Advertisement

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 Brenier
Article

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, 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.

Keywords

Diffusion Equation Euler Equation Heat Equation Magnetic Relaxation Admissible Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 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
  3. 3.
    Andreu F., Caselles V., Mazon J.: Finite propagation speed for limited flux diffusion equations. Arch. Ration. Mech. Anal. 269–297 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics. In: Applied Mathematical Sciences, vol. 125. Springer, New York (1998)Google Scholar
  5. 5.
    Born M., Infeld L.: Foundations of the new field theory. Proc. Roy. Soc. Lond. A 425–451 (1934)ADSCrossRefGoogle Scholar
  6. 6.
    Brenier, Y.: Extended Monge–Kantorovich Theory, Lecture Notes in Mathematics, vol. 1813, Springer, pp. 91–122 (2003)Google Scholar
  7. 7.
    Brenier Y.: Hydrodynamic structure of the augmented Born–Infeld equations. Arch. Ration. Mech. Anal. 65–91 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brenier Y., De Lellis C., Székelyhidi László L. Jr: Weak–strong uniqueness for measure-valued solutions. Comm. Math. Phys. 351–361 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Demoulini S., Stuart D., Tzavaras A.: Weak–strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal. 927–961 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Freedman M., He Z.-X.: Divergence-free fields: energy and asymptotic crossing number. Ann. Math.(2), 189–229 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gigli N.: On the Heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Part. Diff. Equat. 101–120 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gigli N., Kuwada K., Ohta S.I.: Heat flow on Alexandrov spaces. Comm. Pure Appl. Math. 307–331 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 1–17 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 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
  15. 15.
    McCann R., Puel M.: Constructing a relativistic heat flow by transport time steps. Ann. Inst. H. Poincaré Anal. Non Linéaire 2539–2580 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 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
  17. 17.
    Moffatt H.K.: Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. J. Fluid Mech. 359–378 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Nishiyama T.: Construction of the three-dimensional stationary Euler flows from pseudo-advected vorticity equations. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 2393–2398 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nishiyama T.: Magnetohydrodynamic approaches to measure-valued solutions of the two-dimensional stationary Euler equations. Bull. Inst. Math. Acad. Sin. (N.S.) 2, 139–154 (2007)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Rivière T.: High-dimensional helicities and rigidity of linked foliations. Asian J. Math. 505–533 (2002)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 635–664 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Villani, C.: Topics in optimal transportation, Grad. Stud. Math., vol. 58, AMS (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

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

Personalised recommendations