arXiv subjects

Chenmin Sun

Publications and source records attributed to Chenmin Sun.

At least 19 recordsLinked to original sources

arXiv2026

Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the $3d$ ball

We construct probabilistic strong solutions to the cubic Schr\"odinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain--Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng--Nahmod--Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

math.AP↗

arXiv2026

On the pointwise convergence of NLS flow on $ \S^2 $

In this paper, we study the almost everywhere convergence of the cubic nonlinear Schr\"odinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^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-Luc\`a-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 Schr\"odinger equation on $\S^2$. 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].

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arXiv2026

An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding

We give a certain $L^{\infty}(\mathbb{R}^2)$-estimate for the heat semigroup $\{e^{t\Delta}\}_{t \ge 0}$ that is closely related to the fact $H^1(\mathbb{R}^2) \not\subset L^{\infty}(\mathbb{R}^2)$, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallou\"et inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as $t \to 0^+$ is indeed optimal.

math.FA↗

arXiv2025

Almost sure global nonlinear smoothing for the 2D NLS

In this article, we prove an almost-sure global in time nonlinear smoothing effect for NLS on the two-dimensional torus. For deterministic data, this phenomenon was proved for the NLS on the circle by Erdo\u{g}an--Tzirakis, which remains unknown on multidimensional torus. Our argument is based on a quantitative quasi-invariance of Gaussian measures with covariance operator $(1-\Delta)^{-s}$ for $s>2$.

math.AP↗

arXiv2025

Gryphon: Scaling Hyperscale Multi-Tenant Gateways Beyond the Petabit-Era via DPU-Augmented Hierarchical Co-Offloading

At ByteDance, cloud gateway clusters orchestrate petabit-scale aggregate traffic. Traditional ASIC-only gateways fail to meet these escalating demands due to severe on-chip resource constraints and limited programmable flexibility, while pure software solutions or alternatives like disaggregated SmartNICs struggle to match terabit-scale line-rate throughput. To bridge this gap, we present Gryphon, a hyperscale cloud gateway built on a hybrid architecture that integrates DPUs directly into the switching ASIC's forwarding path. This design resolves the fundamental tension between capacity and speed, expanding table scale by up to 1000$\times$ and augmenting programmability, while sustaining 1.6 Tbps line-rate throughput at a cost of only ~8$\mu s$ in additional average latency. To manage this hardware heterogeneity, we introduce Hierarchical Co-Offloading (HLCO) in the data plane, achieving >99.9% fast path hit rate, while retaining software fallback for complex operations. In the control plane, we develop an abstraction layer (P4Bridge) that decouples hardware specifics from policy configuration. Gryphon has been operating at production scale for over a year, deployed on hundreds of nodes across multiple Availability Zones. We also share production measurements and operational experiences that serve as the first hyperscale-proven guidelines for next-generation DPU-augmented cloud gateways.

cs.NI↗

arXiv2025

Geometric condition for the observability of electromagnetic Schr\"odinger operators on $\mathbb{T}^2$

In this article we revisit the observability of the Schr\"odinger equation on the two-dimensional torus. In contrast to the Schr\"odinger operator with a purely electric potential, for which any non-empty open set guarantees observability, the presence of a magnetic potential introduces an additional obstruction. We establish a sufficient and almost necessary geometric condition for the observability of electromagnetic Schr\"odinger operators. This condition incorporates the magnetic potential, which can also be characterized by a geometric control condition for the corresponding magnetic field.

math.AP↗

arXiv2025

Hyperbolic nonlinear Schr\"odinger equations on $\mathbb{R}\times \mathbb{T}$

In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.

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arXiv2024

Local well-posedness for the periodic Boltzmann equation with constant collision kernel

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.

math.AP↗

arXiv2024

Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the $2d$ sphere I: positive regularities

We establish the probabilistic well-posedness of the nonlinear Schr\"odinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.

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arXiv2024

The Second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data

We consider the Wick ordered cubic Schr\"odinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e. a stable closed geodesic), which prevents any regularization in the second Picard iteration.

math.AP↗

arXiv2023

Quantitative observability for one-dimensional Schr\"odinger equations with potentials

In this note, we prove the quantitative observability with an explicit control cost for the 1D Schr\"odinger equation over $\mathbb{R}$ with real-valued, bounded continuous potential on thick sets. Our proof relies on different techniques for low-frequency and high-frequency estimates. In particular, we extend the large time observability result for the 1D free Schrodinger equation in Theorem 1.1 of Huang-Wang-Wang [20] to any short time. As another byproduct, we extend the spectral inequality of Lebeau-Moyano [27] for real-analytic potentials to bounded continuous potentials in the one-dimensional case.

math.AP↗

arXiv2023

Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation

We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-\Delta)^{-s}$, where $\Delta$ is the Laplace operator and $s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $1d$ to higher dimensions.

math.AP↗

arXiv2022

Weak universality results for a class of nonlinear wave equations

We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $\Phi_2^4$, we first establish a \emph{sufficient and almost necessary} criteria for the convergence of invariant measures to the fractional $\Phi_2^4$. Then we prove the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. Our constraint on the fractional index is independent of the degree of the nonlinearity. This extends the result of Gubinelli-Koch-Oh [Renormalisation of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018)] to a situation where we do not have a local Cauchy theory with highly supercritical nonlinearities.

math.AP↗

arXiv2022

Sharp resolvent estimate for the Baouendi-Grushin operator and applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

math.AP↗

arXiv2021

Sharp decay rate for the damped wave equation with convex-shaped damping

We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. We show that if the damping vanishes as a H\"older function $|x|^{\beta}$, and in addition, the boundary of the damped region is strictly convex, the wave is stable at rate $t^{-1+\frac{2}{2\beta+7}}$, which is better than the known optimal decay rate $t^{-1+\frac{1}{\beta+3}}$ for strip-shaped dampings of the same H\"older regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the energy decay rate depends not only on the order of vanishing of the damping, but also on the shape of the damped region. The main ingredient of the proof is the averaging method (normal form reduction) developed by Hitrick and Sj\"ostrand (\cite{Hi1}\cite{Sj}).

math.AP↗

arXiv2020

Refined probabilistic global well-posedness for the weakly dispersive NLS

We continue our study of the cubic fractional NLS with very weak dispersion $\alpha>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $\alpha>\alpha_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to our situation the new resolution ansatz in \cite{Deng2} which captures the most singular frequency interaction parts in the $X^{s,b}$ type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the "almost" transport effect of these singular parts and to exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.

math.AP↗

arXiv2020

Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates

In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $\gamma\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $\Delta_\gamma=\partial_x^2+|x|^{2\gamma}\partial_y^2$, which has step $\gamma+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-\Delta_\gamma)^{s}u=0$ where $s\in N$. We identify three different cases: depending on the value of the ratio $(\gamma+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial_tu+(-\Delta_\gamma)^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta_{\gamma}$.

math.AP↗

arXiv2020

Decays rates for Kelvin-Voigt damped wave equations II: the geometric control condition

We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).

math.AP↗