A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit

We study the following system of the viscous Hamilton-Jacobi and the continuity equations in the limit as epsilon down arrow 0: S(t)(epsilon) + 1/2 vertical bar DS(epsilon)vertical bar(2) + V(x) - epsilon Delta S(epsilon) = 0 in Q(T), S(epsilon)(0, x) = S(0)(x) in R(n); rho(epsilon)(t) + div(rho(epsilon) DS(epsilon)) = 0 in Q(T), rho(epsilon)(0, x) = rho(0)(x) in R(n). Here Q(T) = (0, T] x R(n). The potential V and the initial function S(0) are allowed to grow quadratically while rho(0) is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity-measure solution (S, rho).