# Past PDE CDT Lunchtime Seminar

We consider the Navier-Stokes initial boundary value problem (NS-IBVP) in a smooth exterior domain. We are interested in establishing existence of weak solutions (we mean weak solutions as synonym of solutions global in time) with an initial data in L(3,∞)

energy but involving a conservation statement for topological charge of the line defect field for its evolution will be shown to succeed. This is joint work with Chiqun Zhang, graduate student at CMU.

We study a simple one-dimensional coupled heat wave system, obtaining a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations and moreover extends to problems on higher-dimensional domains. Joint work with C.J.K. Batty (Oxford) and L. Paunonen (Tampere).

The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. If time permits I will sketch details of an equivalence theorem and a proof of smoothness properties of mild bounded ancient solutions in the half space, which is a joint work with Gregory Seregin

^{ }-sense and in the $L^{\infty}$ -sense) to be a halfplane, depending on $R$. This is a joint work with Joaquim Serra and Enrico Valdinoci.