arXiv subjects

José A. Carrillo

Publications and source records attributed to José A. Carrillo.

At least 19 recordsLinked to original sources

arXiv2026

An interacting particle consensus method for constrained global optimization

This paper presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. The proposed method combines components from consensus-based optimization algorithm with a newly introduced forcing term directed at the constraint set. A rigorous mean-field limit is derived for the anisotropic particle system, while convergence to the constrained minimizer is established for the isotropic mean-field dynamics. Additionally, we introduce a linearly implicit discretized algorithm, analyze its well-posedness, stability properties, and consistency with the continuous particle dynamics, and conduct various numerical experiments to demonstrate the performance of the proposed method.

math.OC↗

arXiv2026

A Kinetic Theory Approach To Ordered Fluids: Direct Simulation Monte Carlo Methods

We present a direct simulation Monte Carlo (DSMC) method for the kinetic theory of ordered fluids recently introduced by the authors. The microscopic constituents of these fluids carry an order parameter belonging to a manifold. The method applies Strang splitting to couple a Monte Carlo treatment of the binary collisions with a symplectic integration of the mean-field Vlasov transport on the order-parameter manifold. We use the Nanbu-Babovsky approach for Maxwellian collision kernels, and Bird's acceptance-rejection method for general ones. The mean-field Vlasov force is reconstructed from the particle ensemble at each time step. We also include a Bhatnagar-Gross-Krook (BGK) / Andersen thermostat to control the system temperature. We formulate the method for a generic order-parameter manifold, and specialize it to two-dimensional calamitic (rod-like) molecules. Numerical tests validate the method. We confirm the relaxation to the Maxwellian distribution predicted by the H-theorem, and simulate the mean-field alignment driven by quadratic, Kuramoto, and Onsager interaction potentials. With the Onsager potential we successfully capture the isotropic-nematic phase transition at the correct critical temperature.

math-ph↗

arXiv2026

Hydrodynamic limit from nonlinear Fokker--Planck to barotropic Euler equations

The hydrodynamic limit to the barotropic Euler equations, including power-law pressure $P(ρ)=ρ^γ$, for a kinetic nonlinear Fokker--Planck equation with degenerate diffusion is established. This extends the well-known result of the derivation of isothermal Euler equations via Fokker--Planck equation with linear diffusion. We establish the asymptotic analysis using the relative entropy method by quantifying error estimates for pressures and employing the generalized Log-Sobolev inequality for degenerate diffusion.

math.AP↗

arXiv2026

Non-uniqueness of weak solutions to cross-diffusion systems with advection

A cross-diffusion system with advection is considered on the whole line, describing the dynamics of two segregating population species. Starting from two initial densities supported on the half-lines $x\leq 0$ and $x\geq 0$, respectively, we construct two distinct solutions of the system: one pair of densities remains confined to their initial supports and stay completely segregated, while the second pair of densities begins to invade the opposite half-line after a finite time (mixing). To the best of our knowledge, this provides one of the first examples of non-uniqueness for this class of equations, together with an explicit demonstration of mixing phenomena. The construction produces infinitely many mixing solutions and applies throughout the full range of pressure exponents. For a certain range of exponents, we additionally obtain quantitative estimates on the mixing process.

math.AP↗

arXiv2026

Variational inference via Gaussian interacting particles in the Bures-Wasserstein geometry

Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that stochastically explore the search space and self-organize around global minima via a consensus-based optimization (CBO) mechanism. Its construction relies on the Linearized Bures-Wasserstein (LBW) space, a novel parametrization of Gaussian measures we introduce for efficient computations. LBW is inspired by linearized optimal transport and preserves key geometric features while enabling computational tractability. We establish well-posedness and study the convergence properties of the particle dynamics via a mean-field approximation. Numerical experiments on variational inference tasks demonstrate the algorithm's robustness and superior performance with respect to deterministic gradient-based method in presence of low-dimensional non log-concave targets.

math.OC↗

arXiv2026

Stein Variational Gradient Descent dynamics for highly concentrated kernels

Stein Variational Gradient Descent (SVGD) is a widely used in practice algorithm for scalable sampling with deterministic particle updates. We study its behavior in the singular limit where the kernel bandwidth tends to zero. In this regime, we show that the nonlocal SVGD dynamics converge to a local evolution equation that can be formally interpreted as a Wasserstein gradient flow with quadratic mobility. We analyze this singular limit in two settings: integrable kernels and weighted kernels. In the weighted case, the proof is supported by recently established Stein-log-Sobolev inequalities, which provide the necessary functional control. Overall, our results clarify how SVGD collapses from a nonlocal interacting particle system to a local gradient-flow dynamics as the kernel concentrates.

math.AP↗

arXiv2026

A Proximal Primal-Dual Approach to Generalized JKO Schemes for Doubly Nonlinear Parabolic Equations

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

math.NA↗

arXiv2026

Lagrangian formulation and Eulerian closure in alignment dynamics

We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler--Reynolds--alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler--alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker--Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics.

math.AP↗

arXiv2026

A Kinetic Theory Approach to Ordered Fluids

We develop a unified kinetic theory for ordered fluids, which systematically extends the phase space with the appropriate generalized angular momenta. Our theory yields a uniquely determined mesoscopic model for any continuum with microstructure that is characterized by Capriz's order parameter manifold. We illustrate our theory with three running examples: liquids saturated with non-diffusive gas bubbles, liquids composed of calamitic (rodlike) molecules, and liquids composed of calamitic molecules with additional head-to-tail symmetry. We discuss the symmetries of the microscopic interactions via Noether's theorem, and use them to characterize the conserved quantities mesoscopic dynamics. We derive the mesoscopic model for ordered fluids from a kinetic point of view assuming that the microscopic interactions are of weak nature when it comes to the ordering of the fluid. Lastly, we discuss under which conditions an H-theorem result holds at the mesoscopic scale and for which Vlasov potential we can expect the emergence of collective behavior.

math-ph↗

arXiv2026

A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations

Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$.

math.AP↗

arXiv2026

Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities

The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.

math.AP↗

arXiv2026

Well-posedness and mean-field limit of discontinuous weighted dynamics via the relative entropy method

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.

math.AP↗

arXiv2026

Numerical stationary states for nonlocal Fokker-Planck equations via fixed points of consistency maps

We propose a fixed-point-based numerical framework for computing stationary states of nonlocal Fokker-Planck-type equations. Instead of discretising the differential operators directly, we reformulate the stationary problem as a nonlinear fixed-point map built from the original PDE and its nonlocal interaction terms, and solve the resulting finite-dimensional problem with a matrix-free Newton-Krylov method. We compare implementations using the analytic Frechet derivative of this map with a simple central-difference approximation. Because the method does not rely on time evolution, it is agnostic to dynamical stability and can detect both stable and unstable stationary states. Its accuracy is determined mainly by the numerical treatment of convolutions and quadrature, rather than by differentiation stencils. We apply the approach to three model problems with linear diffusion, use existing analytical results to verify the outputs, and reproduce known bifurcation diagrams, as well as new bifurcation behaviour not previously observed in this kind of problem.

math.NA↗

arXiv2026

Asymptotic stability of steady states for the compressible Navier-Stokes-Riesz system in the presence of vacuum

We consider a one-dimensional physical vacuum free boundary problem on the compressible Navier-Stokes-Riesz system for an attractive Riesz potential $|x|^{2s-1}/(2s-1)$ with $0<s<1/2$. It is proved that for the adiabatic constant $γ$ satisfying $2(1-s)<γ<1+2s/3$ under the additional condition that $3/8<s<1/2$, there exists a unique global-in-time strong solution. Specifically, we establish the Lyapunov-type stability of the compactly supported steady states in the Lagrangian coordinates and we also obtain the time rate of convergence for the strong solution to steady states with the same mass in weighted Sobolev spaces where the weights indicate the behavior of solutions near the vacuum free boundary. The difficulties and challenges in the proof are caused not only by the degeneracy due to the vacuum free boundary but also by the non-local feature of the Riesz potential.

math.AP↗

arXiv2025

Inelastic Boltzmann equation under shear heating

In this paper, we study the spatially homogeneous inelastic Boltzmann equation for the angular cutoff pseudo-Maxwell molecules with an additional term of linear deformation. We establish the existence of non-Maxwellian self-similar profiles under the assumption of small deformation in the nearly elastic regime, and also obtain weak convergence to these self-similar profiles for global-in-time solutions with initial data that have finite mass and finite $p$-th order moment for any $2<p\leq 4$. Our results confirm the competition between shear heating and inelastic cooling that governs the large time behavior of temperature. Specifically, temperature increases to infinity if shear heating dominates, decreases to zero if inelastic cooling prevails, and converges to a positive constant if the two effects are balanced. In the balanced scenario, the corresponding self-similar profile aligns with the steady solution.

math.AP↗

arXiv2025

Exponential and algebraic decay in Euler--alignment system with nonlocal interaction forces

We investigate the large-time behavior of the pressureless Euler system with nonlocal velocity alignment and interaction forces, with the aim of characterizing the asymptotic convergence of classical solutions under general interaction potentials $W$ and communication weights. We establish quantitative convergence in three settings. In one dimension with $(λ,Λ)$-convex potentials, i.e., potentials satisfying uniform lower and upper quadratic bounds, bounded communication weights yield exponential decay, while weakly singular ones lead to sharp algebraic rates. For the Coulomb--quadratic potential $W(x)=-|x|+\frac12 |x|^2$, we prove exponential convergence for bounded communication weights and algebraic upper bounds for singular communication weights. In a multi-dimensional setting with uniformly $(λ,Λ)$-convex potentials, we show exponential decay for bounded weights and improved algebraic decay for singular ones. In all cases, the density converges (up to translation) to the minimizer of the interaction energy, while the velocity aligns to a uniform constant. A unifying feature is that the convergence rate depends only on the local behavior of communication weights: bounded kernels yield exponential convergence, while weakly singular ones produce algebraic rates. Our results thus provide a comprehensive description of the asymptotic behavior of Euler--alignment dynamics with general interaction potentials.

math.AP↗

arXiv2025

Well-posedness of aggregation-diffusion systems with irregular kernels

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential $K$. We are interested in establishing their well-posedness theory when the nonlocal interaction potential $K$ is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that $\nabla K\ast K$ is in $L^2$, we can prove that the solution is smooth and unique. When $K$ is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the $n$-species system.

math.AP↗

arXiv2025

Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus

We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fréchet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.

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