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

Abstract

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities

“…for all k ∈ R and β ∈ C 1 c (0, t j ), β ≥ 0, where t j ∈ (0, T ] is defined by (3.6). By [5,Remark 7] the limits in (3.9a)-(3.9b) exist and are finite. Before stating the basic well-posedness result for the Cauchy problem, we introduce the following singular Cauchy-Dirichlet problems, where m 1 , m 2 = ±∞:…”

“…Problem (CL) was considered in [4,5] in the context of Radon measure-valued entropy solutions . There it was shown that if (1.1) u 0 is a signed Radon measure on R, and the singular part u 0s is a finite superposition of Dirac masses, each initial Dirac mass does not increase in time but, since H is bounded, does not disappear instantaneously, i.e.…”

Discontinuous viscosity solutions of first order Hamilton-Jacobi equations

“…In fact, arguing as in the proof of [31, Proposition 3.1] (see also [1]) and using (6.3) gives (6.7)-(6.8), whence (6.9) easily follows (see [10,Lemma 5.2] for details). By estimates (6.6)-(6.8) the family {u ε } is bounded in L ∞ (Q), and there exists M > 0 such that sup ε∈(0,1) u ε W 1,1 (Q) ≤ M .…”

“…although it is not trivial to make the correspondence rigorous ( [11]; see also [4] for a statement in this direction). If u 0 = U ′ 0 is a Radon measure, it is possible to prove existence of suitably defined measure-valued entropy solutions of problem (CL) ( [10]; see also [8,9] for the case of positive initial measures). Moreover, if the singular part u 0s of u 0 (with respect to the Lebesgue measure) is a finite superposition of Dirac masses, uniqueness of such solutions can be proven, if additional compatibility conditions are satisfied near the support of u 0s .…”

“…Moreover, if the singular part u 0s of u 0 (with respect to the Lebesgue measure) is a finite superposition of Dirac masses, uniqueness of such solutions can be proven, if additional compatibility conditions are satisfied near the support of u 0s . Remarkably, the singularities of u 0 in (CL) have a barrier effect which corresponds to that produced by discontinuities of U 0 in (CP ), and well-posedness of (CL) can be proven using singular Dirichlet problems which are the counterpart of (N ) for HJ equations (see [9,10] for details).…”