We study a Cahn–Hilliard–Hele–Shaw (or Cahn–Hilliard–Darcy) system for an incompressible mixture of two fluids. The relative concentration difference φ is governed by a convective nonlocal Cahn–Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity u obeys a Darcy’s law depending on the so-called Korteweg force μ∇ φ, where μ is the nonlocal chemical potential. In addition, the kinematic viscosity η may depend on φ. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on u are also obtained. Weak–strong uniqueness is demonstrated in the two-dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if η is constant. Otherwise, weak–strong uniqueness is shown by assuming that the pressure of the strong solution is α-Hölder continuous in space for α∈ (1 / 5 , 1).

Nonlocal Cahn–Hilliard–Hele–Shaw Systems with Singular Potential and Degenerate Mobility / C. Cavaterra, S. Frigeri, M. Grasselli. - In: JOURNAL OF MATHEMATICAL FLUID MECHANICS. - ISSN 1422-6928. - 24:1(2022 Feb), pp. 13.1-13.49. [10.1007/s00021-021-00648-1]

Nonlocal Cahn–Hilliard–Hele–Shaw Systems with Singular Potential and Degenerate Mobility

C. Cavaterra
Primo
;
S. Frigeri
Penultimo
;
2022

Abstract

We study a Cahn–Hilliard–Hele–Shaw (or Cahn–Hilliard–Darcy) system for an incompressible mixture of two fluids. The relative concentration difference φ is governed by a convective nonlocal Cahn–Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity u obeys a Darcy’s law depending on the so-called Korteweg force μ∇ φ, where μ is the nonlocal chemical potential. In addition, the kinematic viscosity η may depend on φ. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on u are also obtained. Weak–strong uniqueness is demonstrated in the two-dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if η is constant. Otherwise, weak–strong uniqueness is shown by assuming that the pressure of the strong solution is α-Hölder continuous in space for α∈ (1 / 5 , 1).
Cahn–Hilliard equation; Darcy’s law; Degenerate mobility; Logarithmic potential; Non-constant viscosity; Nonlocal free energy; Regularity; Strong solutions; Uniqueness; Weak solutions;
Settore MAT/05 - Analisi Matematica
feb-2022
18-dic-2021
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/898893
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