Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

Elbar, Charles; Perthame, Benôıt; Poulain, Alexandre

doi:10.4310/cms.2022.v20.n7.a5

Cited by 6 publications

(9 citation statements)

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“…the Cahn-Hilliard equation. The Cahn-Hilliard equation Equation ( 11) is an example of a Cahn-Hilliard type equation that is widely used nowadays to represent phase transitions in fluids and living tissues [7,8,10,[12][13][14][15]23,27,34]. Originally introduced in the context of materials sciences [2,3], it is currently applied in numerous fields, including complex fluids, polymer science, and mathematical biology.…”

Section: Literature Review and Relevancy Of The System Phase Transiti...mentioning

confidence: 99%

“…Note, however, that different scaling between α and ε could possibly be considered. Recalling the mass balance equation ( 9) and the ξ -moment equation (10), our aim is to take the limit ε → 0 in these equations, and establish the relations…”

Section: The Limit ε →mentioning

confidence: 99%

“…The latter estimate on J ε also tells us that ε 2 ∂ t J ε (t, x) converges to 0 in the distributional sense. Therefore, establishing the equation (30) from equation (10), is reduced to proving the two local weak limits in…”

Section: The Limit ε →mentioning

confidence: 99%

“…(1) power-type potential F (u) = |u| γ , γ > 2 used in the context of tumor growth models [27,36,38,66],…”

Section: Introductionmentioning

confidence: 99%

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From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

Elbar

Mason

Perthame

et al. 2023

Commun. Math. Phys.

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We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux.

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Section: Literature Review and Relevancy Of The System Phase Transiti...mentioning

confidence: 99%

Section: The Limit ε →mentioning

confidence: 99%

Section: The Limit ε →mentioning

confidence: 99%

“…(1) power-type potential F (u) = |u| γ , γ > 2 used in the context of tumor growth models [27,36,38,66],…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

Elbar

Mason

Perthame

et al. 2023

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then, one sends δ → 0 and obtains solutions to (1.5)- (1.6). This method is a standard way of proving existence of solutions to Cahn-Hilliard equation with degenerate mobility [28,31,33,47]. Of course, one has to prove estistence to (1.15)- (1.16) by virtue of fixed point method and Schauder's estimates for parabolic equations which is very technical.…”

Section: Introductionmentioning

confidence: 99%