Comparision of the definitions of generalized solution of the Cauchy problem for quasi-linear equation

TL;DR

This paper compares various classical and modern definitions of generalized solutions for the 1D scalar quasilinear conservation law Cauchy problem, analyzing conditions under which these definitions are equivalent.

Contribution

It systematically examines and contrasts different approaches to defining generalized solutions, clarifying their equivalence conditions.

Findings

Classical and modern definitions can be equivalent under certain conditions.

Finite-difference approximation approaches align with classical definitions.

Conditions for equivalence help unify different solution concepts.

Abstract

In preprint we consider and compare different definitions of generalized solution of the Cauchy problem for 1d-scalar quasilinear equation (conservation law). We start from the classical approaches goes back to I.M. Gelfand, O.A. Oleinik, S.N. Kruzhkov and move to the modern finite-difference approximations approaches belongs to A.A. Shananin and G.M. Henkin. We discuss the conditions that provide definitions to be equivalent.