Lebesgue spaces with variable exponent: some applications to the Navier-Stokes equations

TL;DR

This paper explores the use of Lebesgue spaces with variable exponents to analyze the 3D Navier-Stokes equations, addressing unique challenges and establishing results on the existence of mild solutions.

Contribution

It introduces methods to handle the peculiarities of variable exponent Lebesgue spaces in the context of Navier-Stokes equations, advancing the understanding of solutions in this framework.

Findings

Established existence of mild solutions in variable exponent Lebesgue spaces

Developed techniques to overcome analytical difficulties in this setting

Extended classical analysis tools to variable exponent spaces

Abstract

In this article we study some problems related to the incompressible 3D Navier-Stokes equations from the point of view of Lebesgue spaces of variable exponent. These functional spaces present some particularities that make them quite different from the usual Lebesgue spaces: indeed, some of the most classical tools in analysis are not available in this framework. We will give here some ideas to overcome some of the difficulties that arise in this context in order to obtain different results related to the existence of mild solutions for this evolution problem.