Title: on the Stokes Semigroup in Some Non-helmholtz Domains

Title: The fundamental solution of compressible and incompressible fluid flow past a rotating obstacle Abstract: We consider the flow of either an incompressible or a compressible fluid around or past a rotating rigid body in the whole space R 3. Using a global coordinate transform and a linearization the problem reduces to a linear PDE system in a time-independent domain which may admit stationary solutions. For the new system on R 3 – with a virtual obstacle – we are looking for the fundamental solution and its estimates in L q-space. In the incompressible case the fundamental solution has been found by T. Hishida, D. Müller and myself via Fourier transformation and estimated in L q-space for all q ∈ (1, ∞), see [1, 2]. Although in the compressible case the same strategy leads to an explicit solution in Fourier space, the corresponding multiplier function shows a very different behaviour: it is not smooth outside of the origin so that usual L q-estimates are not at all obvious. Actually, we need a restriction on q, see [3]. Abstract: It is well-known that the Stokes operator associated with the Stokes equations generates analytic semigroups in L p type space in various domains. Recent study shows that the Stokes operator not only generates an analytic semigroup but also enjoy maximal regularity property provided that domain admits the Helmholtz decomposition in L p. Unfortunately, there are domains that the Helmholtz decomposition fails. In this talk we would like to give an example that the Stokes operator generates an analytic semigroup in L p type for some two-dimensional domains where the L p Helmholtz decomposition is not allowed. Our method appeals to L ∞ theory, which is recently established in various domains including a bounded domain, an exterior domain and a perturbed half space. It turns out that our method applies to some non-Helmholtz domains as well as cylindrical domains. However, a recent study of Lorenz von Below shows that a three-or higher dimensional layer domain does not permit L ∞ theory. This is a joint work in progress with