Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning “best constants” for smoothing estimates. When combined with suitable “comparison principles” (analogous to those established in [39]), they yield smoothing estimates for classes of functions of the operators.

An important contribution of this study involves the comparison of the spectral density of an operator $H$ and its perturbation $H+V.$ It entails a derivation of global spacetime estimates for $H+V$ on the basis of analogous estimates for $H.$

A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrödinger operators with potentials.

We consider the $3D$ primitive equations and show, that one does need less than horizontal viscosity to obtain a well-posedness result in Sobolev spaces. Furthermore, we will also investigate the primitive equations with horizontal viscosity and show that these equations are well-posed without imposing any boundary condition for the horizontal velocity components on the vertical boundary.

This paper is concerned with the following planar Schrödinger-Poisson system with zero mass \begin{equation*} \begin{cases} -\Delta u+\lambda \phi u=f(x,u), \;\; & x\in {\mathbb R}^{2},\\ \Delta \phi=2\pi u^2, \;\; & x\in {\mathbb R}^{2}, \end{cases} \end{equation*} where $\lambda > 0$ and $f\in \mathcal{C}(\mathbb R^2\times\mathbb R, \mathbb R)$ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. By using some new analytical approaches, we prove that the above system has axially symmetric solutions under weak assumptions on $\lambda$ and $f$. This seems the first result on the planar Schrödinger-Poisson system with zero mass.

We prove a gradient bound for solutions of a large class of oblique derivative problems using the maximum principle. Although many of the results in this work are already known, the advantage to the current approach is that it covers a larger variety of problems than previously considered by any single method. In particular, we can study the capillary equation and the false mean curvature equation with a wide range of boundary conditions.