Sharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations

Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.Comment: 20 page

“The Juggler of Notre Dame.” Beautiful Music at the London Opera House

This poster, illustrated by E. Matania, provides a brief review of the London Opera House production of Le Jongleur de Notre Dame, by French opera composer Jules Massenet (1842-1912), printed in The Sphere, a British newspaper. Massenet based the opera upon Anatole France’s story by the same name, and the opera’s French libretto was written by Maurice Léna (1859-1928). While Massenet original intended the main juggler’s part as a tenor, the London Opera House’s production starred Victoria Fer. Fer’s performance was part of a shift towards women taking up the role, which according to some sources upset the composer. Text at the bottom of the poster: “The London Opera House gave the public an excellent Massenet programme last week which included “Le Jongleur de Notre Dame.” The opera was very elaborately mounted, and Miss Victoria Fer, who took the part of the juggler, acted and sang with great charm. The juggler is reproved by the friar for his loose songs and is invited to become a monk. The sight of the well-filled kitchen decides the half-starved juggler, and he enters the abbey. Later, when all the monks are endeavoring to devote some special work to the Virgin, the little juggler finds that he can do nothing. Boniface, the cook, explains in a passage of great sweetness how the humblest offering is acceptable. The juggler decides to do what he can, and in the last act we see him dancing a heel-and-toe measure before the Virgin. The monks discover him and are shocked at what they consider an outrage, but at the same moment the figure of the Virgin comes to life and blesses the juggler. Weakened by his previous privations, the little juggler expires on the steps of the altar. M. Enzo Pozzano as the monkish painter both sang and acted with excellent effect.”https://ecommons.udayton.edu/ml_juggler/1001/thumbnail.jp

Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to nonlinear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of {\sl geometry-compatible} (as we call it) conservation laws is singled out as an important case of interest, which leads to robust $L^p$ estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the $L^1$ contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.Comment: 30 pages. This is Part 1 of a serie

Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A = --c$\Delta$ acting in domains $\Omega$ := $\Omega$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $\epsilon$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $\epsilon$ $\rightarrow$ 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of ''layered media'' enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of ''flattening out'' of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a ''minimal amplitude hypothesis''. However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem