On the mass-conserving Allen-Cahn approximation for incompressible binary fluids. (English) Zbl 1504.35225
Summary: This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling the motion of an incompressible two-phase fluid mixture in presence of capillarity effects in a bounded smooth domain \(\Omega \subset \mathbb{R}^d\), \(d = 2, 3\). We focus on dissipative mixing effects originating from the mass-conserving Allen-Cahn dynamics with the physically relevant Flory-Huggins potential. More precisely, we study the mass-conserving Navier-Stokes-Allen-Cahn system for nonhomogeneous fluids and the mass-conserving Euler-Allen-Cahn system for homogeneous fluids. We prove existence and uniqueness of global weak and strong solutions as well as their property of separation from the pure states. In our analysis, we combine the energy and entropy estimates, a novel end-point estimate of the product of two functions, a new estimate for the Stokes problem with non-constant viscosity, and logarithmic type Gronwall arguments.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35Q31 | Euler equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D27 | Other free boundary flows; Hele-Shaw flows |
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
| 76T06 | Liquid-liquid two component flows |
| 35D35 | Strong solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
Keywords:
two-phase flow; mass-conserving Allen-Cahn approximation; well-posedness; logarithmic potentialReferences:
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