A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker{Planck equations in space dimensions d>2 is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient ow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient ow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, d = 2. A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution's support

Minimal entropy conditions for Burgers equation

We consider stricly convex, 1-d scalar conservation laws. We show that a single strictly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related Hamilton-Jacobi equation

The polar cone of the set of monotone maps

We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices. © 2014 American Mathematical Society

A simple proof of global existence for the 1D pressureless gas dynamics equations

Sticky particle solutions to the one-dimensional pressureless gas dynamics equations can be constructed by a suitable metric projection onto the cone of monotone maps, as was shown in recent work by Natile and Savaré. Their proof uses a discrete particle approximation and stability properties for first-order differential inclusions. Here we give a more direct proof that relies on a result by Haraux on the differentiability of metric projections. We apply the same method also to the one-dimensional Euler-Poisson system, obtaining a new proof for the global existence of weak solutions. © 2015 Society for Industrial and Applied Mathematics

On the optimality of velocity averaging lemmas

tudying weak solutions of Burgers' equation with finite entropy dissipation we show the sharpness of recent results of Jabin and Perthame on velocity averaging. Similar arguments give bounds on the regularity of asymptotic finite-energy states for some variational problems of Ginzburg–Landau type

Structure of entropy solutions for multi-dimensional scalar conservation laws

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation

HCDTE Lecture Notes. Part II. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations

This is the second of two volumes containing the lecture notes of some of the courses given during the intensive trimester HCDTE, Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: analysis and control, held at SISSA, Trieste (Italy) from May 16th to July 22nd, 2011. The lectures covered a number of difierent topics within the fields of hyperbolic equations, fluid dynamic, dispersive and transport equations, measure theory and control and they were primarily intended for PhD students and young researchers at the beginning of their career. 1. S. Daneri and A. Figalli, Variational models for the incompressible Euler equations. 2. A. Pratelli and S. Puglisi, Elastic deformations on the plane and approximations. 3. G. Staffilani, Periodic Schrodinger equations in Hamiltonian form. 4. L. Sz\ue9kelyhidi, From isometric embeddings to turbulence. 5. M. Westdickenberg, Finite energy solutions to the isentropic Euler equations