Mild solutions to the 3D-Boussinesq system with weakened initial temperature

TL;DR

This paper develops local mild solutions for the 3D Boussinesq system with initial temperature in negative Sobolev spaces, demonstrating how the system's structure allows for weakened regularity assumptions.

Contribution

It introduces a method to construct solutions with less regular initial temperature by exploiting the coupled nature of the Boussinesq system.

Findings

Established local existence of mild solutions with negative Sobolev initial temperature

Demonstrated the influence of the coupled structure on regularity requirements

Extended the understanding of initial data regularity in fluid dynamics systems

Abstract

In this research, the Cauchy problem of the 3D viscous Boussinesq system is studied considering an initial temperature with negative Sobolev regularity. Precisely, we construct local in time mild solutions to this system where the temperature term belongs to Sobolev spaces of negative order. Our main contribution is to show how the coupled structure of the Boussinesq system allows us to considerably weaken the regularity in the temperature term.