Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation

Yann Brenier (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of "dissipative solutions", which shares common features with both P.-L. Lions' dissipative solutions to the Euler equations and the concept of "curves of maximal slopes", a la De Giorgi, recently used to study the scalar heat equation in very general metric spaces.

Mots clés: divergence-free vector fields; degenerate diffusion; dissipative solutions

Domaines: Equations aux dérivées partielles [math.AP]

Yann Brenier. Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation. 2014, pp.757-770. ⟨hal-00814263⟩