Abstract. We consider a coupled system of Navier-Stokes and NernstPlanck equations, describing the evolution of the velocity and the concentration fields of dissolved constituents in an electrolyte solution. Motivated by recent applications in the field of micro-and nanofluidics, we consider the model in such generality that electrokinetic flows are included. This prohibits employing the assumption of electroneutrality of the total solution, which is a common approach in the mathematical literature in order to determine the electrical potential. Therefore we complement the system of mass and momentum balances with a Poisson equation for the electrostatic potential, with the charge density stemming from the concentrations of the ionic species. For the resulting Navier-Stokes-Nernst-Planck-Poisson system we prove the existence of unique local strong solutions in bounded domains in R n for any n ≥ 2 as well as the existence of unique global strong solutions and exponential convergence to uniquely determined steady states in two dimensions.

We prove a maximal regularity result for operators corresponding to rotation invariant (in space) symbols which are inhomogeneous in space and time. Symbols of this type frequently arise in the treatment of half-space models for (free) boundary value problems. The result is obtained by extending the Newton polygon approach to variables living in complex sectors and combining it with abstract results on H ∞calculus and R-bounded operator families. As an application we derive maximal regularity for the linearized Stefan problem with Gibbs-Thomson correction.

We establish a global existence result for the rotating Navier-Stokes equations with nondecaying initial data in a critical space which includes a large class of almost periodic functions. The scaling invariant function space we introduce is given as the divergence of the space of 3 × 3 fields of Fourier transformed finite Radon measures. The smallness condition on initial data for global existence is explicitly given in terms of the Reynolds number. The condition is independent of the size of the angular velocity of rotation.

Abstract. We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a (unique) solution that is analytic in space and time.

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It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed half-space admits a bounded H ∞ -calculus on L q ( ) if q ∈ (1, ∞).

Abstract. Generalized Navier-Stokes equations which were proposed recently to describe active turbulence in living fluids are analyzed rigorously. Results on wellposedness and stability in the L 2 (R n )-setting are derived. Due to the presence of a Swift-Hohenberg term global wellposedness in a strong setting for arbitrary initial data in L 2 σ (R n ) is available. Based on the existence of global strong solutions, results on linear and nonlinear (in-) stability for the disordered steady state and the manifold of ordered polar steady states are derived, depending on the involved parameters.

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