arXiv subjects

Junyong Zhang

Publications and source records attributed to Junyong Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2019

Strichartz estimates and wave equation in a conic singular space

Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$ and let $Δ_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=Δ_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\cite[Remark 2.4]{wang} Wang's conjecture for wave equation. The range of the admissible pair is sharp and is influenced by the smallest eigenvalue of $Δ_h+V_0+(n-2)^2/4$. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension $n\geq3$.

math.AP↗

arXiv2019

Uniform resolvent estimates for Schrödinger operator with an inverse-square potential

We study the uniform resolvent estimates for Schrödinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-Δ+V(x)$ where $Δ$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(θ) r^{-2}$ where $r=|x|, θ=x/|x|$ and $V_0(θ)\in\mathcal{C}^1(\mathbb{S}^{n-1})$ is a real function such that the operator $-Δ_θ+V_0(θ)+(n-2)^2/4$ is a strictly positive operator on $L^2(\mathbb{S}^{n-1})$. We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator $\mathcal{L}_V$.

math.AP↗

arXiv2019

Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from \cite{CH2} and arguments borrowed from \cite{HZ, Zhang}. As an application, we prove the small data global existence for any power $p\in(1, 1+\frac{4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm|u|^p$ or $\pm|u|^{p-1}u$, which answers partially an open question raised in \cite{SSW}.

math.AP↗

arXiv2019

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} where $Δ= -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $λ$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic. We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} g(ω) + O(r^{-ε}) \Big), \text{ as } r \to \infty, \quad ε> 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.

math.AP↗

arXiv2019

Linear adjoint restriction estimates for paraboloid

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity.

math.AP↗

arXiv2018

Nonlinear Schrödinger equation with Coulomb potential

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i\partial_tu+Δu+\frac{K}{|x|}u=λ|u|^{p-1}u$ with $1 0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.

math.AP↗

arXiv2017

Global-in-time Strichartz estimates for Schrodinger on scattering manifolds

We study the global-in-time Strichartz estimates for the Schrödinger equation on a class of scattering manifolds $X^{\circ}$. Let $\mathcal{L}_V=Δ_g+V$ where $Δ_g$ is the Beltrami-Laplace operator on the scattering manifold and $V$ is a real potential function on this setting. We first extend the global-in-time Strichartz estimate in Hassell-Zhang \cite{HZ} on the requirement of $V(z)=O(\langle z\rangle^{-3})$ to $O(\langle z\rangle^{-2})$ and secondly generalize the result to the scattering manifold with a mild trapped set as well as Bouclet-Mizutani\cite{BM} but with a potential. We also obtain a global-in-time local smoothing estimate on this geometry setting.

math.AP↗

arXiv2016

Strichartz estimate and nonlinear Klein-Gordon on non-trapping scattering space

We study the nonlinear Klein-Gordon equation on a product space $M=\R\times X$ with metric $\tilde{g}=dt^2-g$ where $g$ is the scattering metic on $X$. We establish the global-in-time Strichartz estimate for Klein-Gordon equation without loss of derivative by using the microlocalized spectral measure of Laplacian on scattering manifold showed in \cite{HZ} and a Littlewood-Paley squarefunction estimate proved in \cite{Zhang}. We prove the global existence and scattering for a family of nonlinear Klein-Gordon equations for small initial data with minimum regularity on this setting.

math.AP↗

arXiv2015

Three-dimensional array foci of generalized Fibonacci photon sieves

We present a new kind of photon sieves on the basis of the generalized Fibonacci sequences. The required numbers and locations of axial foci can be designed by generalized Fibonacci photon sieves (GFiPS). Furthermore, the three-dimensional array foci can be controllable and adjustable by the optical path difference scaling factor (OPDSF) when the amplitude modulation is replaced with the phase modulation. Multi-focal technologies can be applied to nano-imaging, THZ, laser communications, direct laser writing, optical tweezers or atom trapping, etc.

physics.optics↗

arXiv2015

Scattering theory for the radial $\dot H^{1/2}$-critical wave Equation with a cubic convolution

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution $\partial_{t}^2u-Δu=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial solution $u$ with life-span $I$ obeys $(u,u_t)\in L_t^\infty(I;\dot H^{1/2}_x(\mathbb R^d)\times\dot H^{-1/2}_x(\mathbb R^d))$, then $u$ is global and scatters. By the strategy derived from concentration compactness, we show that the proof of the global well-posedness and scattering is reduced to disprove the existence of two scenarios: soliton-like solution and high to low frequency cascade. Making use of the No-waste Duhamel formula and double Duhamel trick, we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of [Dodson,Lawrie, Anal.PDE, 8(2015), 467-497]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction.

math.AP↗

arXiv2015

Generalized Fibonacci zone plates

We propose a family of zone plates which are produced by the generalized Fibonacci sequences and their axial focusing properties are analyzed in detail. Compared with traditional Fresnel zone plates, the generalized Fibonacci zone plates present two axial foci with equal intensity. Besides, we propose an approach to adjust the axial locations of the two foci by means of different optical path difference, and further give the deterministic ratio of the two focal distances which attributes to their own generalized Fibonacci sequences. The generalized Fibonacci zone plates may allow for new applications in micro and nanophotonics.

physics.optics↗

arXiv2015

Global-in-time Strichartz estimates on non-trapping asymptotically conic manifolds

We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results in earlier papers of Hassell-Tao-Wunsch and Mizutani which were local in time.

math.AP↗

arXiv2014

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the properties of the microlocalized spectral measure of Laplacian on this setting showed in Hassell-Zhang and a Littlewood-Paley squarefunction estimate. As applications, we prove the global existence and scattering for a family of nonlinear wave equations on this setting.

math.AP↗

arXiv2014

Scattering theory for nonlinear Schroedinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schroedinger equations with some critical rough potential of inverse square type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with the Lapalacian with inverse square potential. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schroedinger equation with inverse square potential in energy space.

math.AP↗

arXiv2014

The Defocusing Energy-Critical Wave Equation with a Cubic Convolution

In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-Δu+(|x|^{-4}\ast|u|^2)u=0$ in spatial dimension $d \geq 5$. The main difficulties are the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), which is a fundamental property to show the global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-Δu+|u|^\frac4{d-2}u=0$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. Then, the proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solutions and low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the potential energy concentration to preclude the above three enemies.

math.AP↗

arXiv2014

Linear restriction estimates for Schroedinger equation on metric cones

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schroedinger operator on metric cone $M$, where the metric cone $M$ is of the form $M=(0,\infty)_r\timesΣ$ with the cross section $Σ$ being a compact $(n-1)$-dimensional Riemannian manifold $(Σ,h)$ and the equipped metric is $g=\mathrm{d}r^2+r^2h$. Assuming the initial data possesses additional regularity in angular variable $θ\inΣ$, we show some linear restriction estimates for the solutions. As applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schroedinger equation on two-dimensional metric cones.

math.AP↗

arXiv2014

Maximal estimates for Schroedinger equation with inverse-square potential

In this paper, we consider the maximal estimates for the solution to an initial value problem of the linear Schroedinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schroedinger equation with initial data $u_0\in H^{s}(\R^n)$ for $s>1/2$ or radial initial data $u_0\in H^{s}(\R^n)$ for $s\geq1/4$ and the solution does not converge when $s<1/4$.

math.AP↗

arXiv2013

Strichartz estimates for wave equation with inverse square potential

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential $\partial_t^2 u-Δu+\frac{a}{|x|^2}u=\pm|u|^{p-1}u$ for power $p$ being in some regime when the initial data are radial. This result extends the well-posedness result in Planchon, Stalker, and Tahvildar-Zadeh.

math.AP↗