Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods

Basil Nicolaenko; Alex Mahalov; Timofey Shilkin

Séminaire Équations aux dérivées partielles (2006-2007)

  • Volume: 2006-2007, page 1-19

Abstract

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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.

How to cite

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Nicolaenko, Basil, Mahalov, Alex, and Shilkin, Timofey. "Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods." Séminaire Équations aux dérivées partielles 2006-2007 (2006-2007): 1-19. <http://eudml.org/doc/11157>.

@article{Nicolaenko2006-2007,
abstract = {We demonstrate that there exist no self-similar solutions of the incompressible magnetohydrodynamics (MHD) equations in the space $L^3 (\mathbf\{R\}^3)$. 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.},
author = {Nicolaenko, Basil, Mahalov, Alex, Shilkin, Timofey},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {bowup methods; self-similarity},
language = {eng},
pages = {1-19},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods},
url = {http://eudml.org/doc/11157},
volume = {2006-2007},
year = {2006-2007},
}

TY - JOUR
AU - Nicolaenko, Basil
AU - Mahalov, Alex
AU - Shilkin, Timofey
TI - Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods
JO - Séminaire Équations aux dérivées partielles
PY - 2006-2007
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2006-2007
SP - 1
EP - 19
AB - We demonstrate that there exist no self-similar solutions of the incompressible magnetohydrodynamics (MHD) equations in the space $L^3 (\mathbf{R}^3)$. 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.
LA - eng
KW - bowup methods; self-similarity
UR - http://eudml.org/doc/11157
ER -

References

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  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. Zbl1064.35134MR1992563
  3. L. Escauriaza, G. Seregin, V. Šverak, On backward uniqueness for parabolic equations, Arch. Rational Mech. Anal., 169 (2003) no. 2, pp. 147-157. Zbl1039.35052MR2005639
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  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. Zbl0884.35115
  9. V. Scheffer,Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), pp. 97-112. Zbl0357.35071MR510154
  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. Zbl1059.35101MR1992567
  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. Zbl0916.35084
  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. Zbl1072.35154MR2142366

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