A priori estimates for 3D incompressible current-vortex sheets

Jean-François Coulombel (1) , Alessandro Morando (2) , Paolo Secchi (2) , Paola Trebeschi (2)

1 SIMPAF - SImulations and Modeling for PArticles and Fluids
2 Dipartimento di Matematica [Universita di Brescia]

We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.

Mots clés: Magneto-hydrodynamics; incompressible fluids; current-vortex sheets; free boundary; stability

Domaines: Equations aux dérivées partielles [math.AP]

Jean-François Coulombel, Alessandro Morando, Paolo Secchi, Paola Trebeschi. A priori estimates for 3D incompressible current-vortex sheets. Communications in Mathematical Physics, 2012, 311 (1), pp.247-275. ⟨10.1007/s00220-011-1340-8⟩. ⟨hal-00565759⟩