Smooth and discontinuous junctions in the p-system

TL;DR

This paper proves the existence of a Lipschitz semigroup solution for the p-system modeling subsonic fluid flow in a pipe with variable section, highlighting the importance of bounds on the oscillation of the pipe's cross-section.

Contribution

It establishes conditions under which the p-system's Cauchy problem admits a Lipschitz semigroup, including explicit bounds on the pipe's section oscillation.

Findings

Lipschitz semigroup exists under small initial total variation and oscillation of a.

Explicit bounds on the oscillation of a are provided, depending on fluid speed.

An example demonstrates the necessity of the bounds to prevent unbounded growth of total variation.

Abstract

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a = a(x). We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV(a) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.