Dispersive estimates for principally normal pseudodifferential operators

Smoothing effects of dispersive equations on real rank one symmetric spaces

In this article we prove time-global smoothing effects of dispersive pseudodifferential equations with constant coefficient radially symmetric symbols on real rank one symmetric spaces of noncompact type. We also discuss gain of regularities according to decay rates of initial values for the Schroedinger evolution equation. We introduce some isometric operators and reduce the arguments to the well-known Euclidean case. In our proof, Helgason's Fourier transform and the Radon transform as an elliptic Fourier integral operator play crucial roles.

Dispersive mixed-order systems in $L^p$-Sobolev spaces and application to the thermoelastic plate equation

We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

In this paper we consider linear, time dependent Schrödinger equations of the form ${\rm i} \partial_t ψ= K_0 ψ+ V(t) ψ$, where $K_0$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and $V(t)$ a time periodic potential. We give sufficient conditions on $V(t)$ ensuring that $K_0+V (t)$ generates unbounded orbits. The main condition is that the resonant average of $V(t)$, namely the average with respect to the flow of $K_0$, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations.…

无电阻电子磁流体方程在均匀磁场大扰动下的适定性

Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

本文证明了无电阻电子磁流体方程在均匀磁场大扰动下的局部适定性，通过简化重整化算子并引入新函数空间框架，显著放宽了初始数据条件。