Generalized Solutions to Conservation Laws

Abstract

We introduce a framework for studying degenerate limits for conservation laws of hyperbolic type and initial value problems for conservation laws of mixed hyperbolic-elliptic type.

The purpose of this note is to suggest a framework for analyzing certain aspects of singular limits for conservation laws of hyperbolic type, in particular the zero diffusion limit vs the zero dispersion limit and for studying the Cauchy problem for conservation laws of mixed type. The general setting is provided by a system of n equations in one space dimension,

$${u_t} + f{(u)_x} = 0$$

((1))

where the flux function f is a smooth nonlinear map from Rⁿ to Rⁿ. First let us assume that f is strictly hyperbolic in the sense that its Jacobian ∇f has n real and district eigenvalues With regard to singular limits we are interested in diffusion processes generated by parabolic regularization,

$${u_t} + f{(u)_x} = \varepsilon {u_{xx}},$$

((2))

and in dispersion processes generated by third order regularization,

$${u_t} + f{(u)_x} = \delta {u_{xxx}}.$$

One may of course introduce a more general operator on the right hand side depending upon several parameters and entertain the corresponding process.