Related papers: Inhomogeneous Oscillatory Integrals and Global Smo…
We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(\theta, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type…
We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact…
This paper concerns the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lam\'{e} operator…
We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$ {\mathcal I}(t) = \int_{\mathbb R}({\rm e}^{it\phi(x)}-1) {\rm d} \mu(x) $$ where $\mu$ is a probability measure on…
A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
We consider the anharmonic oscillator with an arbitrary-degree anharmonicity, a damping term and a forcing term, all coefficients being time-dependent: u" + g_1(x) u' + g_2(x) u + g_3(x) u^n + g_4(x) = 0, n real. Its physical applications…
We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…
We present a detailed study of the spherically symmetric solutions in Lorentz breaking massive gravity. There is an undetermined function $\mathcal{F}(X, w_1, w_2, w_3)$ in the action of St\"{u}ckelberg fields…
A basic version of the P\'olya-Szeg\H{o} inequality states that if $\Phi$ is a Young function, the $\Phi$-Dirichlet energy -- the integral of $\Phi(\|\nabla f\|)$ -- of a suitable function $f\in \mathcal{V}(\mathbb{R}^n)$, the class of…
We introduce a fairly general dispersive-dissipative nonlinear equation, which is characterized by fractional Laplacian operators in both the dispersive and dissipative terms. This equation includes some physically relevant models of fluid…
We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDEs) driven by space-time noise, for multiplicative and additive noise. We examine convergence of…
In this note we generate two local median oscillation decompositions of an arbitrary measurable function and discuss some applications to Calder\'{o}n-Zygmund singular integral operators $T$. These applications rely on the inequality…
In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs)…
We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +\Psi(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\:…
In this paper, we shall prove the uniform sharp $L^p$ decay estimates for a class of oscillatory integral operators with polynomial phases. By this one-dimensional result, we can use the rotation method to obtain uniform sharp $L^p$…
In this work, we are concerned with the regularities of the solutions to Boltzmann equation with the physical collision kernels for the full range of intermolecular repulsive potentials, $r^{-(p-1)}$ with $p>2$. We give the new and…
In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin…
Using the Ricci and scalar curvatures of the configuration manifold of gravitational N-body systems, we study the exponential instability in their trajectories. It is found that the exponentiation time-scale for isotropic Plummer spheres…
We relate the $L_p$-variation, $2\le p < \infty$, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness…