97 research outputs found
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)
Global existence of solutions to the incompressible Navier-Stokes-Vlasov equations in a time-dependent domain
International audienceIn this article, we prove the existence of global weak solutions for the in-compressible Navier-Stokes-Vlasov system in a three-dimensional time-dependent domain with absorption boundary conditions for the kinetic part. This model arises from the study of respiratory aerosol in the human airways. The proof is based on a regularization and approximation strategy designed for our time-dependent framework
A RAINBOW INVERSE PROBLEM
Abstract. We consider the radiative transfer equation (RTE) with reflection in a three-dimensional domain, infinite in two dimensions, and prove an existence result. Then, we study the inverse problem of retrieving the optical parameters from boundary measurements, with help of existing results by Choulli and Stefanov. This theoretical analysis is the framework of an attempt to model the color of the skin. For this purpose, a code has been developed to solve the RTE and to study the sensitivity of the measurements made by biophysicists with respect to the physiological parameters responsible for the optical properties of this complex, multi-layered material
A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs
International audienceWe 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
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
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