# Pointwise wave behavior of the Navier-Stokes equations in half space

## TL;DR

This paper analyzes the pointwise behavior of solutions to the compressible Navier-Stokes equations in a half space, revealing wave structures involving heat kernels and diffusion interactions that influence decay rates.

## Contribution

It provides a detailed analysis of wave propagation and decay in the compressible Navier-Stokes equations with boundary effects, highlighting the complex wave interactions and their impact on solution behavior.

## Key findings

- Green's function exhibits heat kernel propagation with sound speed in opposite directions
- Reflected heat kernel due to boundary effects propagates with positive sound speed
- Solutions decay algebraically due to wave interactions

## Abstract

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.