Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

Choi, Jongkeun; Kim, Seick

doi:10.1016/j.jfa.2014.08.011

Cited by 10 publications

(4 citation statements)

References 23 publications

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“…which follow from the uniform bounds established in [5]. We define ρ n (x) = ´Ω η n (x − y)ρ(y) dy where η n (x) = n d ωn χ B 1/n (x) and ϕ n (x) = ´Ω G(x, y)ρ n (y) dy.…”

Section: Uniquenessmentioning

confidence: 99%

A density-constrained model for chemotaxis

Kim

Mellet

2023

Nonlinearity

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We consider a model of congestion dynamics with chemotaxis: the density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incompressibility constraint results in the formation of patches, describing regions where the maximal density has been reached. The dynamics of these patches can be described by either Hele-Shaw or Richards equation type flow (depending on whether we consider the model with diffusion or the model with pure advection). Our focus in this paper is on the construction of weak solutions for this problem via a variational discrete time scheme of JKO type. We also establish the uniqueness of these solutions. In addition, we make more rigorous the connection between this incompressible chemotaxis model and the free boundary problems describing the motion of the patches in terms of the density and associated pressure variable. In particular, we obtain new results characterising the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.

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“…which follow from the uniform bounds established in [5]. We define ρ n (x) = ´Ω η n (x − y)ρ(y) dy where η n (x) = n d ωn χ B 1/n (x) and ϕ n (x) = ´Ω G(x, y)ρ n (y) dy.…”

Section: Uniquenessmentioning

confidence: 99%

A density-constrained model for chemotaxis

Kim

Mellet

2023

Nonlinearity

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“…For another approach, we refer to Dong-Kim [17], where they constructed Green functions for the elliptic systems by using heat kernel estimates, the argument in which requires first establishing pointwise bounds for the heat kernels. See also [3,10,12] and the references therein for work in this direction. Lastly, we would like to mention two papers [28,27] on the Green functions for the mixed problems in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

Green functions for stationary Stokes systems with conormal derivative boundary condition in two dimensions

Choi¹,

Kim²

2023

Preprint

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“…For another approach, we refer to Dong-Kim [18], where they constructed Green functions for the elliptic systems by using heat kernel estimates, the argument in which requires first establishing pointwise bounds for the heat kernel. See also [3,11,13] and the references therein for work in this direction. Lastly, we would like to mention a paper by Talyor et al [29] for another adaptation of the approach in [20].…”

Section: Introductionmentioning

confidence: 99%