arXiv subjects
Publications and source records attributed to Jiqiang Zheng.
In this paper, We obtain dispersive estimates for solutions to the Schrödinger equation with a uniform magnetic field on the hyperbolic plane \(\mathbb{H}\). The key ingredient is an explicit representation formula for the kernel of the corresponding Schrödinger propagator. As a consequence, we prove the corresponding Strichartz estimates for all admissible pairs on \(\mathbb{H}\).
We consider the three-dimensional defocusing cubic nonlinear Schrödinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schrödinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.
In this paper, we study the stabilization property and large time controllability for a class of fourth-order Schrödinger equations on a compact manifold without boundary in dimensions $1\leq d\leq5$: \begin{align} i\partial_tu+(Δ_g^2-βΔ_g)u=-|u|^{2k}u, \,\,x\in M\tag{4NLS}\label{4NLS1} \end{align} where $k\in\mathbb{N}$ and $β\in\mathbb{R}_{+}$ when $1\leq d\leq 4$ but $β\in\mathbb{Q}_{+}$ for $d=5$. We adapt the strategy in Macia [Vietnam J. Math. (2021)] to establish observability and the propagation of singularities. Moreover, we use these propagation estimates to deduce the unique continuation property for $\eqref{4NLS1}$. By the classical Hilbert Uniqueness Method (HUM) and the Picard iteration, the stabilization and large time controllability hold under the Geometric Control Condition (GCC) and the Unique Continuation Property (UCP) for the linearized equation. To obtain the controllability and stabilization at the $H^2$ level with $d=5$, we will focus on $M=\Bbb S^5$ with $k=1$. Our results extend those of Laurent [SIAM J. Math. Anal. (2009)] and Capistrano Filho-Pampu [Math. Z., 2022].
In this article, we investigate the local well-posedness of the nonlinear Schrödinger equation on the two-dimensional sphere $\mathbb{S}^2$: \begin{align*} i\partial_tu+Δ_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to be gauge-invariant. More presicely, there exists a function $V\in C^\infty(\mathbb{C},\mathbb{R})$ such that $F=\frac{\partial V}{\partial \bar{z}}$. Moreover, $V(z)$ obeys \begin{equation}\label{H-11} V(e^{iθ}z)=V(z),\,\,θ\in\Bbb R,\,\,z\in\Bbb C, |\partial_z^{k_1}\partial_{\bar{z}}^{k_2}V(z)|\leq C_{k_1,k_2}(1+|z|)^{1+α-k_1-k_2},\tag{H-1} \end{equation} for some $α\geq3.$ The main contribution of this paper is the new lower bound of threshold of local well-posedness $s_c(\mathbb{S}^2,α)$. Specifically, under assumption \eqref{H-11}, we prove that for $α\geq 3$, the equation is ill-posed in $H^s(\mathbb{S}^2)$ with $s < 1 - \frac{2}{α-1}$ in the sense that the norm inflation occurs. Combined with the well-posedness in Yang [Sci. China Math. 58 (2015), 1023-1046], the exact threshold $s_c(\mathbb{S}^2,α)$ for $α\geq5$ is $1-\frac{2}{α-1}$, which matches the scaling-critical regularity as the Euclidean setting. Moreover, for $α\in [3, \frac{11}{3})$, we show that the solution map is not uniformly continuous in the range $0 < s < \frac14$ for the power-type nonlinearity $F(u)=|u|^{α-1}u$, which lies strictly above the scaling-invariant threshold. This provides a new characterization of the ill-posedness regime for all $α\geq 3$, extending an earlier result of Burq-Gérard-Tzvetkov [Math. Res. Lett. 9 (2002), 323-335]. Our result can also be regarded as a Schrödinger counterpart of Xia [Int. Math. Res. Not. (2021), 15533-15554].
In this article, we show that the solution to defocusing cubic nonlinear Schrödinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+Δ_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in $H^s(\R\times\T)$ with $s>\frac{1}{2}$. The proof is based on the $I$-method. Inspired by Colliander-Keel-Staffilani-Takaoka-Tao [Discrete Contin. Dyn. Syst. 21 (2008), 665-686], we construct the modified energy to improve the energy increment. The main difficulty lies in controlling the resonant interactions caused by the modified energy. To this end, we establish refined bilinear Strichartz estimates with angular truncation on the rescaled waveguide, thereby generalizing results previously obtained by Takaoka [J. Differ. Equa. 394 (2024), 296-319]. Furthermore, we demonstrate polynomial growth of $H^s$ with $\frac{1}{2} < s < 1$. Our result extends the recent work of Deng-Fan-Yang-Zhao-Zheng [J. Func. Anal. 287 (2024), 110595].
We consider the 4$d$ mass-energy double critical NLS \[ (i\partial_t+Δ)u = -|u|^2 u + |u| u. \] In Luo (2024) and Cheng--Miao--Zhao (2016), the authors established a scattering/blowup dichotomy for solutions satisfying the energy constraint $E(u_0)< E^c(W)$, where $W$ is the energy-critical NLS ground state and $E^c$ is the energy for the underlying cubic NLS. We prove that the scattering/blowup dichotomy persists even at the energy threshold $E(u_0)=E^c(W)$.
In this paper, we study Hardy-type uncertainty principles and unique continuation properties for linear covariant Schrodinger equations with variable coefficients in the presence of bounded electric and magnetic potentials. Under suitable smallness assumptions on the leading coefficients, we prove that any solution exhibiting super-quadratic exponential decay at two distinct times must vanish identically. Under an additional structural assumption on the coefficient matrix $G$, we further establish a Hardy-type result at the quadratic exponential scale. We also obtain an analogous uniqueness result for the heat equation with variable-coefficient magnetic perturbations. Our results unify and extend previous works in two directions: they recover the constant-coefficient covariant case treated by Barcelo-Fanelli-Gutierrez-Ruiz-Vilela when $G=I$, and the variable-coefficient non-magnetic case considered by Federico-Li-Yu when $A=0$. The proofs combine logarithmic convexity arguments with Carleman estimates adapted to variable-coefficient covariant Schrödinger and parabolic flows. Although our approach follows the general strategy introduced by Escauriaza-Kenig-Ponce-Vega, substantial new difficulties arise from the interaction between the variable metric and the magnetic structure, which requires new weight functions and refined commutator estimates.
In this paper, we study the almost everywhere convergence of the cubic nonlinear Schrödinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + Δ_g u = |u|^2u, \quad (t,x)\in\R\times §^2. \end{equation*} Inspired by the randomization method and the ansatz introduced by Burq, Camps, Sun, and Tzvetkov [Preprint, arXiv:2404.18229], we prove almost sure pointwise convergence almost everywhere for the nonlinear solution at very low regularity. This extends Compaan-Lucà-Staffilani [Int. Math. Res. Not. IMRN, (1) (2021), 596--647] to the spherical setting. We also provide a new necessary condition for the associated $L^p$ maximal estimate for the linear Schrödinger equation on $§^2$. More precisely, we show that the $L^p$ maximal estimate fails for $s<\frac{1}{2}-\frac{1}{2p}$ with $p\ge 2$. In the special case $p=3$, our result matches the corresponding range in the $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449].
In this paper, we establish a Paley-Wiener type uncertainty principle for Schrödinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+Δ_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where $Δ_A=(\nabla-iA)^2$ denotes the magnetic Schrödinger operator. Specifically, under suitable assumptions on $A$ and $V$, we show that if a solution $u$ exhibits linear exponential decay and support property in one spatial direction at times $t=0$ and $t=1$ respectively, then $u$ must vanish identically. This result extends the theorem of Kenig-Ponce-Vega [Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 539-557] to the case $A\neq0$. We overcome the difficulty brought by the magnetic potential which breaks the translation invariance in the leading term of Hamiltonian $H=Δ_A+V$. As a direct consequence, we also obtain a uniqueness result for a class of semi-linear Schrödinger equation with electromagnetic potentials.
We study the following focusing intercritical nonlinear Schrödinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+Δ_{z}u-y^2 u =- |u|^αu,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where $d \geq 1 $ is an integer and the exponent $α$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< α<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold.
We prove a sharp $L^p$-boundedness criterion for Bochner-Riesz multipliers on flat cones $X = (0,\infty) \times \mathbb{S}_σ^1$. The operator $S_λ^δ(Δ_X)$ is bounded on $L^p(X)$ for $1 \leq p \leq \infty$, $p \neq 2$, if and only if $δ> δ_c(p,2) = \max\left\{ 0, 2\left| 1/2 - 1/p \right| - 1/2 \right\}$. This result is also applicable to the infinite sector domain with Dirichlet or Neumann boundary, resolving the critical exponent problem in this wedge setting.
This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^δ_λ(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schrödinger operator $\mathcal{L}_{\mathbf{A}}$ on $\mathbb{R}^2$, where $δ\in (-3/2, 0)$. We determine the conditions on the exponents $p$ and $q$ under which the operator is bounded from $L^p(\mathbb{R}^2)$ to $L^q(\mathbb{R}^2)$. Our main result characterizes the boundedness region as a pentagonal subset $Δ(δ)$ of the $(1/p, 1/q)$-plane, extending previous uniform resolvent result in Fanelli, Zhang and Zheng[Int. Math. Res. Not., 20(2023), 17656-17703].
We study dynamics of the 4$d$ energy-critical nonlinear Schrödinger equation at the ground state energy. Previously, Duyckaerts and Merle [Geom. Funct. Anal. (2009)] proved that any radial solution with kinetic energy less than that of the ground state either scatters in both time directions or coincides (modulo symmetries) with a heteroclinic orbit, which scatters in one time direction and converges to the ground state in the other. We extend this result to the non-radial setting.
In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three.
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.
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 $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.
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.
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.