Japanese - German International Workshop on Mathematical Fluid Dynamics Lecture Notes on Stokes and Navier - Stokes initial

In the ambit of the mini course, to which I have been kindly invited to participate, I would like to develop some results concerning the Stokes and Navier-Stokes initial boundary value problems with nondecaying data. To better explain the aims, we split the case of the Stokes problem from the one of the Navier-Stokes. The Stokes initial boundary value problem with nondecaying data is essentially meant as the well posedeness of the IBVP in exterior domain with an initial data in L∞. The question concerns the existence and the properties of a possible analytic semigroup in L∞. Usually, in literature this topic is known as the Maximum Modulus theorem. The problems related to the topic, which is one of the classical in partial differential equations, have been open for long time for the equations of the hydrodynamics. Just in the last ten years, especially for a bounded domain, the problem has had contributes of some authors: [Solonnikov (2002)1]-[Solonnikov (2006)1], [Abe & Giga (2011), Abe & Giga (2012)]). I will discuss the case of the initial boundary value problem in exterior domains. I would like to point out that the result has to be read in the following sense. Given any maximum modulus theorem for solutions to the Stokes IBVP in bounded domains, then, the theorem proves a maximum modulus theorem for the solutions of the Stokes IBVP in exterior domains with an initial data just belonging to L∞(Ω). For the Navier-Stokes initial boundary value problem with non decaying data also it is meant the IBVP in exterior domains with a data in L∞. I start saying that this topic had contributes concerning the uniqueness several years ago, but it has had a systematic study of the well posedeness essentially in the last 15