In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem family nonlocal interaction equations. These equations are continuum models interacting particle systems with attractive/repulsive pairwise potentials. The main phenomenon interest is that, even smooth initial data, can concentrate mass in finite time. We develop an existence that enables one to go beyond blow-up time classical norms and allows form atomic parts shown be unique exist globally...

We study the system$ c_t+u \cdot \nabla c = \Delta c- nf(c) $$ n_t + u n n^m- (n \chi(c)\nabla c) u_t P - \eta\Delta \phi=0 $ $\nabla 0. arising in modelling of motion swimming bacteria under effect diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers literature lies presence nonlinear porous--medium--like diffusion equation for density $n$ bacteria, motivated by a finite size effect. prove that, constraint $m\in(3/2, 2]$...

This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems non-local interaction PDEs with two species, which are the PDE counterpart deterministic interacting particles species. The main motivations behind those models arise in cell biology, pedestrian movements, opinion formation. In case symmetrizable (i.e. cross-interaction potentials one multiple other), we provide complete within (a suitable generalization of) Wasserstein gradient flow...

We prove some Schauder-type estimates and an invariant Harnack inequality for a class of degenerate evolution operators Kolmogorov type. also Gaussian lower bound the fundamental solution operator uniqueness result Cauchy problem. The proof is obtained by solving suitable optimal control problem using inequality.

The aim of this paper is to establish rigorous results on thelarge time behavior nonlocal models for aggregation, includingthe possible presence nonlinear diffusion terms modeling localrepulsions. We show that, as expected from the practicalmotivation well numerical simulations, one obtainsconcentrated densities (Dirac $\delta$ distributions) asstationary solutions and large limits in absence ofdiffusion. In addition, we provide a comparison aggregationkernels with infinite respectively...

The aim of this paper is to discuss the effects linear and nonlinear diffusion in large time asymptotic behavior Keller–Segel model chemotaxis with volume filling effect. In case we provide several sufficient conditions for part dominate yield decay zero solutions. We also an explicit rate towards self–similarity. Moreover, prove that no stationary solutions positive mass exist. fully determined by whether diffusivity constant larger or smaller than threshold value $\varepsilon =1$. Below...

The aim of this paper is to investigate the mathematical properties a continuum model for diffusion multiple species incorporating size exclusion effects. system two leads nonlinear cross-diffusion terms with double degeneracy, which creates significant novel challenges in analysis system. We prove global existence weak solutions and well-posedness strong close equilibrium. further study some asymptotics model, particular we characterize large-time behavior solutions.

In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds.In particular consider the case of a large human crowd trying to room as possible.The motion every is determined by minimizing cost functional,which depends on his/her position, velocity, time and overall density people.This microscopic setup leads mean-field limit parabolic problem.We discuss macroscopic show how conditions relateto Hughes model for flow. Furthermore provide results...

We adapt the Levi parametrix method to prove existence, estimates, and qualitative properties of a global fundamental solution ultraparabolic partial differential equations Kolmogorov type. Existence uniqueness results for Cauchy problem are also proved.

We prove the equivalence between notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and entropy solution Burgers-type scalar conservation law other. The former is obtained by spatially differentiating latter. proof uses an intermediate step, namely L2 pseudo-inverse distribution function solution. use this to provide rigorous particle-system approximation flow, avoiding regularization effect due...

We consider a class of non-local conservation laws with an interaction kernel supported on the negative real half-line and featuring decreasing jump at origin. provide, for first time, existence uniqueness theory said model initial data in space probability measures. Our concept solution allows us to sort lack problem which we exhibit specific example. approach uses so-called quantile , or pseudo-inverse formulation PDE, has been largely used similar types transport equations one dimension....

We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts population dynamics.The is Wasserstein gradient flow generated by energy E, which sum free interaction energy.The kernel taken radial attractive, nonnegative, integrable, further technical smoothness assumptions.The vs. nonexistence such ruled threshold phenomenon, namely steady states exist if only diffusivity constant strictly smaller...

We consider the follow-the-leader approximation of Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. prove rigorous convergence to weak solutions ARZ system many particle limit presence vacuum. The result is based on uniform BV estimates discrete velocity. complement our with numerical simulations method compared some exact Riemann problem system.

We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics predators and prey, in which all agents interact through attractive/repulsive forces gradient type. In order to model predator–prey interaction, we prescribed proportional potentials (with opposite signs) for cross-interaction part. The has particle-based discrete (ODE) version continuum PDE version. investigate structure particle stationary solution their stability ODE systematic form, then simple...

Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend analysis to a system two species interacting with each other according different inner- intra-species attractions. Under suitable conditions on self- crosswise an interesting effect can be observed, namely phase separation into neighboring regions, of which contains only one species. We prove that intersection support...

We study the obstacle problem for a class of degenerate parabolic operators with continuous coefficients. This arises in Black–Scholes framework when considering path-dependent American options. prove existence unique strong solution u to Cauchy and Cauchy–Dirichlet problems, under rather general assumptions on function. also show that is viscosity sense.