Discontinuous Solutions of Hamilton–Jacobi Equations Versus Radon Measure-Valued Solutions of Scalar Conservation Laws: Disappearance of Singularities

Noncoercive diffusion equations with Radon measures as initial data

“…In the framework of quasilinear parabolic equations, a strictly positive waiting time was shown to exist for logarithmic 𝜙 and space dimension 𝑁 = 2 already in [52,Theorem 3.4] (see also Propositions 3.9 and 3.10 below). For first-order hyperbolic conservation laws in one space dimension with bounded flux, the existence of finite, strictly positive waiting times and their estimates were proven in [14,15].…”

“…In the framework of quasilinear parabolic equations, a strictly positive waiting time was shown to exist for logarithmic $$\phi$$ and space dimension $$N=2$$ already in [52, Theorem 3.4] (see also Propositions 3.9 and 3.10 below). For first‐order hyperbolic conservation laws in one space dimension with bounded flux , the existence of finite, strictly positive waiting times and their estimates were proven in [14, 15]. More than that, it is known that Radon measure‐valued solutions arise even with a regular initial data function $$u_0$$ in models of aggregation phenomena , where an $$L^1$$‐$$\mathcal {M}$$ deregularizing effect takes place.…”