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

Clément Cancès (1) , Daniel Matthes (2)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich

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 [C. Canc\`es, D. Matthes & F. Nabet, Arch. Ration. Mech. Anal., 233:837–866, 2019] is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases, leading to new theoretical difficulties.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Clément Cancès, Daniel Matthes. Construction of a two-phase flow with singular energy by gradient flow methods. Journal of Differential Equations, 2023, 344, pp.439-470. ⟨10.1016/j.jde.2022.11.010⟩. ⟨hal-02510535v2⟩