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Reports Communications in Mathematical Physics Year : 2014

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

Yann Brenier

Abstract

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.
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Dates and versions

hal-00814263 , version 1 (16-04-2013)

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