  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.
