×

Weak and classical solutions of the two-dimensional magnetohydrodynamic equations. (English) Zbl 0683.76103

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\) with smooth boundary \(\partial\Omega\). In \(Q_ T:=\Omega \times (0,T)\), we consider the following magnetohydrodynamic equations for an ideal incompressible fluid coupled with magnetic field: \[ \begin{cases} \partial_ t u + (u,\nabla)u - (B,\nabla)B + \nabla ((1/2) | B|^2) + \nabla\pi = f \qquad&\text{in }Q_ T,\\ \partial_ t B - \Delta B + (u,\nabla)B - (B,\nabla)u =0 \qquad&\text{in }Q_ T,\\ \text{div}u = 0,\quad \text{div}B = 0 \qquad&\text{in }Q_ T,\\ u\cdot v = 0,\quad B\cdot v = 0,\quad \text{rot}B = 0 \qquad&\text{on } \partial\Omega \times (0,T),\\ u |_{t=0} = u_0,\quad B |_{t=0} = B_0. \end{cases} \tag{\(*\)} \] Here \(u = u(x,t) = (u^1(x,t),u^2(x,t))\), \(B = B(x,t) = (B^1(x,t),B^2(x,t ))\) and \(\pi = \pi(x,t)\) denote the unknown velocity field of the fluid, magnetic field and pressure of the fluid, respectively; \(f = f(x,t) = (f^1(x,t),f^2(x,t))\) denotes the given external force, \(u_0 = u_0(x) = (u^1_0(x),u^2_0(x))\) and \(B_0=B_0(x)=(B^1_0(x),B^2_ 0(x))\) denote the given initial data and \(\nu\) denotes the unit outward normal on \(\partial\Omega\).
The first purpose of this paper is to show the existence and uniqueness of a global weak solution of (\(*\)) without restriction on the data. Our second purpos e is to show the existence and uniqueness of a local classical solution of (\(*\)). Although the method of characteristic curves for the vorticity equation plays an important role in a global classical solution of the two-dimensional Euler equations, such a method seems to give us only a local classical solution of (\(*\)) because of the additional terms \((B,\nabla)B\) and \((u,\nabla)B- (B,\nabla)u\). We devote Section 1 to preliminaries and definition of a weak solution of (\(*\)). Two main theorems will then be stated. Sections 2 and 3 are devoted to the proofs of the main theorems.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
35Q99 Partial differential equations of mathematical physics and other areas of application
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. BARDOS, Existence etunicite de lasolution de equation duler en dimension deux, J. Math. Anal. Appl. 40(1972), 769-790. · Zbl 0249.35070 · doi:10.1016/0022-247X(72)90019-4
[2] R. COURANT AND D. HILBERT, Methoden der Mathematischen Physik II, Springer-Verlag, Berlin Heidelberg-New York, 1968. · Zbl 0156.23201
[3] G. DUVAUT AND J. L. LIONS, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-Heidelberg New York, 1976. · Zbl 0331.35002
[4] A. FRIEDMAN, Partial Differential Equations of Parabolic Type, Prentice-Hall, London, 1963 · Zbl 0144.34903
[5] V. GEORGESCU, Someboundary valueproblems fordifferential forms oncompact Riemannian manifolds, Annali di Matematica Pura ed.Applica, Serie 4, 122(1979), 159-198. · Zbl 0432.58026 · doi:10.1007/BF02411693
[6] D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0361.35003
[7] N. M. GUNTER, Potential Theory and Its Application to Basic Problems of Mathematical Physics, Frederick Ungar Publish Co., New York, 1967. · Zbl 0164.41901
[8] T. KATO, On Classical Solutions of the Two-Dimensional Non-Stationary Euler Equation, Arch.Rat Mech. Anal. 25(1967), 188-200. · Zbl 0166.45302 · doi:10.1007/BF00251588
[9] K. KIKUCHI, Exterior problem for the two-dimensional Euler equation, J. Fac.Sci.Univ. Tokyo, Sec IA 30(1983), 63-92. · Zbl 0517.76024
[10] O. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, Gordon reach, New York, 1969. · Zbl 0184.52603
[11] O. A. LADYZHENSKAYA, V. A. SOLONNIKOV AND N. N. URAL’CEVA, Linear and Quasilinear Equation of Parabolic Type, Translation of Mathematical Monographs Vol. 23, Amer. Math. Soc, Providence, Rhode Island (1968). · Zbl 0174.15403
[12] J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin-Heidelberg-New York, 1972. · Zbl 0223.35039
[13] T. MiYAKAWA, The LPapproach to the Navier-Stokes equations with the Neumann boundary condition, Hiroshima Math. J. 10(1980), 517-537. · Zbl 0455.35099
[14] M SERMANGE AND R. TEMAM, Some Mathematical Questions Related to the MHD Equations, Comm Pure Appl. Math. 36(1983), 635-664. · Zbl 0524.76099 · doi:10.1002/cpa.3160360506
[15] H. TANABE, Equations of Evolution, Pitman, London, 197 · Zbl 0123.31903
[16] R. TEMAM, Navier-Stokes Equations, 2nd Ed., North-Holland, Amsterdam, 1979 · Zbl 0426.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.