The initial value problem for motion of incompressible viscous and heat-conductive fluids in Banach spaces

TL;DR

This paper investigates the initial value problem for incompressible viscous and heat-conductive fluids in Banach spaces, establishing existence and uniqueness of solutions using analytic semigroup theory, without neglecting viscous dissipation.

Contribution

It extends the analysis of fluid motion equations in Banach spaces by including viscous dissipation, providing local and global solution results under various initial data conditions.

Findings

Existence of a unique mild solution locally in time.

Global solutions for small initial data.

Solutions can be strong or classical under certain conditions.

Abstract

We consider the abstract initial value problem for the system of evolution equations which describe motion of incompressible viscous and heat-conductive fluids in a bounded domain. It is difficulty of our problem that we do not neglect the viscous dissipation function in contrast to the Boussinesq approximation. This problem has uniquely a mild solution locally in time for general initial data, and globally in time for small initial data. Moreover, a mild solution of this problem can be a strong or classical solution under appropriate assumptions for initial data. We prove the above properties by the theory of analytic semigroups on Banach spaces.