arXiv subjects

Ayman Moussa

Publications and source records attributed to Ayman Moussa.

18 recordsLinked to original sources

arXiv2026

The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on $L^2$ and then explore the endpoint Besov case $B_{p,1}^{d/p}$. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

math.AP↗

arXiv2025

Stability of non-conservative cross diffusion model and approximation by stochastic particle systems

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species move, reproduce and die with rates sensitive to the local densities of the two species. Quantitative estimates are given and convergence is obtained soon as the population per site and the number of sites go to infinity. The proofs rely on the extension of stability estimates via duality approach under a smallness condition and the development of large deviation estimates for structured population models, which are of independent interest. The proofs also involve martingale estimates in H^{-1} and improve the approximation results in the conservative case as well.

math.AP↗

arXiv2025

Analysis of a toy model for optimal crop protection

In this paper we investigate an optimal control problem involving a toy model for the protection on a crop field. Precisely, we consider a protection on a crop field and we want to place intervention zones represented by a control, in order to maximise the protection on the field during a given period. Using a relaxation method, we prove that there exists a control which maximises the protection and, moreover, it must be a bang-bang control. Furthermore, with additional assumptions on the crop field geometry, some results on the shape of the optimal intervention are proved using comparison results for elliptic equations via Schwarz and Steiner symmetrizations. Finally, some numerical simulations are performed in order to illustrate those results.

math.OC↗

arXiv2024

Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach

We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a non-perturbative smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to cross-diffusion systems. More precisely, we prove quantitative estimates for the gap between the stochastic process and the cross-diffusion system. We give first rough but general estimates; then we use the duality approach to obtain fine estimates under less general conditions.

math.AP↗

arXiv2022

Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $Ω\times \mathbb{R}^3$, for a smooth bounded domain $Ω$ of $\mathbb{R}^3$, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to $0$ while the distribution function concentrates towards a Dirac mass in velocity centered at $0$, with an exponential rate. The proof, which follows the methods introduced in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.

math.AP↗

arXiv2021

Persisting entropy structure for nonlocal cross-diffusion systems

For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional)we study nonlocal versions and exhibit sufficient conditions to ensure that thenonlocal version inherits the entropy structure. These nonlocal systems can beunderstood as population models per se or as approximation of the classical ones.With the preserved entropy, we can rigorously link the approximating nonlocalversion to the classical local system. From a modelling perspective this gives away to prove a derivation of the model and from a PDE perspective this providesa regularisation scheme to prove the existence of solutions. A guiding example isthe SKT model [22]. In this context we answer positively the question raised byFontbona and M{é}l{é}ard [12] for the derivation and thus complete the derivation.

math.AP↗

arXiv2019

Large time behavior of the Vlasov-Navier-Stokes system on the torus

We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on $\mathbb{T}^3 \times \mathbb{R}^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing to get global bounds on moments.

math.AP↗

arXiv2018

From non-local to classical SKT systems: triangular case with bounded coefficients

This paper solves partially a question suggested by Fontbona and Méléard in a paper published in 2015. The issue is to obtain rigorously cross-diffusion systems à la Shigesada-Kawasaki-Teramoto as the limit of relaxed systems in which the cross-diffusion and reaction coefficients are non-local. We depart from the existence result established by Fontbona Méléard for a general class of non-local systems and study the corresponding asymptotic as the convolution kernels tend to Dirac masses, but only in the case of (strictly) triangular systems, with bounded coefficients. Our approach is based on a new result of compactness for the Kolmogorov equation, which is reminiscent of the celebrated duality lemma of Michel Pierre.

math.AP↗

arXiv2017

Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system

We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation.

math.AP↗

arXiv2017

On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

In this paper, we are interested in the collective friction of a cloud of particles on the viscous incompressible fluid in which they are moving. The particles velocities are assumed to be given and the fluid is assumed to be driven by the stationary Stokes equations. We consider the limit where the number N of particles goes to infinity with their diameters of order 1/N and their mutual distances of order (1/N)^{1/3}. The rigorous convergence of the fluid velocity to a limit which is solution to a stationary Stokes equation set in the full space but with an extra term, referred to as the Brinkman force, was proven by Desvillettes, Golse and Ricci when the particles are identical spheres in prescribed translations. Our result here is an extension to particles of arbitrary shapes in prescribed translations and rotations. The limit Stokes-Brinkman system involves the particle distribution in position, velocity and shape, through the so-called Stokes' resistance matrices.

math.AP↗

arXiv2016

The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria

In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase.

math.AP↗

arXiv2016

Entropic structure and duality for multiple species cross-diffusion systems

This paper deals with the existence of global weak solutions for a wide class of (multiple species) cross-diffusions systems. The existence is based on two different ingredients: an entropy estimate giving some gradient control and a duality estimate that gives naturally L^2 control. The proof relies on a semi-implicit scheme tailored for cross-diffusion systems firstly defined by the two authors and collaborators. These results are applied to models having an entropy relying on the detailed balance condition recently exhibited by Chen et. al.

math.AP↗

arXiv2015

A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs

We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl. 1987). Our approach allows to handle various numerical discretizations both in the space variables and in the time variable. In particular, we can cope quite easily with variable time steps and with multistep time differentiation methods like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We illustrate our approach by proving the convergence of a two-point flux Finite Volume in space and BDF2 in time approximation of the porous medium equation.

math.NA↗

arXiv2014

Some variants of the classical Aubin-Lions Lemma

This paper explores two generalizations of the classical Aubin-Lions Lemma. First we give a sufficient condition to commute weak limit and multiplication of two functions. We deduce from this criteria a compactness Theorem for degenerate parabolic equations. Secondly, we state and prove a compactness Theorem for non-cylindrical domains, including the case of dual estimates involving only divergence-free test functions.

math.AP↗

arXiv2014

On the entropic structure of reaction-cross diffusion systems

This paper is devoted to the study of systems of reaction-cross diffusion equations arising in population dynamics. New results of existence of weak solutions are presented, allowing to treat systems of two equations in which one of the cross diffusions is convex, while the other one is concave. The treatment of such cases involves a general study of the structure of Lyapunov functionals for cross diffusion systems, and the introduction of a new scheme of approximation, which provides simplified proofs of existence.

math.AP↗

arXiv2013

Existence theory for a kinetic-fluid coupling when small droplets are treated as part of the fluid

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incom- pressible Navier-Stokes equation and of the Vlasov-Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations.

math.AP↗

arXiv2013

Entropy, Duality and Cross Diffusion

This paper is devoted to the use of the entropy and duality methods for the existence theory of reaction-cross diffusion systems consisting of two equations, in any dimension of space. Those systems appear in population dynamics when the diffusion rates of individuals of two species depend on the concentration of individuals of the same species (self-diffusion), or of the other species (cross diffusion).

math.AP↗

arXiv2011

A 2d spray model with gyroscopic effects

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from a joint work of the second author with O. Glass and C. Lacave, where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.

math.AP↗