Ayman Moussa

The Navier–Stokes–Vlasov–Fokker–Planck System near Equilibrium

This paper is concerned with a system that couples the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. Such a system arises in the modeling of sprays, where a dense phase interacts with a disperse phase. The coupling arises from the Stokes drag force exerted by a phase on the other. We study the global-in-time existence of classical solutions for data close to an equilibrium. We investigate further regularity properties of the solutions as well as their long time behavior. The proofs use energy estimates and the hypocoercive/hypoelliptic structure of the system.

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).

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.

EXISTENCE THEORY FOR THE 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 incompressible 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, and a compactness lemma specifically tailored for our study.

Analytical examples of diffusive waves generated by a traveling wave

We construct analytical solutions for a system composed of a reaction–diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction–diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher–KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.

Backward Parabolicity, Cross-Diffusion and Turing Instability

We show that the ill-posedness observed in backward parabolic equation, or cross-diffusion systems, can be interpreted as a limiting Turing instability for a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of nonlinear parabolic and cross-diffusion systems from semi-linear reaction–diffusion systems with fast reaction rates. We illustrate our observation with two generic examples for

Some variants of the classical Aubin–Lions Lemma

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A NONLINEAR TIME COMPACTNESS RESULT AND APPLICATIONS TO DISCRETIZATION OF DEGENERATE PARABOLIC-ELLIPTIC PDES

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