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 $L(t, x, v) \geq \frac{1}{2} m({\mid v \mid}^2 - \omega^{2}{\mid x \mid}^{2})-C$ 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 $Q_T = (0, T)\times {\mathbb{R}}^{n}, S\mid_{t=0} = S_0$, 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) of

$$ S_t + \frac{1}{2}{\mid\nabla S\mid}^2 + U(x) = 0 \quad {\rm in}\, Q_T, \quad S(0,x) = S_0(x) \quad {\rm in} \, {\mathbb{R}}^n; $$

$$ \rho_t + {\rm div}(\rho\nabla S) = 0 \quad {\rm in}\, Q_{T}, \quad \rho(0,x) = \rho_0(x) \quad {\rm in} \, {\mathbb{R}}^n. $$

This system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.