Communications in Mathematical Physics

, Volume 330, Issue 2, pp 757–770 | Cite as

Topology-Preserving Diffusion of Divergence-Free Vector Fields and Magnetic Relaxation

  • Yann BrenierEmail author


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, one 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’s dissipative solutions to the Euler equations and the concept of “curves of maximal slopes”, à la De Giorgi, recently used to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits such global solutions, at least in the two space variable case, and they are unique whenever they are smooth.


Diffusion Equation Euler Equation Heat Equation Magnetic Relaxation Admissible Pair 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNRS, Centre de mathématiques Laurent SchwartzEcole PolytechniquePalaiseauFrance

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