Méthodes multipas pour des équations paraboliques non linéaires.
Mixed problem for the Bellman equation with measurable coefficients.
Modelling the electromagnetic behaviour of SiFe alloys using the Preisach theory and the principle of loss seperation.
Models for phase separation and their mathematics.
Models of Self-Organizing Bacterial Communities and Comparisons with Experimental Observations
Bacillus subtilis swarms rapidly over the surface of a synthetic medium creating remarkable hyperbranched dendritic communities. Models to reproduce such effects have been proposed under the form of parabolic Partial Differential Equations representing the dynamics of the active cells (both motile and multiplying), the passive cells (non-motile and non-growing) and nutrient concentration. We test the numerical behavior of such models and compare...
Monotone iteration for infinite systems of parabolic equations with functional dependence
We consider the initial value problem for an infinite system of differential-functional equations of parabolic type. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. The solutions are obtained by the monotone iterative method. We prove theorems on weak partial differential-functional inequalities as well the existence and uniqueness theorems in the class of continuous bounded functions and in the class of functions satisfying...
Monotonicity in time and stationary solutions for a quasilinear heat equation with source.
Motion by curvature of planar networks
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...
Motion of concentration sets in Ginzburg-Landau equations
Moving mesh for the axisymmetric harmonic map flow
We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the -gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the...
Moving mesh for the axisymmetric harmonic map flow
We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L2-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the...
Multicomponent flow in a porous medium. Adsorption and Soret effect phenomena : local study and upscaling process
Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions on state functions. Then, we present a numerical scheme based on a “mixed finite element” method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to...
Multicomponent flow in a porous medium. Adsorption and Soret effect phenomena: local study and upscaling process
Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions on state functions. Then, we present a numerical scheme based on a "mixed finite element"method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to...
Multidimensional exact solutions to a quasilinear parabolic equation with anisotropic heat conductivity.
Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion
This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or ⊃-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our...
Navier-Stokes equations on unbounded domains with rough initial data
We consider the Navier-Stokes equations in unbounded domains of uniform -type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded -calculus on such domains, and use a general form of Kato’s method. We also obtain information on the corresponding pressure term.
Neumann and second boundary value problems for hessian and Gauß curvature flows
New a priori estimates for nondiagonal strongly nonlinear parabolic systems
We consider nondiagonal elliptic and parabolic systems of equations with quadratic nonlinearities in the gradient. We discuss a new description of regular points of solutions of such systems. For a class of strongly nonlinear parabolic systems, we estimate locally the Hölder norm of a solution. Instead of smallness of the oscillation, we assume local smallness of the Campanato seminorm of the solution under consideration. Theorems about quasireverse Hölder inequalities proved by the author are essentially...
New results on the Burgers and the linear heat equations in unbounded domains.
We consider the Burgers equation and prove a property which seems to have been unobserved until now: there is no limitation on the growth of the nonnegative initial datum u0(x) at infinity when the problem is formulated on unbounded intervals, as, e.g. (0 +∞), and the solution is unique without prescribing its behaviour at infinity. We also consider the associate stationary problem. Finally, some applications to the linear heat equation with boundary conditions of Robin type are also given.
New results on the cauchy problem for parabolic systems and equations with strongly non linear sources.