# Analytical smoothing effect of solution for the boussinesq equations

## TL;DR

This paper proves that solutions to the incompressible Boussinesq equations become analytic over time, demonstrating a smoothing effect similar to that observed in the Navier-Stokes equations, using Fourier analysis.

## Contribution

It establishes the analytical smoothing effect for the Boussinesq equations, showing they share properties with the Navier-Stokes equations in a periodic domain.

## Key findings

- Solutions become analytic for any positive time
- The smoothing effect is identical to that of Navier-Stokes equations
- Fourier method is used to prove analyticity

## Abstract

In this paper, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H 1-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equation admet exactly same smoothing effect properties of incompressible Navier-Stokes equations.