arXiv subjects

José A. Carrillo

Publications and source records attributed to José A. Carrillo.

At least 37 records · Page 2Linked to original sources

arXiv2025

Finite Element Approximation of the Fractional Porous Medium Equation

We construct a finite element method for the numerical solution of a fractional porous medium equation on a bounded open Lipschitz polytopal domain $Ω\subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined as the solution of a fractional Poisson equation, involving the fractional Neumann Laplacian in terms of its spectral definition. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero and show that a subsequence of the sequence of finite element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem under consideration. This result can be therefore viewed as a constructive proof of the existence of a nonnegative, energy-dissipative, weak solution to the initial-boundary-value problem for the fractional porous medium equation under consideration, based on the Neumann Laplacian. The convergence proof relies on results concerning the finite element approximation of the spectral fractional Laplacian and compactness techniques for nonlinear partial differential equations, together with properties of the equation, which are shown to be inherited by the numerical method. We also prove that the total energy associated with the problem under consideration exhibits exponential decay in time.

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arXiv2025

A frame approach for equations involving the fractional Laplacian

Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, $s \in (0,1)$, on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to $\mathbb{R}^d$, $d \in \{1,2\}$. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a $6^\text{th}$-order Runge--Kutta time discretization), a fractional heat equation with a time-dependent exponent $s(t)$, and a two-dimensional problem, observing spectral convergence in the spatial dimension for sufficiently smooth data.

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arXiv2025

A minimax method for the spectral fractional Laplacian and related evolution problems

We present a numerical method for the approximation of the inverse of the fractional Laplacian $(-Δ)^{s}$, based on its spectral definition, using rational functions to approximate the fractional power $A^{-s}$ of a matrix $A$, for $0<s<1$. The proposed numerical method is fast and accurate, benefiting from the fact that the matrix $A$ arises from a finite element approximation of the Laplacian $-Δ$, which makes it applicable to a wide range of domains with potentially irregular shapes. We make use of state-of-the-art software to compute the best rational approximation of a fractional power. We analyze the convergence rate of our method and validate our findings through a series of numerical experiments with a range of exponents $s \in (0,1)$. Additionally, we apply the proposed numerical method to different evolution problems that involve the fractional Laplacian through an interaction potential: the fractional porous medium equation and the fractional Keller-Segel equation. We then investigate the accuracy of the resulting numerical method, focusing in particular on the accurate reproduction of qualitative properties of the associated analytical solutions to these partial differential equations.

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arXiv2025

A nonlocal-to-local approach to aggregation-diffusion equations

Over the past decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models, and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn-Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models, while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. Additionally, we discuss recent analytical results based on the gradient-flow structure of the model, along with open problems and future research directions.

q-bio.CB↗

arXiv2025

Rate of Convergence for a Nonlocal-to-local Limit in One Dimension

We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. Our proof is very simple, exploiting the so-called Evolutionary Variational Inequality for both the nonlocal and local equations as well as a priori estimates. We also present numerical simulations using the finite volume method, which suggests that the obtained rate can be improved - this will be addressed in a forthcoming work.

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arXiv2025

Spatial segregation across travelling fronts in individual-based and continuum models for the growth of heterogeneous cell populations

We consider a partial differential equation model for the growth of heterogeneous cell populations subdivided into multiple distinct discrete phenotypes. In this model, cells preferentially move towards regions where they feel less compressed, and thus their movement occurs down the gradient of the cellular pressure, which is defined as a weighted sum of the densities (i.e. the volume fractions) of cells with different phenotypes. To translate into mathematical terms the idea that cells with distinct phenotypes have different morphological and mechanical properties, both the cell mobility and the weighted amount the cells contribute to the cellular pressure vary with their phenotype. We formally derive this model as the continuum limit of an on-lattice individual-based model, where cells are represented as single agents undergoing a branching biased random walk corresponding to phenotype-dependent and pressure-regulated cell division, death, and movement. Then, we study travelling wave solutions whereby cells with different phenotypes are spatially segregated across the invading front. Finally, we report on numerical simulations of the two models, demonstrating excellent agreement between them and the travelling wave analysis. The results presented here indicate that inter-cellular variability in mobility can provide the substrate for the emergence of spatial segregation across invading cell fronts.

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arXiv2025

Global minimizers for fast diffusion versus nonlocal interactions on negatively curved manifolds

We investigate the existence of ground states for a free energy functional on Cartan-Hadamard manifolds. The energy, which consists of an entropy and an interaction term, is associated to a macroscopic aggregation model that includes nonlinear diffusion and nonlocal interactions. We consider specifically the regime of fast diffusion, and establish necessary and sufficient conditions on the behaviour of the interaction potential for global energy minimizers to exist. We first consider the case of manifolds with constant bounds of sectional curvatures, then extend the results to manifolds with general curvature bounds. To establish our results we derive several new Carlson-Levin type inequalities for Cartan-Hadamard manifolds.

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arXiv2025

Singular flows with time-varying weights

We study the mean field limit for singular dynamics with time evolving weights. Our results are an extension of the work of Serfaty \cite{duerinckx2020mean} and Bresch-Jabin-Wang \cite{bresch2019modulated}, which consider singular Coulomb flows with weights which are constant time. The inclusion of time dependent weights necessitates the commutator estimates of \cite{duerinckx2020mean,bresch2019modulated}, as well as a new functional inequality. The well-posedness of the mean field PDE and the associated system of trajectories is also proved.

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arXiv2025

Global Existence and Nonlinear Stability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with Large Initial Data of Spherical Symmetry

The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy solutions of the multidimensional Euler-Riesz equations with large initial data of spherical symmetry. We consider both attractive and repulsive interactions for a wide range of Riesz and logarithmic potentials for dimensions larger than or equal to two. This is achieved by the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier-Stokes-Riesz equations. The strong convergence of the vanishing viscosity solutions is achieved through delicate uniform estimates in $L^p$. It is observed that, even if the attractive potential is super-Coulomb, no concentration is formed near the origin in the inviscid limit. Moreover, we prove that the nonlinear stability of global finite-energy solutions for the Euler-Riesz equations is unconditional under a spherically symmetric perturbation around the steady solutions. Unlike the Coulomb case where the potential can be represented locally, the singularity and regularity of the nonlocal radial Riesz potential near the origin require careful analysis, which is a crucial step. Finally, unlike the Coulomb case, a Grönwall type estimate is required to overcome the difficulty of the appearance of boundary terms in the sub-Coulomb case and the singularity of the super-Coulomb potential. Furthermore, we prove the nonlinear stability of global finite-energy solutions for the compressible Euler-Riesz equations around steady states by employing concentration compactness arguments. Steady states properties are obtained by variational arguments connecting to recent advances in aggregation-diffusion equations.

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arXiv2024

The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between $H^{-1}(\mathbb{R}^d)$ and $H^{1}(\mathbb{R}^d)$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

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arXiv2024

The graph limit for a pairwise competition model

This paper is aimed at extending the graph limit with time dependent weights obtained in [1] for the case of a pairwise competition model introduced in [10], in which the equation governing the weights involves a weak singularity at the origin. Well posedness for the graph limit equation associated with the ODE system of the pairwise competition model is also proved.

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arXiv2024

Existence of ground states for free energies on the hyperbolic space

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space $\bbh^\dm$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space $\bbh^\dm$. To prove our results we derived several Hardy-Littlewood-Sobolev (HLS)-type inequalities on general Cartan-Hadamard manifolds of bounded curvature, which have an interest in their own.

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arXiv2024

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear nonlocal, diffusive or not, system of PDEs without any variational structure.

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arXiv2024

Aggregation-Diffusion Equations for Collective Behaviour in the Sciences

This is a survey article based on the content of the plenary lecture given by José A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.

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arXiv2024

Linking discrete and continuous models of cell birth and migration

Self-organisation of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of biological intuition. Discrete models provide straightforward interpretations by tracking each individual yet can be computationally expensive. Alternatively, continuous models supply a large-scale perspective by representing the "effective" dynamics of infinite agents, but their results are often difficult to translate into experimentally relevant insights. We address this challenge by quantitatively linking spatio-temporal dynamics of continuous models and individual-based data in settings with biologically realistic, time-varying cell numbers. Specifically, we introduce and fit scaling parameters in continuous models to account for discrepancies that can arise from low cell numbers and localised interactions. We illustrate our approach on an example motivated by zebrafish-skin pattern formation, in which we create a continuous framework describing the movement and proliferation of a single cell population by upscaling rules from a discrete model. Our resulting continuous models accurately depict ensemble average agent-based solutions when migration or proliferation act alone. Interestingly, the same parameters are not optimal when both processes act simultaneously, highlighting a rich difference in how combining migration and proliferation affects discrete and continuous dynamics.

q-bio.CB↗

arXiv2024

Weak-strong uniqueness and high-friction limit for Euler-Riesz systems

In this work we employ the relative energy method to obtain a weak-strong uniqueness principle for a Euler-Riesz system, as well as to establish its convergence in the high-friction limit towards a gradient flow equation. The main technical challenge in our analysis is addressed using a specific case of a Hardy-Littlewood-Sobolev inequality for Riesz potentials.

math.AP↗

arXiv2024

Quantifying cell cycle regulation by tissue crowding

The spatiotemporal coordination and regulation of cell proliferation is fundamental in many aspects of development and tissue maintenance. Cells have the ability to adapt their division rates in response to mechanical constraints, yet we do not fully understand how cell proliferation regulation impacts cell migration phenomena. Here, we present a minimal continuum model of cell migration with cell cycle dynamics, which includes density-dependent effects and hence can account for cell proliferation regulation. By combining minimal mathematical modelling, Bayesian inference, and recent experimental data, we quantify the impact of tissue crowding across different cell cycle stages in epithelial tissue expansion experiments. Our model suggests that cells sense local density and adapt cell cycle progression in response, during G1 and the combined S/G2/M phases, providing an explicit relationship between each cell cycle stage duration and local tissue density, which is consistent with several experimental observations. Finally, we compare our mathematical model predictions to different experiments studying cell cycle regulation and present a quantitative analysis on the impact of density-dependent regulation on cell migration patterns. Our work presents a systematic approach for investigating and analysing cell cycle data, providing mechanistic insights into how individual cells regulate proliferation, based on population-based experimental measurements.

q-bio.QM↗

arXiv2024

A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy-Fokker-Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

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