Initial-boundary value problem for 1D pressureless gas dynamics

TL;DR

This paper develops a method to solve the initial-boundary value problem for 1D pressureless gas dynamics, introducing measure-valued solutions and analyzing boundary behavior, a novel approach in this context.

Contribution

It presents the first explicit construction of solutions for the initial-boundary value problem in pressureless gas dynamics using generalized potentials.

Findings

Solutions satisfy an entropy condition

Mass is conserved and may accumulate at boundaries

Boundary behavior influences momentum conservation

Abstract

The paper considers the system of pressureless gas dynamics in one space dimension. The question of solvability of the initial-boundary value problem is addressed. Using the method of generalized potentials and characteristic triangles, extended to the boundary value case, an explicit way of constructing measure-valued solutions is presented. The prescription of boundary data is shown to depend on the behavior of the generalized potentials at the boundary. We show that the constructed solution satisfies an entropy condition and it conserves mass, whereby mass may accumulate at the boundary. Conservation of momentum again depends on the behavior of the generalized boundary potentials. There is a large amount of literature where the initial value problem for the pressureless gas dynamics model has been studied. To our knowledge, this paper is the first one which considers the initial-boundary value problem.