×

A global existence result in Sobolev spaces for MHD system in the half-plane. (English) Zbl 1058.35175

The authors prove a global existence and uniqueness theorem in suitably chosen Sobolev spaces for the 2-dimensional incompressible MHD system in the half-plane. The method is based on a priori estimates (derived from energy identities) and classical fixed point arguments.

MSC:

35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] G. V. ALEXSEEV, Solvability of a homogeneous initial-boundary value problem for equations of magnetohydrodynamics of an ideal fluid, (Russian), Dinam. Sploshn. Sredy, 57 (1982), pp. 3-20. Zbl0513.76106 MR752597 · Zbl 0513.76106
[2] H. BEIRÃO DA VEIGA, Boundary-value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow, Rend. Sem. Mat. Univ. Padova, 79 (1988), pp. 247-273. Zbl0709.35082 MR964034 · Zbl 0709.35082
[3] H. BEIRÃO DA VEIGA, Kato’s perturbation theory and well posedness for the Euler equations in bounded domains, Arch. Rat. Mech Anal., 104 (1988), pp. 367-382. Zbl0672.35044 MR960958 · Zbl 0672.35044
[4] H. BEIRÃO DA VEIGA, A well posedness theorem for non-homogeneous inviscid fluids via a perturbation theorem, (II) J. Diff. Eq., 78 (1989), pp. 308-319. Zbl0682.35012 MR992149 · Zbl 0682.35012
[5] E. CASELLA - P. SECCHI - P. TREBESCHI, Global classical solutions for MHD system, to appear on Journal of Math. Fluid Mech., Mathematic. Zbl1037.76068 MR1966645 · Zbl 1037.76068
[6] T. KATO, On Classical Solutions of Two-Dimensional Non-Stationary Euler Equation, Arch. Rat. Mech. Anal., 25 (1967), pp. 188-200. Zbl0166.45302 MR211057 · Zbl 0166.45302
[7] T. KATO - C. Y. LAI, Nonlinear evolution equations and the Euler flow, J. Funct. Analysis, 56 (1984), pp. 15-28. Zbl0545.76007 MR735703 · Zbl 0545.76007
[8] K. KIKUCHI, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo, Sec IA 30 (1983), pp. 63-92. Zbl0517.76024 MR700596 · Zbl 0517.76024
[9] H. KOZONO, Weak and Classical Solutions of the Two-dimensional magnetohydrodynamic equations, Tohoku Math. J., 41 (1989), pp. 471-488. Zbl0683.76103 MR1007099 · Zbl 0683.76103
[10] L. LICHTENSTEIN, Grundlagen der Hydromechanik, Edition of 1928 Springer, Berlin, 1968. Zbl0157.56701 MR228225 JFM55.1124.01 · Zbl 0157.56701
[11] P. G. SCHMDT, On a magnetohydrodynamic problem of Euler type, J. Diff. Eq., 74 (1988), pp. 318-335. Zbl0675.35080 MR952901 · Zbl 0675.35080
[12] P. SECCHI, On the Equations of Ideal Incompressible Magneto-Hydrodynamics, Rend. Sem. Mat. Univ. Padova, 90 (1993), pp. 103-119. Zbl0808.35110 MR1257135 · Zbl 0808.35110
[13] R. TEMAM, Navier-Stokes Equations, 2nd Ed., North-Holland, Amsterdam, 1979. Zbl0426.35003 MR603444 · Zbl 0426.35003
[14] R. TEMAM, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), pp. 32-43. Zbl0309.35061 MR430568 · Zbl 0309.35061
[15] W. WOLIBNER, Un théorèm sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment longue, Math. Z., 37 (1933), pp. 698-726. Zbl0008.06901 MR1545430 · Zbl 0008.06901
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.