Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On the proof of the solvability of a linear problem arising in magnetohydrodynamics with the method of integral equations


Authors: Sh. Sahaev and V. A. Solonnikov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 6.
Journal: St. Petersburg Math. J. 26 (2015), 985-1003
MSC (2010): Primary 45B05
DOI: https://doi.org/10.1090/spmj/1371
Published electronically: September 21, 2015
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper is concerned with a linear system of Fredholm-Volterra singular integral equations arising in the study of a linearized initial-boundary value problem of magnetohydrodymnamics for a fluid surrounded by an infinite vacuum region. It is proved that this system is solvable in the class of continuous functions satisfying the Hölder condition with respect to the spatial variables, which yields a classical solution of the problem in question.


References [Enhancements On Off] (What's this?)

  • 1. O. A. Ladyzhenskaya and V. A. Solonnikov, Solution of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid, Tr. Mat. Inst. Steklov 59 (1960), 115-173. (Russian) MR 0170130 (30:371)
  • 2. -, On linearization principle and invariant manifolds for problems of magnetohydrodynamics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 46-93. (Russian) MR 0377310 (51:13482)
  • 3. S. Mosconi and V. A. Solonnikov, On a problem of magnetohydrodynamics in a multi-connected domain, Nonlinear Anal. 74 (2011), no. 2, 462-478. MR 2733223 (2011g:35320)
  • 4. Sh. Sahaev, Solution of the first initial-bondary value problem for nonstationary systems of Maxwell's equations (potential theory). I, Akad. Nauk Kazah. SSR Tr. Inst. Mat. i Meh. 2 (1971), 69-77. (Russian) MR 0454424 (56:12675)
  • 5. G. Grubb, Functional calculus of pseudodifferential boundary problems, Progr. Math., vol. 65, Birkhäuser, Boston, MA, 1996. MR 1385196 (96m:35001)
  • 6. V. A. Solonnikov, An initial-boundary value problem for a generalized system of Stokes equations in a half-space, Zap. Nauchn. Sem. S.-Petgerburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 224-275; English transl., J. Math. Sci. (N. Y.) 115 (2003), no. 6, 2832-2861. MR 1810619 (2002b:76045)
  • 7. V. I. Smirnov, A course of higher mathematics. Vol. IV, Gosudarstv. Izd. Tehn.-Teor. Lit., Moscow, 1951; English transl., Pergamon Press, Oxford, 1964. MR 0177069 (31:1333)
  • 8. -, A course of higher mathematics. Vol. II, Nauka, Moscow, 1974; English transl., Pergamon Press, Oxford, 1964. MR 0182688 (32:171), MR 0182688 (32:173)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 45B05

Retrieve articles in all journals with MSC (2010): 45B05


Additional Information

Sh. Sahaev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
Email: Sahaev@pdmi.ras.ru

V. A. Solonnikov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
Email: solonnik@pdmi.ras.ru

DOI: https://doi.org/10.1090/spmj/1371
Keywords: Fredholm--Volterra singular integral equations, classical solution
Received by editor(s): August 4, 2014
Published electronically: September 21, 2015
Dedicated: Dedicated to Professor G. Grubb on the occasion of her jubilee
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society