Regularity criteria for 3D MHD flows in terms of spectral components

Abstract

We extend the spectral regularity criteria of the Prodi-Serrin kind for the Navier-Stokes equations in a torus to the MHD equations. More precisely, the following is established: for any $ N > 0 $, let $ {{\boldsymbol x}}_{N} $ and $ {{\boldsymbol y}}_{N} $ be the sum of all spectral components of the velocity and magnetic field whose wave numbers possess absolute value greater that $ N $; then, it is possible to show that for any $ N $ the finiteness of the Prodi-Serrin norm of $ {{\boldsymbol x}}_{N} $ implies the regularity of the weak solution $ ({{\boldsymbol u}}, {{\boldsymbol h}}) $; thus, no restriction on the magnetic field is needed.

Keywords

• mhd equations
• prodi-serrin criteria
• spectral galerkin approximations
• regularity