Radon measure-valued solutions of first order scalar conservation laws

Abstract

We study nonnegative solutions of the Cauchy problemwhere u 0 is a Radon measure and ϕ ∶ [0, ∞) ↦ R is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on ϕ, we prove their uniqueness if the singular part of u 0 is a finite superposition of Dirac masses.In terms of the behaviour of ϕ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case ϕ(u) = u this happens for all times). In the latter case we describe the evolution of the singular parts.

Signed Radon measure-valued solutions of flux saturated scalar conservation laws

“…We shall prove below (see Remark 5.4) that, if (A 0 )-(A 1 ) hold, for every entropy solution u of (CL) the limits Remark 3.2. It was already observed in [2] that in general measure-valued entropy solutions are not unique. This is essentially a consequence of the elementary observation that there exists a unique entropy solution for which [u s (t)] = u 0s for a.e.…”

“…In [3] we considered the case of nonnegative initial measures u 0 . In the present paper we consider the case of signed measures (see [2,5,7,9] for motivations and related remarks). A specific motivation is the link between measure-valued solutions of (CL) and discontinuous solutions of the Cauchy problem for the Hamilton-Jacobi equation…”

“…Existence of a solution is proven by a constructive approach which can be outlined as follows. By (A 0 )-(A 1 ) there exists a positive time τ until which all singularities persist (see [2,Theorem 3.5]), thus the real line is the disjoint union of p+1 intervals. In each interval we solve the initial-boundary value problem for the conservation law in (CL), the initial data being the restriction of u 0r to that interval, with "boundary conditions equal to infinity".…”

A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws

“…The above considerations suggest a constructive approach to address problem (P ) under assumption (H 0 ). By the results in [3] there is a positive time τ until which all singularities persist, thus the real line is the disjoint union of p+1 intervals. In each interval we solve the initial-boundary value problem for the conservation law in (P ), the initial data being the restriction of u 0r to that interval, with "boundary conditions equal to infinity" -or, equivalently, by imposing the analogue of (1.22) to be satisfied at each point x j , j = 1, .…”

“…Without loss of generality one may assume that C ϕ = 0 (otherwise replace x by x − C ϕ t, see [3]). If u 0 is any positive bounded Radon measure, an approximation approach can be used to construct suitably defined entropy solutions of (P ) in a space of bounded Radon measures on S (see Definitions 3.1-3.2 below and [3,Theorem 3.2]; in the present section we call such solutions "constructed solutions"). However, an additional condition on solutions is needed for the well-posedness of (P ), since examples of nonuniqueness can be easily produced (see [3,5]).…”