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The Journal of Dynamical Systems and Geometric Theories (JDSGT) is a world leading journal publishing high quality, rigorously peer-reviewed original research on dynamical systems and geometry, including the interactions between these two subjects and interdisciplinary research with other branches of knowledge since 2003. Topics published by the Journal include but are not limited to: Random dynamical systems Geometry and physics Dynamical systems with hyperbolic behaviour Real and complex geometry Ergodic theory Distance geometry Topological dynamics Global differential geometry Infinite-dimensional Hamiltonian systems Symplectic geometry, contact geometry The journal considers original research articles, survey articles, and book reviews for publication. Responses to articles and correspondence will also be considered at the Chief Editor’s discretion.

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Open Access Research Article

A special class of solutions for three dimensional steady state magnetohydrodynamics equations with nonzero boundary velocity

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pp. 87–98Vol. 22Issue 1 & 2November 2024DOI: 10.47974/JDSGT-2024-010XML
Received:
11 Jun 2024
Published Online:
01 Nov 2024
Article type:
Research Article
Language:
EN
Article no.:
JDSGT-2024-010
Pages:
87–98

Abstract

Following the work of Farwig, Galdi, Simader and Sohr [1, 2], we prove existence of a special class of solutions for three-dimensional steady state magnetohydrodynamic flows with non-zero boundary velocity. We compare our results with previously known results on stationary magnetohydrodynamic equations. We also make some remarks on uniqueness of solutions.

Keywords

Subject Classifications

35B3535D3047H10

References

[1] R. Farwig, G. P. Galdi and H. Sohr, “Very Weak Solutions of Stationary and Instationary Navier-Stokes Equations with Nonhomogeneous Data,” Nonlinear Differential Equations and Their Applications, Vol. 64, pp. 113–136 (2005). [2] G. P. Galdi, C. G. Simader, and H. Sohr, “A Class of Solutions to Stationary Stokes and Navier-Stokes Equations with Boundary Data in W-1/q,q(∂Ω),” Math. Ann., 331, pp. 41-74 (2005).[3] H. Kim, “Existence and Regularity of Very Weak Solutions of the Stationary Navier–   Stokes Equations,” Arch. Rational Mech. Anal. 193, pp. 117–152 (2009).[4] Y. Giga, “Analyticity of the semigroup generated by the Stoke’s operator in Lr – spaces,” Math. Z. vol. 178, pp. 297-329 (1981). [5] R. A. Adams, “Sobolev Spaces”, 1st ed. Academic Press, New York (1975).[6] G. P. Galdi, “ An Introduction to the Mathematical Theory of the Navier-Stokes Equations Linearized Steady Problems”, Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, Revised Edition (1998).[7] H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Birkh¨auser Advanced Texts, Birkh¨auser Verlag, Basel (2001).[8] C. G. Simader  and H. Sohr H, The Dirichlet problem for the Laplacian in bounded and unbounded domains,  Pitman Research notes in Mathematics Series, Longman, vol. 360 (1997).[9] G. P. Galdi, C. G. Simader, H. Sohr, “ On the stokes problem in Lipschitz domains,” Ann. Math. Pur Appl. Vol. 167, pp. 147-163 (1994).[10] R. Temam, Navier –Stokes Equations, North – Holland Pub. Co. Amsterdam – New York- Tokyo (1977).[11] S. V. Uddhao, P. D. Raiter, R. V. Saraykar, “ Stability of steady state solutions with finite energy for the magnetohydrodynamic flows in the whole space R3”, Indian Journals of Mathematics Vol.63, No.3, pp. 393-41 (2021).
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