Nonlocal PDE Research Articles

Least energy sign-changing solution to a fractional $p$-Laplacian problem involving singularities

In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

On étudie certaines EDS à sauts et les équations de Fokker–Planck (ou Kolmogorov progressives) correspondantes, qui sont des EDP non-locales. On suppose seulement que les coefficients sont mesurables et à croissance au plus linéaire. On montre que pour toute solution faible $(f_{t})_{t\in[0,T]}$ de l’EDP, il existe une solution faible à l’EDS, dont les lois marginales sont données par $(f_{t})_{t\in[0,T]}$. On en déduit que pour toute donnée initiale, l’existence pour l’EDP est équivalente à l’existence faible pour l’EDS, et que l’unicité en loi pour l’EDS implique l’unicité pour l’EDP. Nous étendons ainsi des idées de Figalli (J. Funct. Anal. 254 (2008) 109–153) concernant des EDS continues et des EDP locales.

Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

In this short note we consider a nonlinear and spatially nonlocal PDE modeling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.

The replicator dynamics for multilevel selection in evolutionary games.

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner's Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner's Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.

Extremals for fractional order Hardy–Sobolev–Maz’ya inequality

In this article, we derive the existence of positive solution of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy–Sobolev–Maz’ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.

Scaling Limit of the Stein Variational Gradient Descent: The Mean Field Regime

We study an interacting particle system in $\mathbf{R}^d$ motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a non-local and nonlinear PDE. We also prove global existence, uniqueness and regularity of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in long time limit.

Traveling waves for nonlocal models of traffic flow

We consider several non-local models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with non-local flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition.

Numerical specified homogenization of a discrete model with a local perturbation and application to traffic flow

We present techniques for the analysis and numerical analysis of non-local non-linear PDEs. We apply these techniques to an equation derived from the modelling of traffic flow. We introduce a macroscopic model in the form of a Hamilton--Jacobi equation with a junction condition. More precisely, the goal of this work is to obtain the numerical homogenization of a non-local PDE deriving from a first order discrete model for traffic flow that simulates the presence of a local perturbation. Previously we showed that the solution of the discrete microscopic model converges to the (unique) solution of a Hamilton--Jacobi equation posed on a network and with a junction condition (it can be seen as a flux limiter that keeps the memory of the local perturbation). The goal of this article is to provide a numerical scheme able to obtain an approximation of this flux-limiter. We prove the convergence of this scheme and we give some numerical results.

Option pricing in a regime switching stochastic volatility model

We consider a regime switching stochastic volatility model where the stock volatility dynamics is a semi-Markov modulated square root mean reverting process. Under this model assumption, we find the locally risk minimizing price of European type vanilla options. The price function is shown to satisfy a non-local degenerate parabolic PDE which can be viewed as a generalization of the Heston PDE. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of solution to the PDE is carried out by studying the IE and using the semigroup theory.

A local saddle point theorem and an application to a nonlocal PDE

We prove a local saddle point theorem that can be viewed as a generalization of the saddle point theorem of Rabinowitz. A difficulty to overcome is that there isn’t any linking. We then apply the theorem to show the existence of solutions of a nonlocal partial differential equations.

Standing waves for a gauged nonlinear Schrödinger equation with a vortex point

This paper is motivated by a gauged Schrödinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.

Multistep Schemes for Forward Backward Stochastic Differential Equations with Jumps

In this work, we are concerned with multistep schemes for solving forward backward stochastic differential equations with jumps. The proposed multistep schemes admit many advantages. First of all, motivated by the local property of jump diffusion processes, the Euler method is used to solve the associated forward stochastic differential equation with jump, which reduce dramatically the entire computational complexity, however, the quantities of interests in the backward stochastic differential equations (with jump) are still of high order rate of convergence. Secondly, in each time step, only one jump is involved in the computational procedure, which again reduces dramatically the computational complexity. Finally, the method applies easily to partial-integral differential equations (and some nonlocal PDE models), by using the generalized Feynman---Kac formula. Several numerical experiments are presented to demonstrate the effectiveness of the proposed multistep schemes.

Nonlocal PDE morphology: a generalized shock operator on graph

This paper presents an adaptation and extension of the shock filters on weighted graphs using the formalism of partial difference equations. This adaptation leads to new morphological operators that alternate between nonlocal dilation and nonlocal erosion filter type on graphs. Furthermore, this adaptation extends the shock filter applications to any data that can be represented by graphs. This approach is illustrated through image, data defined on region maps, and 3D point cloud processing.

Turing-type instabilities in bulk–surface reaction–diffusion systems

In this contribution we consider a coupled bulk–surface reaction–diffusion system and for infinitely fast bulk diffusion its formal reduction to a non-local surface PDE model. Thereby, we review results of linear stability analyses for both models showing that Turing-type instabilities can occur for equal lateral diffusion coefficients. The stability results are confirmed by new numerical results. As a specific application, we study a model for a spatial and reaction cycle of signalling molecules in a cell.

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

We prove the equivalence between the notion of Wasserstein gradient flow for a one- dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L 2 gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the subdifferential of the functional involved in the Wasserstein gradient flow.

Mean-Field Limit of a Microscopic Individual-Based Model Describing Collective Motions

This paper is mainly concerned with a mean-field limit and long time behavior of stochastic microscopic interacting particles systems. Specifically we prove that a class of ODE modeling collective interactions in animals or pedestrians converges in the mean-field limit to the solution of a non-local kinetic PDE. The mathematical analysis, performed by weak measure solutions arguments, shows the existence of measure-valued solutions, asymptotic stability and chaos propagation that are relevant properties in the description of collective behaviors that emerge in animals and pedestrians motions.

Regularity Theory and Adjoint-Based Optimality Conditions for a Nonlinear Transport Equation with Nonlocal Velocity

In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal hyperbolic PDE is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for $W^{1,p}$-data. Based on a general $L^2$ tracking type cost functional, the existence, uniqueness, and regularity of the adjoint system in $W^{1,p}$ is derived using the special structure induced from the nonlocal flux function of the state equation. The assumption of $W^{1,p}$- and not $L^p$-regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined. This problem is discussed in detail, and we give a solution by defining a special type of cost functional, such that the corresponding optimality system is well defined.

A molecular method applied to a non-local PDE in stratified Lie groups

In this article we study a transport-diffusion equation in the framework of the stratified Lie groups. For this equation we will study the existence of the solutions, a maximum principle, a positivity principle and Hölder regularity.

HYDRO-ELASTIC WAVES IN A COCHLEAR MODEL: NUMERICAL SIMULATIONS AND AN ANALYTICALLY REDUCED MODEL

A three-dimensional model of the hydro-elastic waves in the mammalian cochlea is presented along with numerical simulations. The cochlear fluid is treated as linear, incompressible, and inviscid. The cochlear partition is treated as a massless thin plate loaded by the fluid. This model is then reformulated by analytically removing the fluid variable with the use of a Dirichlet-to-Neumann operator. The resulting fifth-order nonlocal PDE for the motion of the partition is simulated using a novel implicit numerical scheme. Simulations demonstrate that this model exhibits traveling wave characteristics and a clear place principle. Asymptotic analysis in the small aspect ratio of the cochlea is performed on the given model equations with energetic concerns in mind. The results of simulations along with these asymptotic arguments suggest a relationship between the form and function of the cochlea which we compare to physiological data.

The global attractor of a non-local PDE model with delay for population dynamics in ℝ n

In this paper, we consider a non-local PDE model with delay for population dynamics in ℝn. First, we prove the existence and uniqueness of weak solutions under some suitable decayed assumptions on non-local term at infinity. Then, we obtain the global attractor by proving ω-limit compactness property of the solution operator semigroup.