We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic...

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in $C^n$ whose differentials have one-dimensional family of resonances in the first $m$ eigenvalues, $m\le n$ (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.

We obtain some microlocal estimates of the resonant states associated to a resonance $z_0$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are $𝒪(\sqrt{|\mathrm{Im}{z}_0|/h}$$+h^{\infty }...$

A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_0)\ll 1$ and $(a/R)\ll 1$ (where $C_0$ is the velocity of the sound), we show that at the rapid time scale $R/C_0$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated to...

In this paper we consider an elliptic system at resonance and bifurcation type with zero Dirichlet condition. We use a Lyapunov-Schmidt approach and we will give applications to Biharmonic Equations.