Construction of a two-phase flow with singular energy by gradient flow methods

Clément Cancès, Daniel Matthes

Research output: Contribution to journalArticlepeer-review


We prove the existence of weak solutions to a system of two diffusion equations that are coupled by a pointwise volume constraint. The time evolution is given by gradient dynamics for a free energy functional. Our primary example is a model for the demixing of polymers, the corresponding energy is the one of Flory, Huggins and de Gennes. Due to the non-locality in the equations, the dynamics considered here is qualitatively different from the one found in the formally related Cahn-Hilliard equations. Our angle of attack is from the theory of optimal mass transport, that is, we consider the evolution equations for the two components as two gradient flows in the Wasserstein distance with one joint energy functional that has the volume constraint built in. The main difference to our previous work [6] is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases, and thus requires a more sophisticated analysis.

Original languageEnglish
Pages (from-to)439-470
Number of pages32
JournalJournal of Differential Equations
StatePublished - 25 Jan 2023


  • Degenerate parabolic equation
  • Minimizing movement scheme
  • Spinodal decomposition
  • Wasserstein gradient flow


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