MHD equations in a bounded domain
Abstract
We consider the MHD system in a bounded domain Ω⊂ℝN, N = 2, 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L2(Ω))N × (L2(Ω))N. Solutions to MHD system are obtained next as a limits of that fractional approximations.
Keywords
MHD equations; abstract parabolic problem; fractional approximations
References
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18. H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285–346.
19. H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J. (2) 41 (1989), 471–488.
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21. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
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27. R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.
28. H. Yu, Global regularity to the 3D MHD equations with large initial data in bounded domains, J. Math. Phys. 57 (2016), 083102, 8 pp.
29. Y. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl. 17 (2014), 245–251.
30. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), 284–312.
2. J.W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.
3. J.W. Cholewa and T. Dlotko, Fractional Navier–Stokes equation, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), 2967–2988.
4. T. Dlotko, Navier–Stokes equation and its fractional approximations, Appl. Math. Optim. 77 (2018), 99–128.
5. T. Dlotko, M.B. Kania, and C. Sun, Pseudodifferential parabolic equations in uniform spaces, Appl. Anal. 93 (2014), 14–34.
6. T. Dlotko, M.B. Kania, and C. Sun, Pseudodifferential parabolic equations; two examples, Topol. Methods Nonlinear Anal. 43 (2014), 463–492.
7. T. Dlotko, M.B. Kania, and C. Sun, Quasi-geostrophic equation in R^2, J. Differential Equations 259 (2015), 531–561.
8. G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal. 46 (1972), 241–279.
9. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
10. H. Fujita and T. Kato, On the Navier–Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315.
11. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_r spaces, Math. Z. 178 (1981), 297–329.
12 Y. Giga, Domains of fractional powers of the Stokes operator in L_r spaces, Arch. Rational Mech. Anal. 89 (1985), 251–265.
13. Y. Giga and T. Miyakawa, Solutions in L_r of the Navier–Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267–281.
14. C. He, X. Huang, and Y. Wang, On some new global existence results of 3D magnetohydrodynamic equations, Nonlinearity 27 (2014), 343–352.
15. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
16. M.B. Kania, Fractional Burgers equation in a bounded domain, Colloq. Math. 151 (2018), 57–70.
17. H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), 22–35.
18. H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285–346.
19. H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J. (2) 41 (1989), 471–488.
20. Y. Lin, H. Zhang, and Y Zhou, Global smooth solutions of MHD equations with large data, J. Differential Equations 261 (2016), 102–112.
21. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
22. C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001.
23. C. Miao and B. Yuan, Well-posedness of the MHD system in critical Besov spaces, Methods Appl. Anal. 13 (2006), 89–106.
24. C. Miao, B. Yuan, and B. Zhang, Well-posedness for the incompressible magnetohydrodynamic system, Math. Methods Appl. Sci. 30 (2007), 961–976.
25. J.C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.
26. M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), 635–664.
27. R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.
28. H. Yu, Global regularity to the 3D MHD equations with large initial data in bounded domains, J. Math. Phys. 57 (2016), 083102, 8 pp.
29. Y. Wang and K. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. Real World Appl. 17 (2014), 245–251.
30. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), 284–312.
KaniaM. B. (2021). MHD equations in a bounded domain. Annales Mathematicae Silesianae, 35(2), 211-235. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13448
Maria B. Kania
maria.kania@us.edu.pl
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
https://orcid.org/0000-0001-8881-9421
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
https://orcid.org/0000-0001-8881-9421
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