arXiv subjects
Publications and source records attributed to Antoine Mellet.
In this paper, we focus on a new type of non-linear kinetic Fokker-Planck equation where the non-linearity comes from a non-linear diffusion in the velocity variable. The existence of solutions in suitable Lebesgue spaces is proved, together with important entropy estimates on these solutions. We then study the diffusive limit of such equation.
We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of $N$ distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the $N\times N$ matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general $\Gamma$-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele--Shaw problem with surface tension (and contact angle conditions at triple junctions).
We introduce here a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work. These colonies display long-range shuffling induced by congestion, and are also subject to jamming. We start by showing the well-posedness of the general evolution equation. We then derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We establish the main mathematical properties of this singular problem by proving a comparison property and describing the dynamics of the free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.
We study the collective dynamics of a population of particles/organisms subject to self-consistent attraction-repulsion interactions and an external velocity field. The starting point of our analysis is a mean-field kinetic model and we investigate the singular limit corresponding to strong interaction forces. For well-prepared initial data, we show that the population asymptotically concentrates within a domain $\Omega(t)=\Omega_0+X(t)$ whose shape $\Omega_0$ is determined by the minimization of the interaction energy while the evolution of the domain's center of mass $X(t)$ is determined by the external force field. In addition, we show that the internal flow of organisms within this moving domain is described by a classical hydrodynamic model (the lake equation). The first part of our result relies only on the existence and uniqueness of minimizers for the interaction energy and holds for rather general interaction kernels. The second part is proved using a modulated energy method under more restrictive conditions on the nature of the interactions, and assuming that the limiting lake equation admits strong solutions.
The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature driven free boundary problems. In this paper, we focus on the Elliptic-Parabolic PKS model and we obtain the first unconditional convergence result in dimension $2$ and $3$ towards the volume preserving mean-curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on Brakke type formulation of the mean-curvature flow and a reinterpretation of the problem as an Allen-Cahn equation with a nonlocal forcing term.
The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the Parabolic-Elliptic PKS model have shown that when the repulsion is modeled by a nonlinear diffusion term $\rho \nabla \rho^{m-1}$ with $m>2$, this competition leads to phase separation phenomena. Furthermore, in some asymptotic regime corresponding to a large population observed over a long enough time, the interface separating regions of high and low density evolves according to the Hele-Shaw free boundary problem with surface tension. In the present paper, we consider the counterpart of that model, namely the Elliptic-Parabolic PKS model and we prove that the same phase separation phenomena occurs, but the motion of the interface is now described (asymptotically) by a volume-preserving mean-curvature flow.
This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constraint (maximum value of the density). The main goal of the paper is to show that, in one spatial dimension, this continuum PDE can be derived as the mean-field limit of a system of ordinary differential equations that describes the motion of a large number of particles constrained to stay at some finite distance from each others. To show that these two models describe the same dynamics at different scale, we will rely on both the Eulerian and Lagrangian points of view and use two different approximations for the density and pressure variables in the continuum limit.
This paper reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a result of the competition between attractive and repulsive phenomena - especially (but not exclusively) in the context of attractive chemotaxis phenomena. At microscopic scales, particles (or other agents) are represented by spheres of radius $\delta>0$ and we discuss both soft-sphere models (with a pressure term penalizing the overlap of the particles) and hard-sphere models (in which overlap is prohibited). The first case leads to so-called ``blob models" which have received some attention recently as a tool to approximate non-linear diffusion by particle systems. The hard-sphere model is similar to a classical model for congested crowd motion. We review well-posedness results for these models and discuss their relationship to classical continuum description of aggregation-diffusion phenomena in the limit $\delta\to0$: the classical nonlinear drift diffusion equation and its incompressible counterpart. In the second part of the paper, we discuss recent results on the emergence and evolution of sharp interfaces when a large population of particles is considered at appropriate space and time scales: At some intermediate time scale, phase separation occurs and a sharp interface appears which evolves according to a Stefan free boundary problem (and the density function eventually relaxes to a characteristic function - metastable steady state for the original problem). At a larger time scale the attractive forces lead to surface tension phenomena and the evolution of the sharp interface can be described by a Hele-Shaw free boundary problem with surface tension. At that same time scale, we will also discuss the emergence of contact angle conditions for problems set in bounded domains.
We consider a Kantorovich potential associated to an optimal transportation problem between measures that are not necessarily absolutely continuous with respect to the Lebesgue measure, but are comparable to the Lebesgue measure when restricted to balls with radius greater than some $δ>0$. Our main results extend the classical regularity theory of optimal transportation to this framework. In particular, we establish both Hölder and Sobolev regularity results for Kantorovich potentials up to some critical length scale depending on $δ$. Our assumptions are very natural in the context of the numerical computation of optimal maps, which often involves approximating by sums of Dirac masses some measures that are absolutely continuous with densities bounded away from zero and infinity on their supports.
We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the $Γ$ convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the incompressible PKS model (which includes a density constraint $ρ\leq 1$ and a pressure term): It shows that the incompressibility constraint is not necessary to observe phase separation and surface tension phenomena.
We study the incompressible limit of the porous medium equation with a reaction term that is non-monotone with respect to the pressure variable. More specifically we consider reaction terms that are either bistable or monostable. We show that this type of reaction term generates many interesting differences in the qualitative behavior of solutions, in contrast to the problem with monotone reaction terms that have been extensively studied in recent literature. After characterizing the limit problem, we embark on a comprehensive study of the problem in one space dimension, to illustrate the delicate nature of the problem, including the generic nature of non-uniqueness and instability. For compactly supported initial data, we show that the density can either perish or thrive, even if it starts from the same initial data, depending on its initial pressure configuration. When the initial pressure is a characteristic function, we establish the existence of the sharp threshold separating the two behaviors. Lastly we present a detailed analysis of the behavior of traveling waves in this incompressible limit. We study the existence of traveling waves for the limiting model and prove convergence results in the incompressible limit (depending on the reaction term).
We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint. We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we show that in this limit the density of cell converges to the characteristic function of a set whose evolution is described by a Hele-Shaw free boundary problem with surface tension. Our problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions for the density function. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface.
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 characterizing the pressure variable as the solution of an obstacle problem and prove that in the pure advection case the dynamic preserves patches.
We establish the $Γ$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary interest of our results is to identify the effects that the boundary conditions imposed on the potential have on the limiting functional. We consider general Robin boundary conditions, which include Dirichlet and Neumann conditions as particular cases. Depending on the order of the elliptic operator the limiting functional involves the usual perimeter or some fractional perimeter. We also consider the $Γ$-convergence of a related energy functional combining the usual perimeter functional and the nonlocal repulsive interaction energy.
We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (\cite{AKY}, \cite{PQV}, \cite{KP}, \cite{MPQ}). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressure, which solves an obstacle problem at each time in the evolution
We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the {\it unique solution} of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established.
In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. Finally, we connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality (a Gagliardo-Nirenberg type inequality for fractional perimeter).
We carry out the homogenization of time-harmonic Maxwell's equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale, $d$. Starting with the weak formulation of the requisite boundary value problem, we prove the convergence of its solution to a homogenization limit as $d$ approaches zero. The effective permittivity and cell problem express a bulk average from the host dielectric and a surface average germane to the 2D material (metallic layer). We discuss implications of this analysis in the modeling of plasmonic crystals.