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Ruoyuan Liu

Publications and source records attributed to Ruoyuan Liu.

16 recordsLinked to original sources
arXiv2026

On the Smoluchowski-Kramers approximation for the hyperbolic $O(N)$ linear sigma model and its mean-field limit

We study the hyperbolic $O(N)$ linear sigma model, i.e. a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities, posed on the two-dimensional torus and indexed by a parameter $\varepsilon > 0$. We show that as $\varepsilon$ goes to zero (Smoluchowski-Kramers approximation) and $N$ goes to infinity (mean-field limit), each component of the solution to the SdNLW system converges to the solution to the stochastic nonlinear heat equation (SNLH) with a mean-field nonlinearity. We prove such convergence via two regimes: first with $\varepsilon$ going to zero to obtain the parabolic $O(N)$ linear sigma model, i.e. a system of $N$ coupled SNLH, and then with $N$ going to infinity; or first with $N$ going to infinity for each component to obtain the mean-field SdNLW and then with $\eps$ going to zero. As a result, we obtain a commutative diagram regarding the convergence from the hyperbolic $O(N)$ linear sigma model to the mean-field SNLH.

math.AP↗
arXiv2026

The Korteweg-de Vries limit for the global dynamics of the Toda lattice

It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of the Toda lattice with H^1 initial data converges to that of the KdV equation globally in time. Our proof relies on tools from harmonic analysis and also on the construction and the conservation of mass and energy of the Toda lattice, the latter of which are derived from the completely integrable structure of the Toda lattice. As a consequence, we obtain long-wave KdV limits for the Toda lattice.

nlin.SI↗
arXiv2025

Hyperbolic $O (N)$ linear sigma model and its mean-field limit

We study large $N$ limits of the hyperbolic $O(N)$ linear sigma model ($\text{HLSM}_N$) on the two-dimensional torus $\mathbb T^2$, namely, a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities. After establishing (pathwise) global well-posedness of $\text{HLSM}_N$ and the limiting equation, called the mean-field SdNLW, we first establish global-in-time convergence of $\text{HLSM}_N$ to the mean-field SdNLW with general initial data (under a suitable assumption). In particular, for the local-in-time convergence, we obtain an optimal convergence rate of order $N^{- \frac 12}$ under an additional integrability assumption on initial data. We then show that the invariant Gibbs dynamics for $\text{HLSM}_N$ converges to that for the mean-field SdNLW with a convergence rate of order $N^{- \frac 12}$ on any large time intervals.

math.AP↗
arXiv2025

Large torus limit of global dynamics of the two-dimensional dispersive Anderson model

We continue the study of the two-dimensional dispersive Anderson model (DAM), i.e. the nonlinear Schr\"odinger equation with multiplicative spatial white noise. For this model, global well-posedness on the periodic domain was established by Visciglia and the second author (2023), and global well-posedness on the full space was established by Debussche, Visciglia, and the authors (2024). We show that, under suitable initial conditions and suitable periodization procedure of the noise, the periodic global dynamics of the DAM converges in spaces of local domains to that of the DAM on the full space as the period goes to infinity. In order to control the growth of the noise and obtain a priori bounds for solutions independent of the periodicity, we introduce periodic weights and construct weighted function spaces on periodic domains. In Appendix, we also discuss the same problem for the parabolic Anderson model.

math.AP↗
arXiv2024

Global well-posedness of the energy-critical stochastic nonlinear wave equations

We consider the Cauchy problem for the defocusing energy-critical stochastic nonlinear wave equations (SNLW) with an additive stochastic forcing on $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by Bényi-Oh-Pocovnicu (2015) and Pocovnicu (2017) and in the context of stochastic PDEs by Oh-Okamoto (2020), we prove global well-posedness of the defocusing energy-critical SNLW. In particular, on $\mathbb{T}^d$, we prove global well-posedness with the stochastic forcing below the energy space.

math.AP↗
arXiv2023

Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus

We study global-in-time dynamics of the stochastic nonlinear beam equations (SNLB) with an additive space-time white noise, posed on the four-dimensional torus. The roughness of the noise leads us to introducing a time-dependent renormalization, after which we show that SNLB is pathwise locally well-posed in all subcritical and most of the critical regimes. For the (renormalized) defocusing cubic SNLB, we establish pathwise global well-posedness below the energy space, by adapting a hybrid argument of Gubinelli- Koch-Oh-Tolomeo (2022) that combines the $I$-method with a Gronwall-type argument. Lastly, we show almost sure global well-posedness and invariance of the Gibbs measure for the stochastic damped nonlinear beam equations in the defocusing case.

math.AP↗
arXiv2023

Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces

We study the Cauchy problem for the cubic fractional nonlinear Schrödinger equation (fNLS) on the real line and on the circle. In particular, we prove global well-posedness of the cubic fNLS with all orders of dispersion higher than the usual Schrödinger equation in negative Sobolev spaces. On the real line, our well-posedness result is sharp in the sense that a contraction argument does not work below the threshold regularity. On the circle, due to ill-posedness of the cubic fNLS in negative Sobolev spaces, we study the renormalized cubic fNLS. In order to overcome the failure of local uniform continuity of the solution map in negative Sobolev spaces, by applying a gauge transform and partially iterating the Duhamel formulation, we study the resulting equation with a cubic-quintic nonlinearity. In proving uniqueness, we present full details justifying the use of the normal form reduction for rough solutions, which seem to be missing from the existing literature. Our well-posedness result on the circle extends those in Miyaji-Tsutsumi (2018) and Oh-Wang (2018) to the endpoint regularity.

math.AP↗
arXiv2023

Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $\Phi^4_3$-model

We study the fractional $\Phi^4_3$-measure (with order $\alpha > 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $\Phi^4_3$-model. We first construct the fractional $\Phi^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $\alpha \leq \frac{9}{8}$, this fractional $\Phi^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $\Phi^4_3$-model and invariance of the fractional $\Phi^4_3$-measure for all $\alpha > 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $\Phi^3_3$-model. Furthermore, when $\alpha > \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $\Phi^4_3$-model by utilizing the convergence of Gibbs measures.

math.AP↗
arXiv2023

Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces

We study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $\overline{u}^2$, posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the $X^{s, b}$-space is known to fail when the regularity $s$ is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the $X^{s, b}$-space.

math.AP↗
arXiv2023

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (2019) with a sub-quadratic nonlinearity.

math.AP↗
arXiv2022

On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with the quadratic nonlinearity $|u|^2$

We study the two-dimensional periodic nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$. In particular, we study the quadratic NLS with random initial data distributed according to a fractional derivative (of order $α\geq 0$) of the Gaussian free field. After removing the singularity at the zeroth frequency, we prove that the quadratic NLS is almost surely locally well-posed for $α< \frac{1}{2}$ and is probabilistically ill-posed for $α\geq \frac{3}{4}$ in a suitable sense. These results show that in the case of rough random initial data and a quadratic nonlinearity, the standard probabilistic well-posedness theory for NLS breaks down before reaching the critical value $α= 1$ predicted by the scaling analysis due to Deng, Nahmod, and Yue (2019), and thus this paper is a continuation of the work by Oh and Okamoto (2021) on stochastic nonlinear wave and heat equations by building an analogue for NLS.

math.AP↗
arXiv2022

Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity $|u|^2$

We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version of the $L^3$-Strichartz estimate without any derivative loss.

math.AP↗
arXiv2022

On the two-dimensional singular stochastic viscous nonlinear wave equations

We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb T^2$, forced by a fractional derivative of the space-time white noise $ξ$. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}ξ$ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.

math.AP↗
arXiv2019

A note on the extremal noncentral sections of the cross-polytope

We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.

math.MG↗