Related papers: Inhomogeneous Oscillatory Integrals and Global Smo…
In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier…
We study the scattering for the energy-subcritical stochastic nonlinear Schr\"odinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise…
We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{it(-\Delta)^\frac m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence…
In mathematical physics it is of interest to study Schr\"odinger equations with friction and possessing an invariant measure. The focus of this paper is the Cauchy problem for the Schr\"odinger equation $\p_t f - i \mathscr L f = 0$, where…
We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where…
In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is…
Finite size effects in the equilibrium phase space density distribution function are taken into account for alculations of the relaxation of collective motion in finite nuclei. Memory effects in the collision integral and the diffusivity…
We show a smoothing effect of near full derivative for low-regularity global-in-time solutions of the periodic Korteweg-de Vries (KdV) equation. The smoothing is given by slightly shifting the space-time Fourier support of the nonlinear…
We derive damping estimates and asymptotics of $L^p$ operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into $L^2$ almost orthogonality technique…
In this manuscript, we investigate the oscillatory behaviour of the anisotropy in the diagonal Bianchi-I spacetimes. Our starting point is a simplification of Einstein's equations using only observable or physical variables. As a…
We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical…
Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized $\Phi$-Laplacian type term. We work in the Carath\'eodory setting.…
Let $\Delta_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{it\Delta_{\mathbb{T}^m\times…
In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves,…
A numerical study of fractional Camassa-Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localised initial data from the…
We consider the Cauchy problem with smooth and compactly supported initial data for the wave equation in a general class of spherically symmetric geometries which are globally smooth and asymptotically flat. Under certain mild conditions on…
We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach…
Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how…
Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a $\si$-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(\mu)$. By using an It\^o type inequality for the $H$-norm…
In this paper, we prove global in time Strichartz estimates for the fractional Schr\"odinger operators, namely $e^{-it\Lambda_g^\sigma}$ with $\sigma \in (0,\infty)\backslash \{1\}$ and $\Lambda_g:=\sqrt{-\Delta_g}$ where $\Delta_g$ is the…