Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

Abstract We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in R n , n = 2 , 3 , namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier–Stokes equations in R n . In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.

[1]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[2]  Christiane Marliani,et al.  Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D , 2000 .

[3]  A. Majda,et al.  Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit , 1981 .

[4]  Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations , 2006, math/0601060.

[5]  J. Gibbon,et al.  Evidence for singularity formation in a class of stretched solutions of the equations for ideal MHD , 2001 .

[6]  K. Ohkitani,et al.  A note on regularity conditions on ideal magnetohydrodynamic equations , 2006 .

[7]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[8]  Roger Temam,et al.  Some mathematical questions related to the MHD equations , 1983 .

[9]  C. Fefferman,et al.  Behavior of several two-dimensional fluid equations in singular scenarios , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[10]  P. Constantin,et al.  An Eulerian-Lagrangian approach for incompressible fluids: Local theory , 2000 .

[11]  Y. Giga,et al.  Asymptotically self‐similar blow‐up of semilinear heat equations , 1985 .

[12]  R. Grauer.  Adaptive Mesh Refinement for Singular Structures in Incompressible Hydro- and Magnetohydrodynamic Flows , 1999 .

[13]  J. Necas,et al.  On Leray's self-similar solutions of the Navier-Stokes equations , 1996 .

[14]  Tai-Peng Tsai.  On Leray's Self‐Similar Solutions of the Navier‐Stokes Equations Satisfying Local Energy Estimates , 1998 .

[15]  S. Chandrasekhar.  Hydrodynamic and Hydromagnetic Stability , 1961 .

[16]  Tosio Kato.  Nonstationary flows of viscous and ideal fluids in R3 , 1972 .

[17]  Russel E. Caflisch,et al.  Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD , 1997 .

[18]  D. Chae.  Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations , 2006, math/0604234.