Geodesic
On the local smoothness of some class of axi-symmetric solutions to the MHD equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 127-148 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper we consider a special class of weak axi-symmetric solutions to the MHD equations for which the velocity field has only poloidal component and the magnetic field is toroidal. We prove local regularity for such solutions. The global strong solvability of the initial-boundary value problem for the corresponding system in a cylindrical domain with non-slip boundary conditions for the velocity on the cylindrical surface is established as well. 
@article{ZNSL_2017_459_a7,
     author = {T. Shilkin},
     title = {On the local smoothness of some class of axi-symmetric solutions to the {MHD} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {127--148},
     year = {2017},
     volume = {459},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/}
}
TY  - JOUR
AU  - T. Shilkin
TI  - On the local smoothness of some class of axi-symmetric solutions to the MHD equations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 127
EP  - 148
VL  - 459
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/
LA  - en
ID  - ZNSL_2017_459_a7
ER  - 
%0 Journal Article
%A T. Shilkin
%T On the local smoothness of some class of axi-symmetric solutions to the MHD equations
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 127-148
%V 459
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/
%G en
%F ZNSL_2017_459_a7
T. Shilkin. On the local smoothness of some class of axi-symmetric solutions to the MHD equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 127-148. http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/

[1] 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 | DOI | MR | Zbl

[2] Ch. He, Zh. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations”, J. Differential Equations, 213:2 (2005), 235–254 | DOI | MR | Zbl

[3] K. Kang, “Regularity of axially symmetric flows in a half-space in three dimensions”, SIAM Journal of Math. Analysis, 35:6 (2004), 1636–1643 | DOI | MR | Zbl

[4] K. Kang, J. Lee, “Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations”, J. Differential Equations, 247 (2009), 2310–2330 | DOI | MR | Zbl

[5] G. Koch, N. Nadirashvili, G. Seregin, V. Sverak, “Liouville theorems for the Navier-Stokes equations and applications”, Acta Math., 203:1 (2009), 83–105 | DOI | MR | Zbl

[6] O. A. Ladyzhenskaya, “On the unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry”, Zapiski Nauchn. Sem. LOMI, 7, 1968, 155–177 (in Russian) | MR | Zbl

[7] 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), 115–173 (in Russian) | MR

[8] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, 1968 | DOI | MR

[9] Zh. Lei, “On axially symmetric incompressible magnetohydrodynamics in three dimensions”, J. Differential Equations, 259 (2015), 3202–3215 | DOI | MR | Zbl

[10] S. Leonardi, J. Malek, J. Necas, M. Pokorny, “On axially symmetric flows in $\mathbb R^3$”, J. for Analysis and its Applications, 18:3 (1999), 639–649 | MR | Zbl

[11] A. I. Nazarov, N. N. Uraltseva, “The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients”, St. Petersburg Math. J., 23:1 (2012), 93–115 | DOI | MR | Zbl

[12] B. Nowakowski, W. Zajaczkowski, “On global regular solutions to magnetohydrodynamics in axi-symmetric domains”, Z. Angew. Math. Phys., 67:6 (2016), Art. 142, 22 pp. | DOI | MR

[13] G. A. Seregin, “Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary”, J. Mathematical Fluid Mechanics, 4:1 (2002), 1–29 | DOI | MR | Zbl

[14] G. A. Seregin, “Differentiability properties of weak solutions of the Navier–Stokes equations”, St.-Petersburg Math. Journal, 14:1 (2003), 147–178 | MR | Zbl

[15] G. Seregin, V. Sverak, “On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations”, Comm. Partial Differential Equations, 34:1–3 (2009), 171–201 | DOI | MR | Zbl

[16] J. Math. Sci. (N.Y.), 132:3 (2006), 339–358 | DOI | MR | Zbl

[17] G. Seregin, T. Shilkin, “The local regularity theory for the Navier-Stokes equations near the boundary”, Proceedings of the St. Petersburg Mathematical Society, v. 15, Amer. Math. Soc. Transl. Ser. 2, 232, Advances in mathematical analysis of partial differential equations, Amer. Math. Soc., Providence, RI, 2014, 219–244 | MR | Zbl

[18] Journal of Mathematical Sciences (New York), 178:3 (2011), 243–264 | DOI | MR | Zbl

[19] V. Vyalov, “On the regularity of weak solutions to the MHD system near the boundary”, J. Math. Fluid Mech., 16:4 (2014), 745–769 | DOI | MR

[20] E. Zadrzyrnska, W. Zajaczkowski, “Global regular solutions with large swirl to the Navier–Stokes equations in a cylinder”, J. Mathematical Fluid Mechanics, 11 (2009), 126–169 | DOI | MR

[21] W. Zajaczkowski, “Stability of two-dimensional solutions to the Navier–Stokes equations in cylindrical domains under Navier boundary conditions”, J. Math. Anal. Appl., 444:1 (2016), 275–297 | DOI | MR | Zbl