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DCDS-S

Consider a domain $\Omega \subset \R^n$ with uniform $C^3$-boundary and assume
that the Helmholtz projection $P$ exists on $L^p(\Omega)$ for some $ 1 < p < \infty$.
Of concern are recent results on the Stokes operator in $L^p(\Omega)$ generating an analytic
semigroup on $L^p(\Omega)$ and admitting maximal $L^p$-$L^q$-regularity.

DCDS

Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on
open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper
discusses recent well-posedness and ill-posedness results for both situations.

DCDS-S

Consider the set of equations describing Oldroyd-B fluids with finite Weissenberg numbers
in exterior domains. In this note, we describe the main ideas of the proofs of two
recent results on global existence for this set of equations on exterior domains subject to
Dirichlet boundary conditions. The methods described here are quite different from the
techniques used in the Lagrangian approach which is often used in the case of
infinite Weissenberg numbers.

EECT

Consider the system of equations describing the motion of a rigid body immersed in a viscous,
compressible fluid within the barotropic regime. It is shown that this system admits a unique, local strong
solution within the $L^p$-setting.

CPAA

Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of
equations admits a unique, global mild solution provided only the horizontal components of the initial
data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in
[1,\infty)$.

CPAA

It is proved the existence of a unique, global strong solution to the two-dimensional
Navier-Stokes initial-value problem in exterior domains in the case where the
velocity field tends, at large spatial distance, to a prescribed velocity field that is
allowed to grow linearly.

DCDS-S

In this paper we prove $L^\infty$-a priori estimates for parabolic evolution
equations in non-divergence form on all of $\R^n$ for bounded coefficients having only
vanishing mean oscillation, thus allowing in particular non continuous coefficients.

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