Matthias Hieber

Bounded Analyticity of the Stokes Semigroup on Spaces of Bounded Functions

Let (Omega subset mathbb{R}^{n}), n ≥ 3, be an exterior domain with smooth boundary. It is shown that the Stokes semigroup on (L_{sigma }^{infty }(Omega )) is a bounded analytic semigroup on this space.

Global strong Lp well-posedness of the 3D primitive equations with heat and salinity diffusion

Abstract Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of H 2 / p , p , 1 p ∞ , satisfying certain boundary conditions. In particular, global well-posedness of the full primitive equations is obtained for initial data having less differentiability properties than H 1 , hereby generalizing a result by Cao and Titi (2007) [5] to the case of non-smooth data. In addition, it is shown that the solutions are exponentially decaying provided the outer forces possess this property.

Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case

The Ericksen–Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress tensor is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally strongly well-posed in the

Strong solutions for two-phase free boundary problems for a class of non-Newtonian fluids

L_p

Stability results for fluids of Oldroyd-B type on exterior domains

Lp-setting, and a dynamical theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

Dynamics of nematic liquid crystal flows: The quasilinear approach

Consider the equations of Navier–Stokes in (mathbb{R}^3) in the rotational setting, i.e., with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided the initial data is small with respect to the norm of the Fourier–Besov space (FB^{{2-3}/p}_{p,r}(mathbb{R}^3)), where (p; in ; (1,infty]; mathrm{and}; r;in [1,infty]). In the two-dimensional setting, a unique, global mild solution to this set of equations exists for non-small initial data (u_{0} ; in ;L^p_{sigma}(mathbb{R}^2);mathrm{for};p ;in;[2,infty).)

A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

Consider the two-phase free boundary problem subject to surface tension and gravitational forces for a class of non-Newtonian fluids with stress tensors Tn of the form