# On Continuity Equations in Space-time Domains

## TL;DR

This paper investigates continuity equations constrained within general space-time domains, analyzing solution stability, well-posedness, and the effects of vanishing viscosity approximations, especially in convex spatial domains.

## Contribution

It introduces a framework for continuity equations in space-time domains, studies their well-posedness, and explores the vanishing viscosity limit with boundary conditions.

## Key findings

- Solution stability depends on initial data decay at infinity.
- Vanishing viscosity limit matches the original solution in convex domains.
- Provides a new interpretation of the equations via viscosity approximation.

## Abstract

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.