Abstract
A model from electro-magneto-hydrodynamics describing a completely ionized gas, a plasma, is studied. Local well posedness of the problem in time-weighted \(L_p\) spaces is obtained by means of maximal regularity of the linearized problem, and the induced local semiflow in the proper state space is constructed. If the underlying domain is bounded and simply connected, based on the principle of linearized stability, it is shown that the trivial solution of the problem is exponentially stable. Moreover, if the orbit of a solution is relatively compact in the state space, the solution exists globally, and it converges to the trivial equilibrium.
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References
C. Amrouche, N.el H. Seoula, \(L^p\)-theory for vector potentials and Sobolev’s inequalities for vector fields. Math. Models & Methods in Applied Sciences 23, 37–92 (2013)
W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics 96, Birkhäuser Verlag (2001).
K-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, Springer, (2000).
H. Fujita and T. Kato, On the non-stationary Navier-Stokes system. Rend. Sem. Mat., Univ. Padova 32, 243–260 (1962)
Y. Giga, Z. Yoshida, On the equations of the two-component theory in magnetohydrodynamics. Comm. Part. Diffential Eqns. 9, 503–522 (1984).
M. Köhne, J. Prüss, M. Wilke, On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces. J. Evol. Eqns. 10, 443–463 (2010).
K. Miyamoto, Plasma Physics for Nuclear Fusion. MIT, Cambridge, MA 1981
J. Prüss, Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser Verlag (1993).
J. Prüss, Maximal regularity for evolution equations in \(L_p\)-spaces. Conf. Sem. Mat. Univ. Bari 285, 1–39 (2003).
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Problems. Monographs in Mathematics 105, Birkhäuser Verlag 2016.
J. Prüss, M. Wilke, Addendum to the paper “On quasi-linear parabolic evolution equations in weighted \(L_p\)-spaces II”. J. Evolution Eqns. to appear (2017).
G. Ströhmer, About an initial-boundary value problem from magneto-hydrodynamics. Math. Z. 209, 345–362 (1992).
G. Ströhmer, An existence result for partially regular weak solutions of certain abstract evolution equations, with an application to magneto-hydrodynamics. Math. Z. 213, 373–385 (1993).
N.G. Van Kampen, B.U. Felderhof, Theoretical Methods in Plasma Physics. North-Holland, Amsterdam 1967.
Z. Yoshida, Y. Giga, On the Ohm-Navier-Stokes system in magnetohydrodynamics. J. Math. Phys. 24, 2860–2864 (1983).
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The research of S.S was partially supported by JSPS Grant-in-Aid for Scientific Research (B)—16H03945, MEXT.
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Prüss, J., Shimizu, S. On a Navier–Stokes–Ohm problem from plasma physics. J. Evol. Equ. 18, 351–371 (2018). https://doi.org/10.1007/s00028-017-0404-4
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DOI: https://doi.org/10.1007/s00028-017-0404-4
Keywords
- Navier–Stokes–Ohm problem
- Electro-magneto-hydrodynamics
- Plasma physics
- Stokes operator
- Maxwell operator
- Maximal regularity
- Critical spaces
- Well posedness
- Nonlinear stability
- Global existence
- Convergence