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Jiqiang Zheng

Publications and source records attributed to Jiqiang Zheng.

At least 55 records · Page 3Linked to original sources
arXiv2023

Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential

In this paper, we study the $2$D cubic nonlinear Schrödinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the partial harmonic oscillator. Then, we give the polynomial bound of the Sobolev norms for the solutions using the method of the Planchon, Tzvetkov, and Visciglia.

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arXiv2022

The $W^{s,p}$-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential $$\mathcal L_a=-Δ+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension $d\geq 2$. We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are $W^{s,p}$-bounded for certain $p$ and $s$ which depend on $a$. As corollaries, we solve some open problems associated with the operator $\mathcal L_a$, which include the dispersive estimates and the local smoothing estimates in dimension $d\geq 2$. We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

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arXiv2021

Dispersive estimates for two-particles Schrödinger and wave equations in the Aharonov-Bohm field

We study the dispersive behaviors of two-particles Schrödinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schrödinger and wave equations in this setting. The key point is to construct spectral measure of Schrödinger operator with an Aharonov-Bohm type potential in $\R^4$. As applications, we finally prove a scattering theory for the nonlinear defocusing subcritical two-particles Schrödinger equation with Aharonov-Bohm potential.

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arXiv2021

Uniform resolvent estimates for critical magnetic Schrödinger operators in 2D

We study the $L^p-L^q$-type uniform resolvent estimates for 2D-Schrödinger operators in scaling-critical magnetic fields, involving the Aharonov-Bohm model as a main example. As an application, we prove localization estimates for the eigenvalue of some non self-adjoint zero-order perturbations of the magnetic Hamiltonian.

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arXiv2021

$\ell^2$ decoupling for certain surfaces of finite type in $\mathbb{R}^3$

In this article, we establish an $\ell^2$ decoupling inequality for the surface $$F_4^2:=\Big\{(ξ_1,ξ_2,ξ_1^4+ξ_2^4): (ξ_1,ξ_2) \in [0,1]^2\Big\}$$ associated with the decomposition adapted to finite type geometry from our previous work. The key ingredients of the proof include the so-called generalized rescaling technique, an $\ell^2$ decoupling inequality for the surfaces $$\Big\{(ξ_1,ξ_2,ϕ_1(ξ_1)+ξ_2^4): (ξ_1,ξ_2) \in [0,1]^2\Big\}$$ with $ϕ_1$ being non-degenerate, reduction of dimension arguments and induction on scales.

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arXiv2021

Scattering theory For Quadratic Nonlinear Schrödinger System in dimension six

In this paper, we study the solutions to the energy-critical quadratic nonlinear Schrödinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${\rm u}_0$ is below or above that of the ground state.

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arXiv2021

The radial defocusing energy-supercritical NLS in dimension four

We consider the radial defocusing nonlinear Schrödinger equations $iu_t+Δu=|u|^{p}u$ with supercritical exponent $p>4$ in four space dimensions, and prove that any radial solution that remains bounded in the critical Sobolev space must be global and scatter.

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arXiv2021

Threshold scattering for the focusing NLS with a repulsive potential

We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.

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arXiv2021

Decay and Strichartz estimates in critical electromagnetic fields

We study the $L^1\to L^\infty$-decay estimates for dispersive equations in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the construction of Schwartz kernels of the spectral measure and heat propagator for the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and show that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the $L^p$-regularity property of wave propagation in the same setting.

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arXiv2020

Dispersive estimates for 2D-wave equations with critical potentials

We study the 2D-wave equation with a scaling-critical electromagnetic potential. This problem is doubly critical, because of the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In particular, the diamagnetic phenomenon allows to consider negative electric potential which can be singular in the same fashion as the inverse-square potential. We prove sharp time-decay estimates in the purely magnetic case, and Strichartz estimates for the complete model, involving a critical electromagnetic field.

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arXiv2020

Restriction estimates for certain surfaces of finite type in $\mathbb{R}^3$

In this paper, we study the restriction estimate for a certain surface of finite type in $\mathbb{R}^3$, and partially improves the results of Buschenhenke-Müller-Vargas. The key ingredients of the proof include the so called generalized rescaling technique based on a decomposition adapted to finite type geometry, a decoupling inequality and reduction of dimension arguments.

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arXiv2020

Restriction estimates in a conical singular space: wave equation

We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$, and consider the operator $\mathcal{L}_V=Δ_g+V$ with $V=V_0r^{-2}$, where $V_0(θ)\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $Δ_h+V_0+(n-2)^2/4$ is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with $\mathcal{L}_V$. The smallest positive eigenvalue of the operator $Δ_h+V_0+(n-2)^2/4$ plays an important role in the result. As an application, for independent of interests, we prove local energy estimates and Keel-Smith-Sogge estimates for the wave equation in this setting.

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arXiv2020

Long time dynamics for defocusing cubic NLS on three dimensional product space

In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.

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arXiv2019

Scattering for the non-radial inhomogeneous NLS

We extend the result of Farah and Guzmán on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.

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arXiv2019

On the dimension of divergence sets of Schrödinger equation with complex time

This article studies the pointwise convergence for the fractional Schrödinger operator $P^{t}_{a,γ}$ with complex time in one spatial dimension. Through establishing $L^2$-maximal estimates for initial datum in $H^{s}(\mathbb{R})$, we see that the solution converges to the initial data almost everywhere with $s>\frac14 a(1-\frac1γ)_+$ when $0 \frac{1}{2}(1-\frac{1}γ)_{+}$ when $a=1$. By constructing counterexamples, we show that this result is almost sharp up to the endpoint. These results extends the results of P. Sjölin, F. Soria and A. Baily. Second, we study the Hausdorff dimension of the set of the divergent points, by showing some $L^1$-maximal estimates with respect to general Borel measure. Our results reflect the interaction between dispersion effect and dissipation effect, arising from the fractional Schrödinger type operator $P^{t}_{a,γ}$ with the complex time.

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arXiv2019

Well-posedness for energy-critical nonlinear Schrödinger equation on waveguide manifold

In this article, we utilize the scale-invariant Strichartz estimate on waveguide which is developed recently by Barron \cite{Barron} based on Bourgain-Demeter $l^2$ decoupling method \cite{BD} to give a unified and simpler treatment of well-posedness results for energy critical nonlinear Schrödinger equation on waveguide when the whole dimension is three and four.

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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 $n\geq3$.

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