arXiv subjects

Shao Liu

Publications and source records attributed to Shao Liu.

7 recordsLinked to original sources

arXiv2026

Refined global well-posedness for the periodic modulated Korteweg-de Vries equation

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s > - \frac 32$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s \in \mathbb R$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb T)$, thus going beyond the barrier of the scaling critical regularity $s = - \frac 32$.

math.AP↗

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↗

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↗

arXiv2021

The sharp constant for truncated Hardy-Littlewood maximal inequality

This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the $L^1$-norm of $M^b_a$, and then the $L^1$-norm of $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$ is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.

math.CA↗

arXiv1995

Transverse Lepton Polarization in Polarized W Decays

Calculations of transverse polarization of leptons in the decay $W\rightarrow lν$ with polarized $W$'s are presented. Planned accelerators will produce enough $W$'s for observation of the Standard Model contributions to this polarization. One loop corrections to the polarization are given; these are too small to be seen at presently available $W$ sources. The exchange of Majorons will contribute to these polarizations; these may provide limits on the couplings of these particles to leptons.

hep-ph↗

arXiv1994

Random Ising Spins in Two Dimensions - A Flat Space Realization of the KPZ Exponents

A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. As a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the KPZ values associated with Ising spins, and with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries.

hep-th↗

arXiv1994

Dynamically Triangulated Ising Spins in Flat Space

A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. It is found that as a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries.

hep-th↗