Self-propagating High temperature Synthesis (SHS) in the high activation energy regime

We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit turns out to be the Stefan problem for supercooled water with spatially inhomogeneous coefficients. Although the present paper leaves open mathematical questions concerning the convergence, our precise form of the limit problem suggest a strikingly simple explanation for the numerically observed pulsating waves

Global continuous solutions to diagonalizable hyperbolic systems with large and monotone data

In this paper, we study diagonalizable hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and nondecreasing initial data. Moreover, we show in particular cases some uniqueness results. We also remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonalizable hyperbolic systems appears naturally in the modelling of the dynamics of dislocation densities

Junction of elastic plates and beams (Preliminary version)

We consider the linearized elasticity system in a multidomain of the three dimensional space. This multidomain is the union of a horizontal plate, with fixed cross section and small thickness "h", and of a vertical beam with fixed height and small cross section of radius "r". The lateral boundary of the plate and the top of the beam are assumed to be clamped. When "h" and "r" tend to zero simultaneously, with "r" much greater than the square of "h", we identify the limit problem. This limit problem involves six junction conditions.Comment: Ceci est la redaction du 3 Mars 2003. Francois Murat souhaite y faire des modification

A class of germs arising from homogenization in traffic flow on junctions

We consider traffic flows described by conservation laws. We study a 2:1 junction (with two incoming roads and one outgoing road) or a 1:2 junction (with one incoming road and two outgoing roads). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time and the traffic can also be slowed down by periodic in time flux-limiters. After a long time, and at large scale in space, we intuitively expect an effective junction condition to emerge. Precisely, we perform a rescaling in space and time, to pass from the mesoscopic scale to the macroscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the effective junction condition, which belongs to a general class of germs (in the terminology of [6, 21, 37]). The identification of this germ and of a characteristic subgerm which determines the whole germ, is the first key result of the paper. The second key result of the paper is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm. This construction is indeed explicit at the level of some mixed Hamilton-Jacobi equations for concave Hamiltonians (i.e. fluxes). The explicit solutions are found in the spirit of representation formulas for optimal control problems

Existence and qualitative properties of multidimensional conical bistable fronts

International audienceTravelling fronts with conical-shaped level sets are constructed for reaction-diffusion equations with bistable nonlinearities of positive mass. The construction is valid in space dimension 2, where two proofs are given, and in arbitrary space dimensions under the assumption of cylindrical symmetry. General qualitative properties are presented under various assumptions: conical conditions at infinity, existence of a sub-level set with globally Lipschitz boundary, monotonicity in a given direction

Stability of travelling waves in a model for conical flames in two space dimensions

This paper deals with the question of the stability of conical-shaped solutions of a class of reaction-diffusion equations in $\R^2$. One first proves the existence of travelling waves solutions with conical-shaped level sets, generalizing earlier results by Bonnet, Hamel and Monneau. One then gives a characterization of the global attractor of these semilinear parabolic equations under some conical asymptotic conditions. Lastly, the global stability of the travelling waves solutions is proved

Conservation law and Hamilton-Jacobi equations on a junction: the convex case

The goal of this paper is to study the link between the solution to an Hamilton-Jacobi (HJ) equation and the solution to a Scalar Conservation Law (SCL) on a special network. When the equations are posed on the real axis, it is well known that the space derivative of the solution to the Hamilton-Jacobi equation is the solution to the corresponding scalar conservation law. On networks, the situation is more complicated and we show that this result still holds true in the convex case on a 1:1 junction. The correspondence between solutions to HJ equations and SCL on a 1:1 junction is done showing the convergence of associated numerical schemes. A second direct proof using semi-algebraic functions is also given. Here a 1:1 junction is a simple network composed of two edges and one vertex. In the case of three edges or more, we show that the associated HJ germ is not a L 1-dissipative germ, while it is the case for only two edges. As an important byproduct of our numerical approach, we get a new result on the convergence of numerical schemes for scalar conservation laws on a junction. For a general desired flux condition which is discretized, we show that the numerical solution with the general flux condition converges to the solution of a SCL problem with an effective flux condition at the junction. Up to our knowledge, in previous works the effective condition was directly implemented in the numerical scheme. In general the effective flux condition differs from the desired one, and is its relaxation, which is very natural from the point of view of Hamilton-Jacobi equations. Here for SCL, this effective flux condition is encoded in a germ that we characterize at the junction

Strictly convex Hamilton-Jacobi equations: strong trace of the gradient

We consider Lipschitz continuous solutions to evolutive Hamilton-Jacobi equations. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the gradient of the solution. This result is based on a Liouville-type result of classification of global solutions on the half space. Under zero Dirichlet boundary condition, we show that the solution only depends on the normal variable. As a consequence, we show that the existence of a pointwise tangential gradient implies existence of a pointwise normal gradient. For the Liouville-type result, and when the Hamiltonian is not convex, we give a counterexample with a solution which is not one-dimensional. We give two applications. On the one hand, for the classical stationary Dirichlet problem on a bounded domain, we show the existence of a closed subset of the boundary of the domain, where Taylor expansion of the solution is uniform. On the other hand, for Hamilton-Jacobi equations on a network, we show that the space derivative of the solution has a trace at each node, which satisfies a natural germ condition

Strictly convex Hamilton-Jacobi equations: strong trace of the derivatives in codimension ≥ 2

We consider Lipschitz continuous viscosity solutions to an evolutive Hamilton-Jacobi equation. The equation arises outside a closed set Γ. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the derivatives of the solution on the Lipschitz boundary Γ of codimension d ≥ 2. The very special case d = 1 is done in a separated work. This result is based on a Liouville-type result of classification of global solutions with zero Dirichlet condition on the boundary Γ, where Γ is an affine subspace. We show in particular that such solutions only depend on the normal variable to Γ. As a consequence, we show more generally that the existence of a pointwise tangential gradient along Γ implies the existence of pointwise directional derivatives in all directions. This result also holds true for Hamiltonians depending on the time-space variables, under an additional Dini condition involving certain moduli of continuity. We also give a counterexample for d = 2 in the stationary case, where the Hamiltonian is only continuous in the space variable, and where the solution has no directional derivatives in any directions normal to Γ. Such phenomenon does not hold for d = 1