On the Faedo-Galerkin method for a free boundary problem for incompressible viscous magnetohydrodynamics
Keywords
Free boundary, incompresible magnetohydrodynamics, Faedo-Galerkin methodAbstract
The motion of incompressible magnetohydrodynamics (mhd) in a domain bounded by a free surface and coupled through it with external electromagnetic field is considered. Transmission conditions for electric currents and magnetic fields are prescribed on the free surface. Although we show the idea of a proof of local existence by the method of successive approximations, we are not going to prove neither local nor global existence of solutions. The existence of solutions of the linearized problems (the Stokes system for velocity and pressure and the linear transmission problem for the electromagnetic fields) is the main step in the proof of existence to the considered problem. This can be done either by the Faedo-Galerkin method or by the technique of regularizer. We concentrate our considerations to the Faedo-Galerkin method. For this we need an existence of a fundamental basis. We have to find the basis for the Stokes system and mhd system. We concentrate our considerations on the mhd system because this for the Stokes system is well known. We have to emphasize that the considered mhd system is obtained after linearization and transformation to the initial domains by applying the Lagrangian coordinates. This is the main aim of this paper.References
E.B. Bykhovsky, Solvability of mixed problem for the Maxwell equations for ideal conductive boundary, Vestn. Len. Univ. Ser. Mat. Mekh. Astr. 13 (1957), 50–66. (in Russian)
G.H.A. Cole, Fluid Dynamics, London & Colchester, 1962.
E. Frolova, Free boundary problem of magnetohydrodynamics, Zap. Nauchn. Sem. POMI 425 (2014), 149–178.
E. Frolova and V.A. Solonnikov, Solvability of a free boundary problem of magnetohydrodynamics in an infinite time interval, Zap. Nauchn. Sem. POMI 410 (2013), 131–167.
P. Kacprzyk, Local existence of solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid, Appl. Math. 30 (2003), 461–488.
P. Kacprzyk, Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid, Appl. Math. 31 (2004), 69–77.
P. Kacprzyk, Free boundary problem for the equations of magnetohydrodynamic incompressible viscous fluid, Appl. Math. 34 (2007), 75–95.
P. Kacprzyk, Local free boundary problem for incompressible magnetohydrodynamics, Dissertationes Math. 509 (2015), 1–52.
P. Kacprzyk, Global free boundary problem for incompressible magnetohydrodynamics, Dissertationes Math. 510 (2015), 1–44.
L. Kapitański and K. Pileckas, On some problems of vector analysis, Zap. Nauchn. Sem. LOMI 138 (1984), 65–85. (in Russian)
N.E. Kochin, Vectorial Calculus and Introduction to Tensor Calculus, Moscow, 1951. (in Russian)
O.A. Ladyzhenskaya, Boundary Value Problems for Mathematical Physics, Moscow, 1973. (in Russian)
O.A. Ladyzhenskaya, Krajewyje Zadaci Matematicieskoj Fizyki, Nauka, Moskwa, 1973. (in Russian)
O.A. Ladyzhenskaya and V.A. Solonnikov, Solvability of some nonstationary problems of magnetohydrodynamics for viscous incompressible fluids, Trudy Mat. Inst. Steklov 59 (1960), 115–173. (in Russian)
L.D. Landau, E.M. Lifshitz and L.P. Pitaevskiı̆, Electrodynamics of Continuous Media, second edition, Landau and Lifshitz Course of Theoretical Physics, Vol. 8.
M. Padula and V.A. Solonnikov, On free boundary problem of mhd, Zap. Nauchn. Sem. POMI 385 (2010); Kraevye Zadachi Matematicheskoj Fiziki i Smezhnye Voprosy Teorii Funktsii 41 (2010), 135–186; English transl.: J. Math. Sci. (N.Y.) 178 (2011), 313–344.
M. Sahaev and V.A. Solonnikov, On some stationary problems of magnetohydrodynamics in general domains, Zap. Nauchn. Sem. POMI 397 (2011), 126–149.
Y. Shibata and W.M. Zajączkowski, On local motion to a free boundary problem for incompressible viscous magnetohydrodynamics in the Lp -approach, Part 1.
Y. Shibata and W.M. Zajączkowski, On local motion to a free boundary problem for incompressible viscous magnetohydrodynamics in the Lp -approach, Part 2.
V.A. Solonnikov, Estimates of solutions to nonstationary linearized Navier–Stokes system, Trudy MIAN 70 (1964), 213–317. (in Russian)
V.A. Solonnikov, Estimates of solutions of an initial-boundary value problem for the linear non-stationary Navier–Stokes system, Zap. Nauchn. Sem. LOMI 59 (1976), 178–254. (in Russian)
V.A. Solonnikov, On an unsteady motion of an isolated volume of a visous incompressible fluid, Izv. Ross. Akad. Nauk Ser. Mat. 51 (1987), 1065–1087. (in Russian)
R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, 2001.
W.M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993), pp. 101.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
