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Denis Serre

Publications and source records attributed to Denis Serre.

At least 19 recordsLinked to original sources
arXiv2024

Compensated Integrability in bounded domains ; Applications to gases

An accurate functional inequality for Div-BV positive symmetric tensors $A$ in a bounded domain $U\subset\mathbb{R}^n$ arises whenever the tangential part of the normal trace $\gamma_\nu A\sim A\vec\nu$ is a finite measure over $\partial U$. The proof involves an extension operator to a neighbourhood of $\bar U$. The resulting inequality depends upon the domain only through the $C^3$-regularity of $\partial U$, some constant involving the curvature and its first derivatives.This abstract statement applies to several models of Gas Dynamics (Euler system, Hard Spheres dynamics), as the boundary condition (slip, or reflection) tells us that $A\vec\nu$ is parallel to $\vec\nu$, where $A$ is the mass-momentum tensor.

math.AP↗
arXiv2023

Mixed determinants, Compensated Integrability and new {\em a priori} estimates in Gas dynamics

We extend the scope of our recent Compensated Integrability theory, by exploiting the multi-linearity of the determinant map over ${\bf Sym}_n(\mathbb{R})$. This allows us to establish new {\em a priori} estimates for inviscid gases flowing in the whole space ${\mathbb R}^d$. Notably, we estimate the defect measure (Boltzman equation) or weighted spacial correlations of the velocity field (Euler system). As usual, our bounds involve only the total mass and energy of the flow.

math.AP↗
arXiv2021

Symmetric Divergence-free tensors in the calculus of variations

Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called "second" variational principle, where the argument of the Lagrangian is a closed differential form. Divergence-free tensors are nothing but the second form of the Euler--Lagrange equations. The symmetry is associated with the invariance of the Lagrangian density upon the action of some orthogonal group.

math.AP↗
arXiv2021

Projective properties of Divergence-free symmetric tensors, and new dispersive estimates in gas dynamics

The class of Divergence-free symmetric tensors is ubiquitous in Continuum Mechanics. We show its invariance under projective transformations of the independent variables. This action, which preserves the positiveness, extends Sophus Lie's group analysis of Newtonian dynamics.When applied to models of gas dynamics --~such as Euler system or Boltzmann equation,~-- in combination with Compensated Integrability, this yields new dispersive estimates. The most accurate one is obtained for mono-atomic gases. Then the space-time integral of $tρ^\frac1d p$ is bounded in terms of the total mass and moment of inertia alone.

math.AP↗
arXiv2021

Asymptotic stability of scalar multi-D inviscid shock waves

In several space dimensions, scalar shock waves between two constant states u $\pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.

math.AP↗
arXiv2020

Hard spheres dynamics: weak vs hard collisions

We consider the motion of a finite though large number $N$ of hard spheres in the whole space $\mathbb{R}^n$. Particles move freely until they experience elastic collisions. We use our recent theory of Compensated Integrability in order to estimate how much the particles are deviated by collisions. Our result, which is expressed in terms of hodographs, tells us that only $O(N^2)$ collisions are significant.

math.AP↗
arXiv2020

Source-solutions for the multi-dimensional Burgers equation

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $\times$ M. The Dirac mass at X $\in$ R n is $δ$ X or $δ$ x=X. If $ν$ $\in$ M (R m) and $μ$ $\in$ M (R q), then $ν$ $\otimes$ $μ$ is the measure over R m+q uniquely defined by $ν$ $\otimes$ $μ$, $ψ$ = $ν$, f $μ$, g whenever $ψ$(x, y) $\not\equiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL \# 5669. 46 all{é}e d'Italie,

math.AP↗
arXiv2019

On the upper semicontinuity of a quasiconcave functional

In the recent paper \cite{SER}, the second author proved a divergence-quasiconcavity inequality for the following functional $ \mathbb{D}(A)=\int_{\mathbb{T}^n} det(A(x))^{\frac{1}{n-1}}\,dx$ defined on the space of $p$-summable positive definite matrices with zero divergence. We prove that this implies the weak upper semicontinuity of the functional $\mathbb{D}(\cdot)$ if and only if $p>\frac{n}{n-1}$.

math.AP↗
arXiv2019

Estimating the number and the strength of collisions in molecular dynamics

We consider the motion of a finite though large number of particles in the whole space R n. Particles move freely until they experience pairwise collisions. We use our recent theory of divergence-controlled positive symmetric tensors in order to establish two estimates regarding the set of collisions. The only information needed from the initial data is the total mass and the total energy.

math.AP↗
arXiv2018

Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\ge0}$ extends uniquely as a continuous semi-group over $L^p(\mathbb{R}^n)$ whenever $1\le p<\infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \infty$ and $t>0$, $S_t$ actually maps $L^p(\mathbb{R}^n)$ into $L^q(\mathbb{R}^n)$. These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration.

math.AP↗
arXiv2018

Multi-dimensional scalar conservation laws with unbounded integrable initial data

We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.

math.AP↗
arXiv2018

Compensated integrability. Applications to the Vlasov--Poisson equation and other models in mathematical physics

We extend our analysis of divergence-free positive symmetric tensors (DPT) begun in a previous paper. On the one hand, we refine the statements and give more direct proofs. Next, we study the most singular DPTs, and use them to prove that the determinant is the only quantity that enjoys an improved integrability. Curiously, these singularities are intimately related to the Minkowski's Problem for convex bodys with prescribed Gaussian curvature. We then cover a list of models of mathematical physics that display a divergence-free symmetric tensor ; the most interesting one is probably that of nonlinear Maxwell's equations in a relativistic frame. The case of the wave equation is the occasion to highlight the role of the positivity assumption. Last, but not least, we show that the Vlasov--Poisson equation for a plasma is eligible for our theory.

math.AP↗
arXiv2017

Divergence-free positive symmetric tensors and fluid dynamics

We consider $d\times d$ tensors $A(x)$ that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of $(\det A)^{\frac1{d-1}}$. We apply them to models of compressible inviscid fluids: Euler equations, Euler--Fourier, relativistic Euler, Boltzman, BGK, etc... We deduce an {\em a priori} estimate for a new quantity, namely the space-time integral of $ρ^{\frac1n}p$, where $ρ$ is the mass density, $p$ the pressure and $n$ the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

math.AP↗
arXiv2015

Expansion of a compressible gas in vacuum

Tai-Ping Liu \cite{Liu\_JJ} introduced the notion of "physical solution' of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a Hölder singularity of the sound speed. We address the question of when this acceleration appears or when the front just move at constant velocity. We know from \cite{Gra,SerAIF} that smooth isentropic flows with a non-accelerated front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\in\R$ ; we say that they are {\em eternal}. We derive a sufficient condition in terms of the initial data, under which the boundary singularity must appear. As a consequence, we show that, in contrast to the even-dimensional case, eternal flows with a non-accelerated front don't exist in odd space dimension. In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free flow, their asymptotics as both ends $t\rightarrow\pm\infty$ are intimately related to each other.

math.AP↗
arXiv2010

The numerical measure of a complex matrix

We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $μ_A$ can be defined as the law of the random variable $ \in \C$ when the vector $X \in \C^n$ is uniformly distributed on the unit sphere. If the matrix $A$ is normal, we show that $μ_A$ has a piecewise polynomial density $f_A$, which can be identified with a multivariate $B$-spline. In the general (nonnormal) case, we relate the Radon transform of $μ_A$ to the spectrum of a family of Hermitian matrices, and we deduce an explicit representation formula for the numerical density which is appropriate for theoretical and computational purposes. As an application, we show that the density $f_A$ is polynomial in some regions of the complex plane which can be characterized geometrically, and we recover some known results about lacunas of symmetric hyperbolic systems in $2+1$ dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix $A \in M_n(\C)$ concentrates to a Dirac mass as the size $n$ goes to infinity.

math.FA↗
arXiv2009

Experimental and theoretical study of diffraction properties of various crystals for the realization of a soft gamma-ray Laue lens

Crystals are the elementary constituents of Laue lenses, an emerging technology which could allow the realization of a space borne telescope 10 to 100 times more sensitive than existing ones in the 100 keV - 1.5 MeV energy range. This study addresses the current endeavor to the development of efficient crystals for the realization of a Laue lens. In the theoretical part 35 candidate-crystals both pure and two-components are considered. Their peak reflectivity at 100 keV, 500 keV and 1 MeV is calculated assuming they are mosaic crystals. It results that a careful selection of crystals can allow a reflectivity above 30% over the whole energy range, and even reaching 40% in its lower part. Experimentally, we concentrated on three different materials (Si_{1-x}Ge_x with gradient of composition, mosaic Cu and Au) that have been measured both at ESRF and ILL using highly-monochromatic beams ranging from 300 keV up to 816 keV. The aim was to check their homogeneity, quality and angular spread (mosaicity). These crystals have shown outstanding performance such as reflectivity up to 31% at ~600 keV (Au) or 60% at 300 keV (SiGe) and angular spread as low as 15 arcsec for Cu, fulfilling very well the requirements for a Laue lens application. Unexpectedly, we also noticed important discrepancies with Darwin's model when a crystal is measured using various energies.

astro-ph.IM↗