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Regis Monneau

Publications and source records attributed to Regis Monneau.

16 recordsLinked to original sources
arXiv2025

Beyond uniqueness: Relaxation calculus of junction conditions for coercive Hamilton-Jacobi equations

A junction is a particular network given by the collection of $N\ge 1$ half lines $[0,+\infty)$ glued together at the origin. On such a junction, we consider evolutive Hamilton-Jacobi equations with $N$ coercive Hamiltonians. Furthermore,we consider a general desired junction condition at the origin, given by some monotone function $F_0:\R^N\to \R$.There is existence and uniqueness of solutions which only satisfy weakly the junction condition (at the origin, they satisfy either the desired junction condition or the PDE).We show that those solutions satisfy strongly a relaxed junction condition $\frak R F_0$ (that we can recognize as an effective junction condition). It is remarkable that this relaxed condition can be computed in three different but equivalent ways: 1) using viscosity inequalities, 2) using Godunov fluxes, 3) using Riemann problems.Our result goes beyond uniqueness theory, in the following sense: solutions to two different desired junction conditions $F_0$ and $F_1$ do coincide if $\frak R F_0=\frak R F_1$.

math.AP↗
arXiv2024

Germs for scalar conservation laws: the Hamilton-Jacobi equation point of view

We prove that the entropy solution to a scalar conservation law posed on the real line with a flux that is discontinuous at one point (in the space variable) coincides with the derivative of the solution to a Hamilton-Jacobi (HJ) equation whose Hamiltonian is discontinuous. Flux functions (Hamiltonians) are not assumed to be convex in the state (gradient) variable. The proof consists in proving the convergence of two numerical schemes. We rely on the theory developed by B.~Andreianov, K.~H.~Karlsen and N.~H.~Risebro (\textit{Arch. Ration. Mech. Anal.}, 2011) for such scalar conservation laws and on the viscosity solution theory developed by the authors (\textit{arxiv}, 2023) for the corresponding HJ equation. This study allows us to characterise certain germs introduced in the AKR theory (namely maximal and complete ones) and relaxation operators introduced in the viscosity solution framework.

math.AP↗
arXiv2024

The twin blow-up method for Hamilton-Jacobi equations in higher dimension

In this paper, we show how to extend the twin blow-up method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.

math.AP↗
arXiv2023

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.

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arXiv2023

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.

math.AP↗
arXiv2023

Coercive Hamilton-Jacobi equations in domains: the twin blow-ups method

In this note, we consider an evolution coercive Hamilton-Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient variables are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017) to extend their comparison principle to more general boundary conditions and to Hamiltonians that are not globally Lipschitz continuous in the time variable. Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared. We refer to our technique as the twin blow-ups method since two blow-ups are performed simultaneously, one for each variable of the doubling variable method. The Lipschitz regularity of the regularized subsolution provides a key Lipschitz inequality satisfied by the two blow-up limits, that are a priori allowed to be infinite. For expository reasons, the result is presented here in the framework of space dimension one and the general case is treated in a companion paper.

math.AP↗
arXiv2023

Non-convex coercive Hamilton-Jacobi equations: Guerand's relaxation revisited

This work is concerned with Hamilton-Jacobi equations of evolution type posed in domains and supplemented with boundary conditions. Hamiltonians are coercive but are neither convex nor quasiconvex. We analyse boundary conditions when understood in the sense of viscosity solutions. This analysis is based on the study of boundary conditions of evolution type. More precisely, we give a new formula for the relaxed boundary conditions derived by J. Guerand (J. Differ. Equations, 2017). This new point of view unveils a connection between the relaxation operator and the classical Godunov flux from the theory of conservation laws. We apply our methods to two classical boundary value problems. It is shown that the relaxed Neumann boundary condition is expressed in terms of Godunov's flux while the relaxed Dirichlet boundary condition reduces to an obstacle problem at the boundary associated with the lower non-increasing envelope of the Hamiltonian.

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arXiv2010

Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.

math.AP↗
arXiv2009

Global Existence Results and Uniqueness for Dislocation Equations

We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all time. Then we discuss the uniqueness of such solutions in several situations, both in the monotone and non monotone case.

math.AP↗
arXiv2009

Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system

math-ph↗
arXiv2008

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.

math-ph↗
arXiv2006

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.

math.AP↗
arXiv2005

An Unstable Elliptic Free Boundary Problem arising in Solid Combustion

We prove a regularity result for the unstable elliptic free boundary problem $Δu = -χ_{\{u>0\}}$ related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular, that is, $\{u=0\}$ is locally an analytic surface and $u|_{\bar{\{u>0\}}}, u|_{\bar{\{u<0\}}}$ are locally analytic functions. Moreover we prove a partial regularity result for solutions that are non-degenerate of second order: here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$. We discuss possible singularities.

math.AP↗
arXiv2004

On the one-dimensional parabolic obstacle problem with variable coefficients

This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results.

math.AP↗
arXiv2003

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.

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