TY - JOUR

T1 - Construction of a two-phase flow with singular energy by gradient flow methods

AU - Cancès, Clément

AU - Matthes, Daniel

N1 - Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/1/25

Y1 - 2023/1/25

N2 - 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.

AB - 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.

KW - Degenerate parabolic equation

KW - Minimizing movement scheme

KW - Spinodal decomposition

KW - Wasserstein gradient flow

UR - http://www.scopus.com/inward/record.url?scp=85141805044&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2022.11.010

DO - 10.1016/j.jde.2022.11.010

M3 - Article

AN - SCOPUS:85141805044

SN - 0022-0396

VL - 344

SP - 439

EP - 470

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -