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.

We prove the existence of weak solutions of a class of multi-species cross-diffusion systems as well as the propagation of chaos result by means of nonlocal approximation of the nonlinear diffusion terms, coupling methods and compactness arguments. We also prove the uniqueness under further structural assumption on the mobilities by combining the uniqueness argument for viscous porous medium equations and linear Fokker-Planck equations. We show that these equations capture the macroscopic behaviour of stochastic interacting particle systems if the localisation parameter is chosen logarithmically with respect to the number of particles.

We focus on evolution equations on co-evolving, infinite, graphs and establish a rigorous link with a class of nonlinear continuity equations, whose vector fields depend on the graphs considered. More precisely, weak solutions of the so-called graph-continuity equation are shown to be the push-forward of their initial datum through the flow map solving the associated characteristics' equation, which depends on the co-evolving graph considered. This connection can be used to prove contractions in a suitable distance, although the flow on the graphs requires a too limiting assumption on the overall flux. Therefore, we consider upwinding dynamics on graphs with pointwise and monotonic velocity and prove long-time convergence of the solutions towards the uniform mass distribution.

This paper is concerned with the Dirichlet initial-boundary value problem of a 2-D parabolic-elliptic system of the following form m_t-κΔ m+|m|^2(γ-1)m=(m·∇_p)∇_p, -∇·[(𝐈+m⊗ m)∇_p]=S, which was proposed to model the formation of biological transport networks. Even if global weak solutions for this system are known to exist, how to improve the regularity of weak solutions is a challenging problem due to the peculiar cubic nonlinearity and the possible elliptic singularity of the system. Global-in-time existence of classical solutions has recently been established, showing that finite time singularities cannot emerge in this problem. However, whether or not singularities in infinite time can be precluded was still pending. In this work, we show that classical solutions of the initial-boundary value problem are uniformly bounded in time as long as γ ⩾ 1 and κ is suitably large, closing this gap in the literature. Moreover, the uniqueness of classical solutions is also achieved based on the uniform-in-time bounds. Furthermore, it is shown that the corresponding stationary problem possesses a unique classical stationary solution which is semi-trivial, and that is globally exponentially stable, i.e., all solutions of the time-dependent problem converge exponentially fast to the semi-trivial steady state for κ large enough.

We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions, introduced by Brenier, Gangbo, Savaré and Westdickenberg, and entropy solutions, studied by Nguyen and Tudorascu for the Euler--Poisson system, are equivalent. For the Euler--Poisson system this can be seen as a generalization to second-order systems of the equivalence between $L^2$-gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier. The key observation is an equivalence between Ole\u{\i}nik's E-condition for conservation laws and a characterization due to Natile and Savaré of the normal cone for $L^2$-gradient flows. This new equivalence allows us to define unique solutions after blow-up for classical solutions of the Euler--Poisson system with quadratic confinement due to Carrillo, Choi and Zatorska, as well as to describe their asymptotic behavior.

We study the mean field limit for singular dynamics with time evolving weights. Our results are an extension of the work of Serfaty [14] and Bresch-Jabin-Wang [8], which consider singular Coulomb flows with weights which are constant time. The inclusion of time dependent weights necessitates the commutator estimates of [8, 14], as well as a new functional inequality. The well-posedness of the mean field PDE and the associated system of trajectories is also proven.

With the recently increased interest in probabilistic models, the efficiency of an underlying sampler becomes a crucial consideration. Hamiltonian Monte Carlo is one popular option for models of this kind. Performance of the method, however, strongly relies on a choice of parameters associated with an integration approach for Hamiltonian equations. Up to date, such a choice remains mainly heuristic or introduces time complexity. We propose a novel computationally inexpensive and flexible approach (we call it Adaptive Tuning or ATune) that, by combining a theoretical analysis of the multivariate Gaussian model with simulation data generated during a burn-in stage of a Hamiltonian Monte Carlo simulation, detects a system specific splitting integrator with a set of reliable sampler’s hyperparameters, including their credible randomization intervals, to be readily used in a production simulation. The method automatically eliminates those values of simulation parameters which could cause undesired extreme scenarios, such as resonance artifacts, low accuracy or poor sampling. The new approach is implemented in the in-house software package HaiCS, with no computational overheads introduced in a production simulation, and can be easily incorporated in any package for Bayesian inference with Hamiltonian Monte Carlo. The tests on popular statistical models reveal the superiority of adaptively tuned standard and generalized Hamiltonian Monte Carlo methods in terms of stability, performance and accuracy over conventional Hamiltonian Monte Carlo tuned heuristically and coupled with the well-established integrators. We also claim that the generalized Hamiltonian Monte Carlo is preferable for achieving high sampling performance. The efficiency of the new methodology is assessed in comparison with state-of-the-art samplers, e.g. the No-U-Turn-Sampler, in real-world applications, such as endocrine therapy resistance in cancer, modeling of cell-cell adhesion dynamics and influenza A epidemic outbreak.

In this paper, we study the spatially homogeneous inelastic Boltzmann equation for angular cutoff pseudo-Maxwell molecules with an additional term of linear deformation. We establish the existence of non-Maxwellian selfsimilar 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 <= 4. Our results confirm the competition between shear heating and inelastic cooling that governs the long-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.

We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.

Numerical approximations of Landau-type operators represent fundamental components of time integration methods for demanding problems such as inhomogeneous Vlasov–Landau-type equations. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb-type interaction. This work is concerned with the introduction and numerical comparison of novel approaches for the reliable and efficient evaluation of Landau-type collision operators, where the focus is on the treatment of integral operators involving general singular kernels. In the spirit of collocation, common tools are the identification of fundamental integrals, series expansions of the integral kernel and the density function on the main part of the velocity domain, and interpolation as well as quadrature approximation nearby the singularity of the kernel. Focusing on the favourable choice of the Fourier spectral method, their practical implementation uses the reduction to basic integrals, fast Fourier techniques, and summations along certain directions. Moreover, an important observation is that a significant percentage of the overall computational effort can be transferred to precomputations which are independent of the density function. For the purpose of exposition and numerical validation, the cases of constant, regular, and singular integral kernels are distinguished, and the procedure is adapted accordingly to the increasing complexity of the problem.

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.

Estimating the statistics of the state of a dynamical system, from partial and noisy observations, is both mathematically challenging and finds wide application. Furthermore, the applications are of great societal importance, including problems such as probabilistic weather forecasting and prediction of epidemics. Particle filters provide a well-founded approach to the problem, leading to provably accurate approximations of the statistics. However these methods perform poorly in high dimensions. In 1994 the idea of ensemble Kalman filtering was introduced by Evensen, leading to a methodology that has been widely adopted in the geophysical sciences and also finds application to quite general inverse problems. However, ensemble Kalman filters have defied rigorous analysis of their statistical accuracy, except in the linear Gaussian setting. In this article we describe recent work which takes first steps to analyze the statistical accuracy of ensemble Kalman filters beyond the linear Gaussian setting. The subject is inherently technical, as it involves the evolution of probability measures according to a nonlinear and nonautonomous dynamical system; and the approximation of this evolution. It can nonetheless be presented in a fairly accessible fashion, understandable with basic knowledge of dynamical systems, numerical analysis and probability.

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. We also establish uniform energy estimates of global solutions for not only the Riesz potential but also the limiting end-point case: the logarithmic potential. 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.

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 del K & lowast;K is in L-2, we can prove that the strong solution is 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.

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.

We derive a class of multi-species aggregation-diffusion systems from stochastic interacting particle systems via relative entropy method with quantitative bounds. We show an algebraic $L^1$-convergence result using moderately interacting particle systems approximating attractive/repulsive singular potentials up to Newtonian/Coulomb singularities without additional cut-off on the particle level. The first step is to make use of the relative entropy between the joint distribution of the particle system and an approximated limiting aggregation-diffusion system. A crucial argument in the proof is to show convergence in probability by a stopping time argument. The second step is to obtain a quantitative convergence rate to the limiting aggregation-diffusion system from the approximated PDE system. This is shown by evaluating a combination of relative entropy and $L^2$-distance.

Cell sorting by differential adhesion is one of the basic mechanisms explaining spatial organization of neurons in early stage brain development of fruit flies. The columnar arrangements of neurons determine the large-scale patterns in the fly visual center. Experimental studies indicate that hexagonal configurations regularly appear in the fly compound eye, which is connected to the visual center by photoreceptor axons, while tetragonal configurations can be induced in mutants. We need a mathematical framework to study the mechanisms of such a transition between hexagonal and tetragonal arrangements. Here, we propose a new mathematical model based on macroscopic approximations of agent-based models that produces a similar behavior changing from hexagonal to tetragonal steady configurations when medium-range repulsion and longer-range attraction between individuals are incorporated in previous successful models for cell sorting based on adhesion and volume constraints. We analyze the angular configurations of these patterns based on angle summary statistics and compare between experimental data and parameter fitted ARA (Adhesion-Repulsion-Attraction) models showing that intermediate patterns between hexagonal and tetragonal configuration are common in experimental data as well as in our ARA mathematical model. Our studies indicate an overall qualitative agreement of ARA models in tile patterning and pave the way for their quantitative studies. Our study opens up a new avenue to explore tile pattern transitions, found not only in the column arrangement in the brain, but also in the other related biological processes.

We consider the Kac model for the space-homogeneous Landau equation with the Coulomb potential. We show that the Fisher information of the Liouville equation for the unmodified N-particle system is monotonically decreasing in time. The monotonicity ensures the compactness to derive a weak solution of the Landau hierarchy.

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space ${\mathbb H}<^>n$ . 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 ${\mathbb H}<^>n$ . 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.