On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

“…In short, (1)-(2) is investigated with respect to viscosity-measure solutions (S, ρ). We refer to [31,4] for general properties of and justifications for measure solutions. Here we mention only that, in decent situations, as ε → 0, S ε → S locally uniformly and ρ ε → ρ in C([0, T ]; weak*-M(R n )) (M(R n ) denoting the space of Borel measures on R n ), where (S, ρ) is the viscosity-measure solution, in either of the following cases of regularization:…”

“…In the central reference [4] (see also [22] for another approach in one space dimension) the existence and uniqueness of viscosity-measure solutions of systems of the HamiltonJacobi and the continuity equations, along with several results on their singularities and other properties, were obtained under certain technical conditions on the Hamiltonian function H and the initial data. The objective of this contribution is to derive existence and uniqueness theorems under weaker or other types of conditions in particular on the growth of the associated Lagrangian functions.…”

“…The basic strategy of this paper is similar to that of [4] inasmuch as the proof of the existence and uniqueness for the system (1)-(2) is divided into the following steps:…”

Semiconcavity estimates for viscous Hamilton–Jacobi equations

“…This system, in the inviscid case where ε = 0, arises in the study of highly oscillatory solutions of Schrödinger's equation as well as in some models for pressureless fluid flow; see [4,16]. The vanishing viscosity transition from (3)-(4) to its inviscid counterpart is the subject of a forthcoming paper [16].…”

On the Hamilton–Jacobi Equation for a Harmonic Oscillator

“…For the author, a source of motivation is the system of the Hamilton-Jacobi equation S t + 1 2 |∇S| 2 + U (x) = 0 and the continuity equation ρ t + div(ρ∇S) = 0 appearing, e.g., in the high frequency approximation of Schrödinger's equation [8,10]. A special feature of this system is that mass accumulates on the singular set of S. Thus, the mass density ρ(t) is a measure even if initially the density is a regular function [2,3,5,7,14,16]. The one-dimensional case has been investigated in some detail in [5].…”