Sufficient and necessary conditions for the existence and uniqueness of classical solutions to the Cauchy problem for the scalar conservation law are found in the class of discontinuous initial data and non-convex flux function. Regularity of rarefaction waves starting from discontinuous initial data and their dependence on the flux function are investigated and illustrated in a few examples.

We study oscillatory solutions of semilinear first order symmetric hyperbolic system $Lu=f(t,x,u,\bar{u})$, with real analytic $f$.The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in $T,X$ with only the natural hypothesis of coherence.In the special case where $L$ has constant coefficients and the phases are linear, the solutions have asymptotic description$$u^{ϵ}=U(t,x,t/ϵ,x/ϵ)+o\left(1\right)$$where the profile $U(t,x,T,X)$ is almost periodic in $(T,X)$.The main novelty in the analysis is the space of profiles which...

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough” coefficient function $k\left(x\right)$. We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, $k^{\text{'}}$ is in $BV$, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations...

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a "rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion...