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Jiqiang Zheng

Publications and source records attributed to Jiqiang Zheng.

At least 37 records · Page 2Linked to original sources
arXiv2025

Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schrödinger system

In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schrödinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schrödinger system, we present a complete classification of minimal-mass blowup solutions. Specifically, we demonstrate that all such solutions must be either solitons or their pseudo-conformal transformations. To prove this result, we develop a modulation analysis that accounts for multi-component interactions to overcome the multiply phase transformations caused by the multi-component. A long time Strichartz estimate for vector-valued solutions is established to solve the difficulty posed by the Galilean transformations and spatial translation, where a new vector-valued bilinear estimate is proven to address the challenges caused by the coupled nonlinear interaction. For the infinitely coupled focusing nonlinear Schrödinger system when the mass is equal or slightly above the scattering threshold in \cite{CGHY}, we show that scattering is the only dynamical behavior of the solutions to the infinitely coupled system.

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arXiv2025

$L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field

In this paper, we study the $L^{p}$-estimates for the solution to the $2\mathrm{D}$-wave equation with a scaling-critical magnetic potential. Inspired by the work of \cite{FZZ}, we show that the operators $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{2})$ for $1 |1/p-1/2|$ and $t>0$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator. In particular, we derive the $L^{p}$-bounds for the sine wave propagator $\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\mathcal{L}^{-\frac12}_{\mathbf{A}}$. The key ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.

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arXiv2025

Decoupling for finite type phases in higher dimensions

In this paper, we establish an $\ell^2$ decoupling inequality for the hypersurface \[\Big\{(ξ_1,...,ξ_{n-1},ξ_1^m+...+ξ_{n-1}^m): (ξ_1,...,ξ_{n-1}) \in [0,1]^{n-1}\Big\}\]associated with the decomposition adapted to hypersufaces of finite type, where $n\geq 2$ and $m\geq 4$ is an even number. The key ingredients of the proof include an $\ell^2$ decoupling inequality for the hypersurfaces \[\Big\{(ξ_1,...,ξ_{n-1},ϕ_1(ξ_1)+...+ϕ_s(ξ_s)+ξ_{s+1}^m+...+ξ_{n-1}^m): (ξ_1,...,ξ_{n-1}) \in [0,1]^{n-1}\Big\},\] $0 \leq s \leq n-1$, with $ϕ_1,...,ϕ_s$ being $m$-nondegenerate.

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arXiv2025

Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation $iu_t + Δ_Ωu = |u|^αu$ in the exterior domain $Ω$ of a smooth, compact and strictly convex obstacle in $\mathbb{R}^3$. It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^3))$ with $s_c := \frac{3}{2} - \frac{2}α \in (0, \frac{3}{2})$, then $u$ is global and scatters. In this paper, assuming that this conjecture holds, we prove that if $u$ is a solution to the nonlinear Schrödinger equation in exterior domain $Ω$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(Ω))$ with $s_c \in \left[\frac{1}{2}, \frac{3}{2}\right)$, then $u$ is global and scatters. The proof of the main results relies on the concentration-compactness/rigidity argument of Kenig and Merle [Invent. Math. {\bf 166} (2006)]. The main difficulty is to construct minimal counterexamples when the scaling and translation invariance breakdown on $Ω$. To achieve this, two key ingredients are required. First, we adopt the approach of Killip, Visan, and Zhang [Amer. J. Math. {\bf 138} (2016)] to derive the linear profile decomposition for the linear propagator $e^{itΔ_Ω}$ in $\dot{H}^{s_c}(Ω)$. The second ingredient is the embedding of the nonlinear profiles. More precisely, we need to demonstrate that nonlinear solutions in the limiting geometries, which exhibit global spacetime bounds, can be embedded back into $Ω$. Finally, to rule out the minimal counterexamples, we will establish long-time Strichartz estimates for the exterior domain NLS, along with spatially localized and frequency-localized Morawetz estimates.

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arXiv2024

Intertwining operators beyond the Stark Effect

The main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion in Fanelli, Felli, Fontelos and Primo [Comm. Math. Phys., 324(2013), 1033-1067; 337(2015), 1515-1533]. Recently, Miao, Su, and Zheng introduced in [Tran. Amer. Math. Soc., 376(2023), 1739--1797] a family of spectrally projected intertwining operators, reminiscent of the Kato's wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in $L^p$. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher. In addition, we investigate the mapping properties between $L^p$-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and $L^p$-bounds of Bochner--Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions.

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arXiv2024

Global well-posedness and scattering for mass-critical inhomogeneous NLS when $d\ge3$

We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with $d\ge3,0 0$) are that the presence of the inhomogeneity $|x|^{-b}$ creates a nontrivial singularity at the origin, and breaks the translation symmetry as well as the Galilean invariance of the equation, which makes the establishment of the profile decomposition and long time Strichartz estimates more difficult. To overcome these difficulties, we perform the concentration compactness/rigidity methods of [Kenig and Merle, Invent. Math. (2006)] in the Lorentz space framework, and reduces the problem to the exclusion of almost periodic solutions. The exclusion of these solutions will utilize fractional estimates and long time Strichartz estimates in Lorentz spaces. In our study, we obseve that the decay of the inhomogeneity $|x|^{-b}$ at infinity prevents the concentration of the almost periodic solution at infinity in either physical or frequency space. Therefore, we can use classical Morawetz estimates, rather than interaction Morawetz estimates, to exclude the existence of the quasi-soliton.

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arXiv2024

On pointwise convergence of multilinear Bochner-Riesz means

We improve the range of indices when the multilinear Bochner-Riesz means converges pointwisely. We obtain this result by establishing the $L^p$ estimates and weighted estimates of $k$-linear maximal Bochner-Riesz operators inductively, which is new when $p<2/k$ in higher dimensions. To prove these estimates, we make use of a variant of Stein's square function and its multilinear generalization.

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arXiv2024

Global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains

In this article, we consider an energy-critical complex Ginzburg-Landau equation in the exterior of a smooth compact strictly convex obstacle. We prove the global well-posedness of energy-critical complex Ginzburg-Landau equation in an exterior domain by the concentration-compactness/rigidity theorem method. As corollaries of our main result, we establish both the existence of global weak solutions to energy-critical defocusing nonlinear Schrödinger equations and the global well-posedness theory for energy-critical semi-linear heat equation in exterior domains.

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arXiv2024

Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

In this paper, we study Strichartz estimates for the Schrödinger equation on a metric cone $X$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. For the metric $g$ on $X$ given by $g=dr^2+r^2h$, let $Δ_g$ be the positive Friedrichs extension Laplacian on $X$ and $V=V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ is a real function such that the operator $P:=Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. We establish the full range of global-in-time Strichartz estimates without loss for the Schrödinger equation associated with the operator $\LL_V=Δ_g+V_0 r^{-2}$ including the endpoint estimate both in homogeneous and inhomogeneous cases. A new finding reveals that the range of admissible pairs at $\dot H^s$-level is influenced by the smallest eigenvalue of the operator $P$. This additionally proves the conjecture in Wang [Ann. Inst. Fourier 2006] and generalizes the results of Ford [Comm. Math. Phys. 2010] and Baskin-Marzuola-Wunsch [Contemp. Math. 2014]. As an application, we show the well-posedness theory and scattering theory for the Schrödinger equation with a cubic nonlinearity on this setting which verifies a conjecture in Baskin-Marzuola-Wunsch [Contemp. Math. 2014].

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arXiv2024

On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement

In this article, we study the scattering theory for the two dimensional defocusing quintic nonlinear Schrödinger equation(NLS) with partial harmonic oscillator which is given by \begin{align}\label{NLS-abstract} \begin{cases}\tag{PHNLS} i\partial_tu+(\partial_{x_1}^2+\partial_{x_2}^2)u-x_2^2u=|u|^4u,&(t,x_1,x_2)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R},\\ u(0,x_1,x_2)=u_0(x_1,x_2). \end{cases} \end{align} First, we establish the linear profile decomposition for the Schrödinger operator $e^{it(\partial_{x_1}^2+\partial_{x_2}^2-x_2^2)}$ by utilizing the classical linear profile decomposition associated with the Schrödinger equation in $L^2(\mathbb{R})$. Then, applying the normal form technique, we approximate the nonlinear profiles using solutions of the new-type quintic dispersive continuous resonant (DCR) system. This allows us to employ the concentration-compactness/rigidity argument introduced by Kenig and Merle in our setting and prove scattering for equation (PHNLS) in the weighted Sobolev space. The second part of this paper is dedicated to proving the scattering theory for this mass-critical (DCR) system. Inspired by Dodson's seminal work [B. Dodson, Amer. J. Math. 138 (2016), 531-569], we develop long-time Strichartz estimates associated with the spectral projection operator $Π_n$, along with low-frequency localized Morawetz estimates, to address the challenges posed by the Galilean transformation and spatial translation.

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arXiv2024

Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds

In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) $ i\partial_tu+Δ_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$.

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arXiv2024

Dynamics of the combined nonlinear Schrödinger equation with inverse-square potential

We consider the long-time dynamics of focusing energy-critical Schrödinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$ \begin{equation}\label{NLS-ab} \begin{cases} i\partial_tu-\mathcal{L}_au=-|u|^{\frac{4}{d-2}}u+|u|^{\frac{4}{d-1}}u, \quad (t,x)\in\mathbb{R}\times\mathbb{R}^d,\tag{CNLS$_a$},\\ u(0,x)=u_0(x)\in H^1_a(\mathbb{R}^d), \end{cases} \end{equation} where $\mathcal{L}_a=-Δ+a|x|^{-2}$ and the energy is below and equal to the threshold $m_a$, which is given by the ground state $W_a$ satisfying $\mathcal{L}_aW_a=|W_a|^{\frac{4}{d-2}}W_a$. When the energy is below the threshold, we utilize the concentration-compactness argument as well as the variatonal analysis to characterize the scattering and blow-up region. When the energy is equal to the threshold, we use the modulation analysis associated to the equation \eqref{NLS-ab} to classify the dynamics of $H_a^1$-solution. In both regimes of scattering results, we do not need the radial assumption in $d=4,5$. Our result generalizes the scattering results of [31-33] and [3] in the setting of standard combined NLS.

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arXiv2024

Remark on the scattering theory of the nonlinear Schrödinger equation on the cylinders

In this article, we consider the nonlinear Schrödinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schrödinger equation on $\mathbb{R}^d \times \mathbb{T}$. In the short range case, we show the decay and scattering of solutions of the nonlinear Schrödinger equation on $\mathbb{R}^d \times \mathbb{T}$ for small data.

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arXiv2024

On Strichartz estimates for many-body Schrödinger equation in the periodic setting

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori $\mathbb{T}^d$, where $d\geq 3$. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in Bourgain-Demeter \cite{BD}. As a comparison, this result can be regarded as a periodic analogue of Hong \cite{hong2017strichartz} though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.

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arXiv2023

Restriction estimates for one class of hypersurfaces with vanishing curvature in $\mathbb{R}^n$

In this paper, we study the restriction problem for one class of hypersurfaces with vanishing curvature in $\mathbb{R}^n$ with $n$ being odd. We obtain an $L^2-L^p$ restriction estimate, which is optimal except at the endpoint. Furthermore, we establish an $L^s-L^p$ restriction estimate for these hypersurfaces, which is achieved by improving the known $L^{\infty}$ restriction estimate for hypersurfaces with $\frac{n-1}{2}$ positive principal curvatures and $\frac{n-1}{2}$ negative principal curvatures.

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arXiv2023

Threshold dynamics for the 3$d$ radial NLS with combined nonlinearity

We consider the nonlinear Schrödinger equation with focusing quintic and defocusing cubic nonlinearity in three space dimensions: \[ (i\partial_t+Δ)u = |u|^2 u - |u|^4 u. \] In [18, 23], the authors classified the dynamics of solutions under the energy constraint $E(u)< E^c(W)$, where $W$ is the quintic NLS ground state and $E^c$ is the quintic NLS energy. In this work we classify the dynamics of $H^1$ solutions at the threshold $E(u)=E^c(W)$.

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