Contractive metrics for scalar conservation laws

We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L^1-contraction property shown by Kruzkov

L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method

We consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L^2 perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L^2 norm of a perturbed solution relative to the shock wave is bounded above by the L^2 norm of the initial perturbation.Comment: 17 page

Precision constrained simulation of the Local Universe

We use the formalism of constrained Gaussian random field to compute a precise large scale simulation of the 60 Mpc/h volume of our Local Universe. We derive the constraints from the reconstructed peculiar velocities of the 2MASS Redshift Survey. We obtain a correlation of 0.97 between the log-density field of the dark matter distribution of the simulation and the log-density of observed galaxies of the Local Universe. We achieve a good comparison of the simulated velocity field to the observed velocity field obtained from the galaxy distances of the NBG-3k. At the end, we compare the two-point correlation function of both the 2MRS galaxies and of the dark matter particles of the simulation. We conclude that this method is a very promising technique of exploring the dynamics and the particularities the Universe in our neighbourhood.Comment: 8 pages, 5 figures, accepted by MNRA

Two limit cases of Born-Infeld equations

International audienceWe study two limit cases \l \rightarrow \infty and \l \rightarrow 0 in Born-Infeld equations. Here the parameter \l >0 is interpreted as the maximal electric field in the electromagnetic theory and the case \l = 0 corresponds to the string theory. Formal limits are governed by the classical Maxwell equations and pressureless magnetohydrodynamics system, respectively. For studying the limit \l \rightarrow \infty, a new scaling is introduced. We give the relations between these limits and Brenier high and low field limits. Finally, using compensated compactness arguments, the limits are rigorously justified for global entropy solutions in $L^\infty$ in one space dimension, based on derived uniform estimates and techniques for linear Lagrangian systems

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology

Scattering defect in large diameter titanium-doped sapphire crystals grown by the Kyropoulos technique

International audienceThe Kyropoulos technique allows growing large diameter Ti doped sapphire for Chirped pulse amplification laser. A scattering defect peculiar to Kyropoulos grown crystals is presented. This defect is characterized by different techniques: luminescence, absorption measurement, X-ray rocking curve. The impact of this defect to the potential application in chirped pulse amplification CPA laser is evaluated. The nature of this defect is discussed. Modified convexity of the interface is proposed to avoid the formation of this defect and increase the quality of the Ti sapphire crystal

Global W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve

Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit

This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately

Mass transportation with LQ cost functions

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier's Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case