Topology-Preserving Diffusion of Divergence-Free Vector Fields and Magnetic Relaxation
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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.
KeywordsDiffusion Equation Euler Equation Heat Equation Magnetic Relaxation Admissible Pair
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- 1.Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser, (2008)Google Scholar
- 2.Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195, 289–391 (2014)Google Scholar
- 4.Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics. In: Applied Mathematical Sciences, vol. 125. Springer, New York (1998)Google Scholar
- 6.Brenier, Y.: Extended Monge–Kantorovich Theory, Lecture Notes in Mathematics, vol. 1813, Springer, pp. 91–122 (2003)Google Scholar
- 14.Lions, P.-L.: Mathematical Topics in Fluid Mechanics. vol. 1. Incompressible models, Oxford Lecture Series in Mathematics and its Applications, pp. 3 (1996)Google Scholar
- 16.Mielke, A.: Differential, energetic, and metric formulations for rate-independent processes. In: Nonlinear PDE’s and Applications. Lecture Notes in Mathematics, vol. 2028, pp. 87–170. Springer, Heidelberg (2011)Google Scholar
- 22.Villani, C.: Topics in optimal transportation, Grad. Stud. Math., vol. 58, AMS (2003)Google Scholar