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

Abstract

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition L(t, x, v) ≥ 1 2 m(|v| 2 −ω 2 |x| 2 )−C where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S t +H (t, x, ∇S) = 0 in Q T = (0, T )×R n , S| t=0 = S 0 , without any conditions at infinity on the solution.Here ωT < π/2. To the Hamilton-Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity-measure solution (S, ρ) ofThis system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.

Exponentially growing solutions of parabolic Isaacs' equations

“…However, uniqueness (without conditions at infinity) nevertheless holds if H corresponds to a sufficiently well-behaved variational problem [19][20][21][22]. In this case, the only solution in the class of all continuous viscosity solutions is the associated value function and comparison holds between viscosity solutions [19,21,22].…”

On the Hamilton–Jacobi Equation for a Harmonic Oscillator

“…Then, for each a ∈ R n , S 0 (a) := lim In the stated generality, Theorems 1 and 2 are special cases of the results of [14].…”

“…Statements (14) through (20) are mutually equivalent by calculation, the definition of the inverse of a multivalued mapping, and relation (10).…”

“…In [14] the well-posedness for the initial-value problem for far more general Hamilton-Jacobi equations than (1) has been proved in the class of all continuous viscosity solutions. The unique solution is the value or action function.…”

“…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].…”