Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 21, 19 p.

We demonstrate that there exist no self-similar solutions of the incompressible magnetohydrodynamics (MHD) equations in the space L 3 (R 3 ). This is a consequence of proving the local smoothness of weak solutions via blowup methods for weak solutions which are locally L 3 . We present the extension of the Escauriaza-Seregin-Sverak method to MHD systems.

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     title = {Local {Smoothness} of {Weak} {Solutions} to the {Magnetohydrodynamics} {Equations} via {Blowup} {Methods}},
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Nicolaenko, Basil; Mahalov, Alex; Shilkin, Timofey. Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 21, 19 p. http://www.numdam.org/item/SEDP_2006-2007____A21_0/

[1] Caffarelli, Kohn, Nirenberg L. Caffarelli, R.V. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. | MR 673830 | Zbl 0509.35067

[2] L. Escauriaza, G. Seregin, V. Šverak, L 3, - solutions to the Navier-Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk, 58 (2003) no. 2, pp. 3-44. | MR 1992563 | Zbl 1064.35134

[3] L. Escauriaza, G. Seregin, V. Šverak, On backward uniqueness for parabolic equations, Arch. Rational Mech. Anal., 169 (2003) no. 2, pp. 147-157. | MR 2005639 | Zbl 1039.35052

[4] O.A. Ladyzhenskaya, G.A. Seregin, On Partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 1 (1999), pp. 356-387. | MR 1738171 | Zbl 0954.35129

[5] O.A. Ladyzhenskaya, V.A. Solonnikov, Mathematical problems of hydrodynamics and magnetohydrodynamics of a viscous incompressible fluid, Proceedings of V.A. Steklov Mathematical Institute, 59 (1960), pp. 115-173 (in Russian). | MR 170130

[6] J. Leray, Sur le mouvement d’un liquide visqueus emplissant l’espace, Acta Math. 63 (1934), pp. 193-248.

[7] F.-H. Lin, A new proof of the Caffarell-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), no. 3, pp. 241-257. | MR 1488514 | Zbl 0958.35102

[8] J. Necas, M. Růžička, V. Šverak, On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (1996), pp. 283-294. | Zbl 0884.35115

[9] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), pp. 97-112. | MR 510154 | Zbl 0357.35071

[10] G. Seregin, Algebra and Analysis, in press.

[11] G. Seregin, Handbook of Mathematical Hydrodynamics, in press.

[12] V.A. Solonnikov, On the estimates of solutions of nonstationary Stokes problem in anisotropic S.L. Sobolev spaces and on the estimate of resolvent of the Stokes problem, Uspekhi Matematicheskih Nauk, 58 (2003) no.2 (350) pp. 123-156. | MR 1992567 | Zbl 1059.35101

[13] Tai-Peng Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying Local Energy Inequality, Arch. Rational Mech. Anal. 143 (1998), pp. 29-51. | Zbl 0916.35084

[14] Cheng He, Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differential Equations 213 (2005), no. 2, pp. 235-254. | MR 2142366 | Zbl 1072.35154