35024 |
| [5] | Biskamp, D., Nonlinear Magnetohydrodynamics (1993), Cambridge University Press: Cambridge University Press Cambridge, UK |
| [6] | Biskamp, D., Magnetic Reconnection in Plasmas (2000), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0891.76094 |
| [7] | Caflisch, R. E.; Klapper, I.; Steele, G., Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184, 443-455 (1997) · Zbl 0874.76092 |
| [8] | Cannone, M.; Chen, Q.-L.; Miao, C.-X., A losing estimate for the Ideal MHD equations with application to blow-up criterion, SIAM J. Math. Anal., 38, 1847-1859 (2007) · Zbl 1126.76057 |
| [9] | Chemin, J.-Y., Perfect Incompressible Fluids (1998), Oxford University Press: Oxford University Press New York · Zbl 0927.76002 |
| [10] | Duvaut, G.; Lions, J. L., Inéquation en thermoélasticite et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46, 241-279 (1972) · Zbl 0264.73027 |
| [11] | Giga, Y., Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62, 182-212 (1986) · Zbl 0577.35058 |
| [12] | He, C.; Xin, Z., On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213, 235-254 (2005) · Zbl 1072.35154 |
| [13] | Kozono, H.; Taniuchi, Y., Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235, 173-194 (2000) · Zbl 0970.35099 |
| [14] | Kozono, H.; Ogawa, T.; Taniuchi, Y., The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242, 251-278 (2002) · Zbl 1055.35087 |
| [15] | Majda, A. J., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci., vol. 53 (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0537.76001 |
| [16] | Núñez, M., Existence theorems for two-fluid magnetohydrodynamics, J. Math. Phys., 46, 083101 (2005) · Zbl 1110.76057 |
| [17] | Priest, E. R.; Forbes, T. G., Magnetic Reconnection: MHD Theory and Applications (2000), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0959.76002 |
| [18] | Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36, 635-664 (1983) · Zbl 0524.76099 |
| [19] | Triebel, H., Theory of Function Spaces, Monogr. Math., vol. 78 (1983), Birkhäuser: Birkhäuser Basel · Zbl 0546.46028 |
| [20] | Wu, J., Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12, 395-413 (2002) · Zbl 1029.76062 |
| [21] | Wu, J., Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10, 543-556 (2004) · Zbl 1055.76062 |
| [22] | Zhang, Z.; Liu, X., On the blow-up criterion of smooth solutions to the 3D Ideal MHD equations, Acta Math. Appl. Sin. E, 20, 695-700 (2004) · Zbl 1138.35382 |
| [23] | Zhou, Y., Remarks on the regularity for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12, 881-886 (2005) · Zbl 1068.35117 |
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