Order Hamilton-Jacobi Equations Research Articles

A system of first order Hamilton–Jacobi equations related to an optimal debt management problem

A system of first order Hamilton–Jacobi equations related to an optimal debt management problem

Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator

We study the well-posedness of second order Hamilton-Jacobi equations with an Ornstein-Uhlenbeck operator in R N and R N ×[0, +∞). As applications, we solve the associated ergodic problem associated to the stationary equation and obtain the large time behavior of the solutions of the evolution equation when it is nondegenerate. These results are some generalizations of the ones obtained by Fujita, Ishii & Loreti (2006) by considering more general diffusion matrices or nonlocal operators of integro-differential type and general sublinear Hamiltonians. Our work uses as a key ingredient the a-priori Lipschitz estimates obtained in Chasseigne, Ley & Nguyen (2017).

High-order filtered schemes for time-dependent second order HJB equations

In this paper, we present and analyse a class of “filtered” numerical schemes for second order Hamilton–Jacobi–Bellman (HJB) equations. Our approach follows the ideas recently introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal. 51 (2013) 423–444, and more recently applied by other authors to stationary or time-dependent first order Hamilton–Jacobi equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by “filtering” them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.

High-order filtered schemes for first order time dependent linear and non-linear partial differential equations

This work is based on high-order “filtered scheme”. Recently filtered scheme has been introduced to solve some first order Hamilton–Jacobi equations. In this paper, we aim to solve some linear and non-linear partial differential equations by a high order filtered scheme. The proposed filtered scheme that is not monotone but still satisfies some ϵ-monotone property with a convergence result and with precise error estimate also has been proven. We will present filtration of different schemes for some linear and non-linear partial differential equations in several dimensions.

Selection problems for a discount degenerate viscous Hamilton–Jacobi equation

We prove that the solution of the discounted approximation of a degenerate viscous Hamilton–Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by using the nonlinear adjoint method and deriving a commutation lemma. This convergence result was first proven by Davini, Fathi, Iturriaga, and Zavidovique for first order Hamilton–Jacobi equations.

Error estimates for approximation schemes of effective Hamiltonians arising in stochastic homogenization of Hamilton-Jacobi equations

We study approximation schemes for effective Hamiltonians arising in the homogenization of first order Hamilton-Jacobi equations in stationary ergodic settings. In particular, we prove error estimates concerning the rate of convergence of the approximated solution to the effective Hamiltonian. Our main motivations are front propagation problems, but our results can be generalized to other types of Hamiltonians.

Continuous-time limit of dynamic games with incomplete information and a more informed player

We study a two-player, zero-sum, dynamic game with incomplete information where one of the players is more informed than his opponent. We analyze the limit value as the players play more and more frequently. The more informed player observes the realization of a Markov process (X, Y) on which the payoffs depend, while the less informed player only observes Y and his opponent’s actions. We show the existence of a limit value as the time span between two consecutive stages goes to zero. This value is characterized through an auxiliary optimization problem and as the unique viscosity solution of a second order Hamilton–Jacobi equation with convexity constraints.

Sobolev regularity for the first order Hamilton–Jacobi equation

We provide Sobolev estimates for solutions of first order Hamilton–Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse Holder inequality. Applications to mean field games are discussed.

Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

We consider the optimal control of solutions of first order Hamilton–Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.

On a class of first order Hamilton–Jacobi equations in metric spaces

We establish well-posedness of a class of first order Hamilton–Jacobi equation in geodesic metric spaces. The result is then applied to solve a Hamilton–Jacobi equation in the Wasserstein space of probability measures, which arises from the variational formulation of a compressible Euler equation.

Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations

In this paper, we consider a numerical scheme to solve first order Hamilton-Jacobi (HJ) equations posed on a junction. The main mathematical properties of the scheme are first recalled and then we give a traffic flow interpretation of the key elements. The scheme formulation is also adapted to compute the vehicles densities on a junction. The equivalent scheme for densities recovers the well-known Godunov scheme outside the junction point. We give two numerical illustrations for a merge and a diverge which are the two main types of traffic junctions. Some extensions to the junction model are finally discussed.

A Hamilton-Jacobi approach to junction problems and application to traffic flows

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

On a class of Hamilton-Jacobi equations and related logarithmic Sobolev inequality

In this paper we establish a new logarithmic Sobolev inequality. The proof is based on the analysis of the Cauchy problem for a first order Hamilton-Jacobi equation and qualitative properties of solutions.

Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary

We consider the weighted mean curvature flow in the plane with a driving term. For certain anisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi equation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi equation with a spatially non-local and discontinuous Hamiltonian. Weprove existence and uniqueness of solutions. On the way we show acomparison principle and a stability theorem for viscosity solutions.

Construction of Simple, Stable, and Convergent High Order Schemes for Steady First Order Hamilton–Jacobi Equations

We develop a very simple algorithm that permits to construct compact, high order schemes for steady first order Hamilton–Jacobi equations. The algorithm relies on the blending of a first order scheme and a compact high order scheme. The blending is conducted in such a way that the scheme is formally high order accurate. A convergence proof without error estimate is given. We provide several numerical illustrations that demonstrate the effective accuracy of the scheme. The numerical examples use triangular unstructured meshes, but our method may be applied to other kind of meshes. Several implementation remarks are also given.

Fractional semi-linear parabolic equations with unbounded data

This paper is devoted to the study of semi-linear parabolic equations whose principal term is fractional, i.e. is integral and eventually singular. A typical example is the fractional Laplace operator. This work sheds light on the fact that, if the initial datum is not bounded, assumptions on the non-linearity are closely related to its behaviour at infinity. The sublinear and superlinear cases are first treated by classical techniques. We next present a third original case: if the associated first order Hamilton-Jacobi equation is such that perturbations propagate at finite speed, then the semi-linear parabolic equation somehow keeps memory of this property. By using such a result, locally bounded initial data that are merely integrable at infinity can be handled. Next, regularity of the solution is proved. Eventually, strong convergence of gradients as the fractional term disappears is proved for strictly convex non-linearity.

Stochastic Differential Games with Asymmetric Information

We investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual viscosity solutions of some second order Hamilton-Jacobi equation.

A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Holder continuous with Holder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Holder inequality.

Homogenization of First Order Equations with (u/ϵ)-Periodic Hamiltonians Part II: Application to Dislocations Dynamics

This paper is concerned with a result of homogenization of a non-local first order Hamilton–Jacobi equation describing the dislocations dynamics. Our model for the interaction between dislocations involves both an integro-differential operator and a (local) Hamiltonian depending periodicly on u/ϵ. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane.

Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a 1D system of a parabolic equation and a first order Hamilton–Jacobi equation that are coupled together. We examine an associated Dirichlet boundary value problem. We prove the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial time. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of entropy solution for the initial value problem of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of bounded-from-below solutions. In order to prove our results on the bounded domain, we use an “extension and restriction” method, and we exploit a relation between scalar conservation laws and Hamilton–Jacobi equations, mainly to get our gradient estimates.