arXiv subjects

Jian-Guo Liu

Publications and source records attributed to Jian-Guo Liu.

At least 19 recordsLinked to original sources

arXiv2026

Bose--Einstein Condensation without an Initial Low-Energy Concentration Assumption

We prove that semi-strong convergence to a Bose--Einstein equilibrium below the critical temperature implies finite-time Bose--Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent $0\leqη<1$. For every admissible initial measure with $\overline T/\overline T_c<1$, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose--Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.

math.AP↗

arXiv2026

Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes

We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax--Friedrichs entropy dissipation and the physical-constraint-preserving Lax--Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge--Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi--Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

math.NA↗

arXiv2026

Optimal control formulation of transition path problems for Markov Jump Processes

We formulate the transition path problem for Markov jump processes as a stochastic optimal control problem on path space. Transitions between metastable sets are induced by an unbounded terminal cost at a stopping time together with controlled modification of jump rates. The running cost takes an entropic form arising naturally from the Girsanov transform for jump processes, allowing both finite- and infinite-horizon formulations to be expressed as optimal changes of measure relative to a reference process. We prove that the optimal path measure admits an explicit representation in terms of the committor function, which solves an elliptic boundary value problem. The singular control induced by the terminal cost is obtained via $Γ$-convergence, yielding a limiting controlled process whose transition rates correspond to a Doob-h transform. The optimally controlled process generates transition paths almost surely while preserving the bridges of the reference process.

math.OC↗

arXiv2026

No Gelation and Global Existence for a Boltzmann Equation with Regularly Varying Mass-Exchange Rates

We develop a no-gelation mechanism that yields global solutions to the spatially homogeneous Boltzmann equation with mass exchange for Grad-cutoff hard potentials $B=E^γb(ξ), 0<γ<1$ and regularly varying mass-exchange rates. Without assuming any higher mass moment, we construct a convex superlinear weight assembled from dyadic hinges. Production of the weighted moment by collisions between large particles of comparable mass is absorbed by dissipation through collisions with a uniformly populated reservoir of bounded-mass particles. Regular variation makes the signed increments in these two collision configurations comparable, while mass conservation and $γ<1$ yield the vanishing factor $L^{γ-1}$. This yields a uniform moment bound of the mass-cutoff approximations on every finite time interval and rules out finite-time mass escape to infinity. Consequently, for every nonnegative initial density with finite physical moments, we obtain a global integral weak solution.

math.AP↗

arXiv2026

On the Spatially Homogeneous Boltzmann Equation with Mass Exchange

We study the spatially homogeneous Boltzmann equation with continuous mass exchange on $X=(0,\infty)_m\times\mathbb R^d_v$, with a Grad cut-off hard-potential collision kernel. For bounded continuous symmetric mass-exchange rates, every nonnegative initial datum with finite number, mass, and kinetic energy admits a global nonnegative $L^1$-integral weak solution in $W^{1,\infty}(0,\infty;L^1(X))$ with number and mass conserved, kinetic energy dissipated. If $\int_X (m|v|^2)^{1+δ} f_0(x)dx<\infty,$ for some $δ>0$, this higher-energy moment propagates on every finite time interval and kinetic energy is conserved through the constructed solution. Moreover, every energy-dissipating solution propagates any such moment. Under the additional $1+γ$ moment assumption, the solution is unique among all energy-dissipating $L^1$-integral weak solutions with the same initial datum. We also establish a local theory for a linearly growing mass-exchange rate. With $H_p(f)=\int_X (1+m+m|v|^2)^pfdx,$ every datum with $H_p(f_0)<\infty$, where $p\ge1+γ$ admits a conservative local $H_p$-solution. Moreover, an $H_p$-solution continues across every finite time $T$ for which $H_{1+γ}(f)\in L^1(0,T).$ This proof requires no detailed-balance or relative-entropy structures. It is based on a new two-stage bootstrap method. The first stage rules out mass concentration at $m=0$, while the second stage combines this control with collision geometry to establish uniform integrability. These estimates provide the compactness needed for the global solution and for the identification of the nonlinear collision form.

math.AP↗

arXiv2026

Phase-Field Models, Sharp Interface Limits, and Numerical Schemes for Contact Line Dynamics

We study phase-field and sharp-interface models for contact line dynamics of a liquid droplet on a solid substrate within a unified variational framework. The motion of the contact line, where liquid, gas, and solid phases meet, poses a fundamental difficulty in continuum modeling due to the classical stress singularity of no-slip hydrodynamics. Phase-field models regularize this singularity by introducing a thin transition layer of thickness and encoding interfacial effects through a Ginzburg-Landau free energy augmented by a wall energy on the substrate. Starting from the total free energy $E = E_b + E_w$, we analyze two phase-field models: the Allen-Cahn equation and the Cahn-Hilliard equation. Using matched asymptotic expansions as $δ\to 0$, we recover their corresponding sharp interface limits. In the Allen-Cahn case, the limit yields motion by mean curvature with a contact line law driven by deviations of the dynamic contact angle from Young's angle. In the Cahn-Hilliard case, the limit leads to a Mullins-Sekerka problem with the same form of contact line dynamics. A central result of this work is the identification of consistent gradient-flow structures across both models. The Allen-Cahn dynamics correspond to an $L^2$-gradient flow, while the Cahn-Hilliard dynamics correspond to an $H^{-1}$-gradient flow, and both converge to sharp-interface evolutions that preserve the same energy-dissipation structure. This provides a unified interpretation of contact line motion as a consequence of a single variational principle. Finally, we develop energy-stable numerical schemes based on the minimizing movement principle and establish discrete energy dissipation and well-posedness of the fully discrete problem. Numerical examples confirm that both schemes relax toward the same stationary sharp interface solution while their dynamics reflect the different dissipation mechanisms.

math.AP↗

arXiv2026

Energetic variational formulation for electrohydrodynamics of surfactant-laden droplets

The coupling between surfactant-laden droplet dynamics and electric fields plays an important role in liquid-handling technologies such as digital microfluidics. We develop an energetic variational framework for the coupled electrohydrodynamics of surfactant-laden droplets, incorporating two-phase Stokes flow, insoluble surfactant transport on a moving interface, and electrostatic effects. Based on Onsager's principle, the governing equations are derived by minimizing the Rayleighian, defined as the sum of the rate of change of the free energy and the dissipation functional, subject to the incompressibility constraint. This formulation simultaneously yields the Stokes equations in each bulk phase, the interfacial stress-balance condition incorporating Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for the surfactant concentration, and the moving contact-line dynamics. By replacing the viscous dissipation functional with Rayleigh dissipation, we further derive a reduced model describing the evolution of surfactant-laden droplets driven by motion by mean curvature. For sessile droplets represented as graph surfaces, the model reduces to a one-dimensional coupled electrohydrodynamic model for the liquid height, surfactant concentration, and electrostatic potential. A first-order implicit-explicit numerical scheme is proposed for the reduced system, and numerical results illustrate the coupled effects of surfactant transport and electric fields on driving droplet deformation and migration.

physics.flu-dyn↗

arXiv2026

Accelerating sampling via asymptotic relaxation enhancing flows

In this paper, we accelerate Langevin Monte Carlo sampling from Gibbs measures $π\propto \exp(-U)$ by adding a large drift that preserves the invariant measure. For warm-start initial data, we characterize the sharp asymptotic decay rate of the relative entropy and introduce asymptotic relaxation enhancing flows: sequences that achieve arbitrarily fast decay. We construct such flows on the torus by scaling cellular flows and pushing them forward via diffeomorphisms, and we extend the construction to the full space using a Lyapunov function method to control behavior at infinity without periodization, obtaining explicit finite energy flows that guarantee arbitrarily fast convergence under natural growth conditions on $U$.

math.PR↗

arXiv2025

Analysis of the adhesion model and the reconstruction problem in cosmology

In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model suppresses multi-streaming by introducing viscosity. We study mass flow in this model by analysis of Lagrangian advection in the zero-viscosity limit. Under mild conditions, we show that a unique limiting Lagrangian semi-flow exists. Limiting particle paths stick together after collision and are characterized uniquely by a differential inclusion. The absolutely continuous part of the mass measure agrees with that of a Monge-Ampère measure arising by convexification of the free-streaming velocity potential. But the singular parts of these measures can differ when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. The use of Monge-Ampère measures and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier & Frisch et al. (Month. Not. Roy. Ast. Soc. 346, 2003). In a neighborhood of merging singular structures in our examples, however, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.

math.AP↗

arXiv2025

On Random Batch Methods (RBM) for interacting particle systems driven by Lévy processes

In many real-world scenarios, the underlying random fluctuations are non-Gaussian, particularly in contexts where heavy-tailed data distributions arise. A typical example of such non-Gaussian phenomena calls for Lévy noise, which accommodates jumps and extreme variations. We propose the Random Batch Method for interacting particle systems driven by Lévy noises (RBM-Lévy), which can be viewed as an extension of the original RBM algorithm in [Jin et al. J Compt Phys, 2020]. In our RBM-Lévy algorithm, $N$ particles are randomly grouped into small batches of size $p$, and interactions occur only within each batch for a short time. Then one reshuffles the particles and continues to repeat this shuffle-and-interact process. In other words, by replacing the strong interacting force by the weak interacting force, RBM-Lévy dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step. Meanwhile, the resulting dynamics converges to the original interacting particle system, even at the appearance of the Lévy jump. We rigorously prove this convergence in Wasserstein distance, assuming either a finite or infinite second moment of the Lévy measure. Some numerical examples are given to verify our convergence rate.

math.NA↗

arXiv2025

On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $\frac{r}{pn+r}\binom{pn+r}{n}$, $n=0,1,\ldots$ are the moments of a probability distribution on some interval $[0,τ]$ {if and only if} $p\ge1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom{pn+r-1}n$, $n=0,1,\ldots$. A corrigendum (Trans. Amer. Math.Soc., to appear) has been included as an appendix, correcting gaps in the proof of Lemma 3.

math.CA↗

arXiv2025

Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo

We study the implicit Langevin Monte Carlo (iLMC) method, which simulates the overdamped Langevin equation via an implicit iteration rule. In many applications, iLMC is favored over other explicit schemes such as the (explicit) Langevin Monte Carlo (LMC). LMC may blow up when the drift field $\nabla U$ is not globally Lipschitz, while iLMC has convergence guarantee when the drift is only one-sided Lipschitz. Starting from an adapted continuous-time interpolation, we prove a time-discretization error bound under the relative entropy (or the Kullback-Leibler divergence), where a crucial gradient estimate for the logarithm numerical density is obtained via a sequence of PDE techniques, including Bernstein method. Based on a reflection-type continuous-discrete coupling method, we prove the geometric ergodicity of iLMC under the Wasserstein-1 distance. Moreover, we extend the error bound to a uniform-in-time one by combining the relative entropy error bound and the ergodicity. Our proof technique is universal and can be applied to other implicit or splitting schemes for simulating stochastic differential equations with non-Lipschitz drifts.

math.NA↗

arXiv2025

Convergence of Random Batch Method with replacement for interacting particle systems

The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems (IPS). In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size $N$. In the RBM-r algorithm, one randomly picks a small batch of size $p \ll N$, and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength $\frac{1}{N-1}$) in the original system is replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. We give a rigorous proof of this approximation with an explicit convergence rate. In detail, we show that the Wasserstein-2 distance between first marginal distributions of IPS and RBM-r has an $O(κ^{1/4})$ upper bound, where $κ$ is the time step for choosing the random batch and the bound is independent of $N$. An improved $O(κ^{1/2})$ rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.

math.NA↗

arXiv2025

Equilibrium Correction Iteration for A Class of Mean-Field Game Inverse Problem

This work investigates the ambient potential identification problem in inverse Mean-Field Games (MFGs), where the goal is to recover the unknown potential from the value function at equilibrium. We propose a simple yet effective iterative strategy, Equilibrium Correction Iteration (ECI), that leverages the structure of MFGs rather than relying on generic optimization formulations. ECI uncovers hidden information from equilibrium measurements, offering a new perspective on inverse MFGs. To improve computational efficiency, two acceleration variants are introduced: Best Response Iteration (BRI), which uses inexact forward solvers, and Hierarchical ECI (HECI), which incorporates multilevel grids. While BRI performs efficiently in general settings, HECI proves particularly effective in recovering low-frequency potentials. We also highlight a connection between the potential identification problem in inverse MFGs and inverse linear parabolic equations, suggesting promising directions for future theoretical analysis. Finally, comprehensive numerical experiments demonstrate how viscosity, terminal time, and interaction costs can influence the well-posedness of the inverse problem.

math.OC↗

arXiv2025

Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes

We study all the ways that a given convex body in $d$ dimensions can break into countably many pieces that move away from each other rigidly at constant velocity, with no rotation or shearing. The initial velocity field is locally constant, but may be continuous and/or fail to be integrable. For any choice of mass-velocity pairs for the pieces, such a motion can be generated by the gradient of a convex potential that is affine on each piece. We classify such potentials in terms of a countable version of a theorem of Alexandrov for convex polytopes, and prove a stability theorem. For bounded velocities, there is a bijection between the mass-velocity data and optimal transport flows (Wasserstein geodesics) that are locally incompressible. Given any rigidly breaking velocity field that is the gradient of a continuous potential, the convexity of the potential is established under any of several conditions, such as the velocity field being continuous, the potential being semi-convex, the mass measure generated by a convexified transport potential being absolutely continuous, or there being a finite number of pieces. Also we describe a number of curious and paradoxical examples having fractal structure.

math.AP↗

arXiv2025

Convergence Analysis and Acceleration of Fictitious Play for General Mean-Field Games via the Best Response

A mean-field game (MFG) seeks the Nash Equilibrium of a game involving a continuum of players, where the Nash Equilibrium corresponds to a fixed point of the best-response mapping. However, simple fixed-point iterations do not always guarantee convergence. Fictitious play is an iterative algorithm that leverages a best-response mapping combined with a weighted average. Through a thorough study of the best-response mapping, this paper develops a simple and unified convergence analysis, providing the first explicit convergence rate for the fictitious play algorithm in MFGs of general types, especially non-potential MFGs. We demonstrate that the convergence and rate can be controlled through the weighting parameter in the algorithm, with linear convergence achievable under a general assumption. Building on this analysis, we propose two strategies to accelerate fictitious play. The first uses a backtracking line search to optimize the weighting parameter, while the second employs a hierarchical grid strategy to enhance stability and computational efficiency. We demonstrate the effectiveness of these acceleration techniques and validate our convergence rate analysis with various numerical examples.

math.OC↗

arXiv2025

On the discrete Poincaré inequality for B-schemes of 1D Fokker-Planck equations in full space

In this paper, we propose two approaches to derive the discrete Poincaré inequality for the B-schemes, a family of finite volume discretization schemes, for the one-dimensional Fokker-Planck equation in full space. We study the properties of the spatially discretized Fokker-Planck equation in the viewpoint of a continuous-time Markov chain. The first approach is based on Gamma-calculus, through which we show that the Bakry-Émery criterion still holds in the discrete setting. The second approach employs the Lyapunov function method, allowing us to extend a local discrete Poincaré inequality to the full space. The assumptions required for both approaches are roughly comparable with some minor differences. These methods have the potential to be extended to higher dimensions. As a result, we obtain exponential convergence to equilibrium for the discrete schemes by applying the discrete Poincaré inequality.

math.NA↗

arXiv2025

Nonsymmetric traveling wave solution to a Hele-Shaw type tumor growth model

We consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. The model is derived by taking the incompressible limit of porous medium type equations, and the boundary instability of this model was recently studied in \cite{feng2022tumor} using asymptotic analysis. In this paper, we further prove the existence of nonsymmetric traveling wave solutions to the model in a two dimensional tube-like domain, which reflect intrinsic boundary instability in tumor growth dynamics.

math.AP↗