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Rongchang Liu

Publications and source records attributed to Rongchang Liu.

15 recordsLinked to original sources
arXiv2026

Polynomial mixing for the 3D damped cubic nonlinear Schr\"odinger equation with degenerate noise

We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schr\"odinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sharp geometric characterizations of saturation. The proof is based on a polynomial mixing criterion built on a stable--compact decomposition of the exact solution differences with polynomial moment control of the logarithmic path amplification. Dense Malliavin range allows the compact defect to be compensated by finite-dimensional Cameron--Martin shifts, producing a block multiplier with negative mean logarithm. A logarithmic transportation gauge, combined with a renewal--reset coupling scheme, then yields mixing at every prescribed polynomial order in the weaker $L^2$ distance. A stationary regularity gain to $H^{2-}$ then enables us to upgrade the convergence to $H^1$.

math.PR↗
arXiv2026

Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise

We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable--compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron--Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.

math.AP↗
arXiv2026

Unique Ergodicity for the Projective Process of the 2D Navier--Stokes Equation with Nondegenerate Noise

We prove unique ergodicity of the projective process associated with the two dimensional Navier--Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg--Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foia\c{s}--Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.

math.PR↗
arXiv2026

Exponential mixing and Freidlin--Wentzell large deviation principle for Markov cocycles

This paper studies the long time statistics and small noise asymptotics of Markov cocycles associated with Markov processes in random environments modeled by measure preserving dynamical systems on a standard Borel probability space. Our first result provides an abstract criterion for exponential mixing of stationary measures for such cocycles, formulated toward SPDE applications with assumptions that can be verified directly from a priori estimates. To overcome the nonuniformity from the environment, we combine generalized coupling arguments with ergodic theoretic methods. This allows us to convert nonuniform estimates along the environment into contraction on a positive density set of times, and then upgrade this to all time contraction by introducing a block gap-counting argument. Our second result establishes a Freidlin--Wentzell large deviation principle(LDP) for the unique stationary measure in the small noise limit with a good rate function. For the upper bound, the noise is allowed to be degenerate, while the deterministic pullback attractor may have nontrivial dynamics. The abstract theory applies to nonautonomous SPDEs. We illustrate it with two examples: the two-dimensional Navier--Stokes equations on bounded domains and damped Sine--Gordon equations, where both the deterministic forcing and the degenerate additive noise depend on the random environment.

math.PR↗
arXiv2026

Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation

We study dual-cascade flux laws for the stochastic forced--dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and an antisymmetrized mixed flux law for the inverse cascade of the Hamiltonian. In particular, the inverse Hamiltonian law appears to be new even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: $B^s_{3,\infty}$-regularity above the Onsager threshold $1/3$ rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.

math.AP↗
arXiv2025

The Restricted Three-Body Problem as a Perturbed Duffing Equation

This paper investigates the restricted circular planar three-body problem. We prove that for every negative Jacobi constant of sufficiently large magnitude, the surface of unperturbed parabolic solutions breaks to induce homoclinic tangle for all but at most finitely many mass ratios of primaries. This result is not covered by \cite{G} as the required large magnitude of the Jacobi constant is uniform across the mass ratios. Our approach consists of three main ingredients. First, by introducing new coordinate transformations, we reformulate the restricted three-body problem as a perturbed Duffing equation. Second, we adopt the method recently introduced in \cite{CW} to derive integral equations for the primary stable and unstable solutions. This enables us to effectively capture the order of the singularities involved and to further establish the existence and analytic dependence of the invariant manifolds on the mass ratio of the primaries and the Jacobi constant. Third, in evaluating the Poincaré-Melnikov integral, we take advantage of the explicit homoclinic solution of the unperturbed Duffing equation. Compared to existing works, our proof is significantly simpler and more direct. Moreover, the paper is self-contained: we do not rely on McGehee's analysis in \cite{Mc} to justify the applicability of the Poincaré-Melnikov method.

math.CA↗
arXiv2025

Averaging principle for the stochastic primitive equations in the large rotation limit

It is known that the unique ergodicity of the viscous primitive equations with additive white-in-time noise remains an open problem. In this work, we demonstrate that, as the rotational intensity approaches infinity, the distribution of any given strong solution, rescaled by the rotation, is attracted to the unique invariant measure of the stochastic limit resonant system. This suggests that the system exhibits nearly unique ergodicity in the large rotation limit. The proof is based on a stochastic averaging principle combined with a coupling argument.

math.PR↗
arXiv2025

On the local well-posedness of fractionally dissipated primitive equations with transport noise

We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by $ (-Δ)^{s/2} $ with $ s \in (1,2)$ and $s = 1$, respectively. For $σ>3$, we establish the local existence of unique pathwise solutions in Sobolev space $H^σ$. This result applies to arbitrary initial data in the subcritical case ($s \in(1,2)$), and to small initial data in the critical case ($s=1$). The analysis is particularly challenging due to the loss of horizontal derivatives in the nonlinear terms and the lack of full dissipation. To address these challenges, we develop novel commutator estimates involving the hydrostatic Leray projection.

math.AP↗
arXiv2024

Regularization by noise for the inviscid primitive equations

The deterministic inviscid primitive equations (also called the hydrostatic Euler equations) are known to be ill-posed in Sobolev spaces and in Gevrey classes of order strictly greater than 1, and some of their analytic solutions exist only locally in time and exhibit finite-time blowup. This work demonstrates that introducing suitable random noise can restore the local well-posedness and prevent finite-time blowups. Specifically, random diffusion addresses the ill-posedness in certain Gevrey classes, allowing us to establish the local well-posedness almost surely and the global existence of solutions with high probability. In the case of random damping (linear multiplicative noise), the noise prevents analytic solutions from forming singularities in finite time, resulting in globally existing solutions with high probability.

math.AP↗
arXiv2024

Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise

In this paper, we consider the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise. By assuming the ionic species have the same diffusivity and opposite valences, we prove the global well-posedness of the system. Furthermore, we illustrate the enhanced dissipation phenomenon in the system with specific transportation noise by establishing that it enables an arbitrarily large exponential convergence rate of the solutions.

math.AP↗
arXiv2023

Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes Equations in the hypoelliptic setting

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a quasi-periodic invariant measure that exponentially attracts the law of all solutions. The result is true for any value of the viscosity $ν>0$ and does not depend on the strength of the external forces. By utilizing this quasi-periodic invariant measure, we establish a quantitative version of the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes with explicit convergence rates. It turns out that the convergence rate in the central limit theorem depends on the time inhomogeneity through the Diophantine approximation property on the quasi-periodic frequency of the quasi-periodic force.

math.PR↗
arXiv2023

Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise

We consider the incompressible 2D Navier-Stokes equations on the torus, driven by a deterministic time periodic force and a noise that is white in time and degenerate in Fourier space. The main result is twofold. Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the time inhomogeneous Feynman-Kac evolution operators with regular potentials associated with the stochastic Navier-Stokes system. The theorem characterizes asymptotic behaviors of the Feynman-Kac operators in terms of the periodic family of principal eigenvalues and corresponding unique eigenvectors. The proof involves a time inhomogeneous version of Ruelle's lower bound technique. Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer's criterion, we establish a Donsker-Varadhan type large deviation principle with a nontrivial good rate function for the occupation measures of the time inhomogeneous solution processes.

math.PR↗
arXiv2021

Statistical Properties of 2D Stochastic Navier-Stokes Equations with Time-Periodic Forcing and Degenerate Stochastic Forcing

We consider the incompressible 2D Navier-Stokes equations with periodic boundary conditions driven by a deterministic time periodic forcing and a degenerate stochastic forcing. We show that the system possesses a unique ergodic periodic invariant measure which is exponentially mixing under a Wasserstein metric. We also prove the weak law of large numbers for the continuous time inhomogeneous solution process. In addition, we obtain the weak law of large numbers and central limit theorem by restricting the inhomogeneous solution process to periodic times. The results are independent of the strength of the noise and hold true for any value of viscosity with a lower bound $ν_1$ characterized by the Grashof number $G_1$ associated with the deterministic forcing. In the laminar case, there is a larger lower bound $ν_2$ of the viscosity characterized by the Grashof number $G_2$ associated with both the deterministic and random forcing. We prove that in this laminar case, the system has trivial dynamics for any viscosity larger than $ν_2$ by demonstrating the existence of a unique globally exponentially stable random periodic solution that supports the unique periodic invariant measure.

math.DS↗
arXiv2014

A type of multiple integral with loggamma function

In this paper, we study the multiple integral $ \displaystyle I= \int_0^1 \int_0^1 \dots \int_0^1 f(x_1+x_2 + \dots +x_n) \, dx_1 \, dx_2 \, \dots \, dx_n$. A general formula of $I$ is presented. As an application, the integral $I$ with $f(x)= \log Γ(x)$ is evaluated. We show that the values of $I$ share a common formula for all $n \in \mathbb{N}$. The subsidiary computational challenges are substantial and interesting in their own right.

math.CA↗