Questions d'examen. Complément à la discussion des lignes courbes représentées par des équations algébriques quelconques
Finite speed of propagation for a non-local porous medium equation
This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and Vázquez (2010).
Second-order elliptic integro-differential equations : viscosity solutions' theory revisited
Estimates on elliptic equations that hold only where the gradient is large
We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
A Hamilton-Jacobi approach to junction problems and application to traffic flows
This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...
An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions
Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations
This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...
Conservative numerical methods for a two-temperature resistive MHD model with self-generated magnetic field term
We propose numerical methods on Cartesian meshes for solving the 2-D axisymmetric two-temperature resistivive magnetohydrodynamics equations with self-generated magnetic field and Braginskii’s [1] closures. These rely on a splitting of the complete system in several subsystems according to the nature of the underlying mathematical operator. The hyperbolic part is solved using conservative high-order dimensionally split Lagrange-remap schemes whereas...
Magnetic equations with FreeFem++: The Grad-Shafranov equation & the current hole
FreeFem++ [11] is a software for the numerical solution of partial differential equations. It is based on finite element method. The FreeFem++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem++ tool on basic magnetic equations arising in Fusion Plasma...
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