Point process with last-arrival-time dependent intensity and 1-dimensional incompressible fluid system with evaporation

TL;DR

This paper models an infinite-component incompressible fluid system with evaporation, using a non-Markovian point process to prove unique solutions for a complex PDE system with boundary conditions.

Contribution

It introduces a novel approach combining PDE analysis with a non-Markovian point process to handle last-arrival-time dependent intensities in fluid dynamics.

Findings

Proved unique existence of solutions for the PDE system.

Developed a representation using a non-Markovian point process.

Established boundary conservation conditions.

Abstract

We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.