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Zihui He

Publications and source records attributed to Zihui He.

14 recordsLinked to original sources
arXiv2026

Gradient flow approach to Landau equation: A cross-product structure

We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form $A(|v-v_*|)\sim \langle v-v_*\rangle^\gamma|v-v_*|^2$ with $\gamma\in(-\infty,1]$, this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted $L^1$-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.

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arXiv2026

The Homogeneous Landau Equation with Regularised Thermal Noise

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-It\^o correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

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arXiv2026

Unified Formulation and Asymptotic Limits of Inhomogeneous Kinetic Models within GENERIC

In this paper, we study a general class of inhomogeneous kinetic models that unifies fundamental models in both the statistical physics of particles and of waves, namely the kinetic Boltzmann equations and the kinetic wave equations, in both classical (non-relativistic), relativistic and quantum settings. We formulate this unified equation into the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework. We then derive the grazing (small-angle) limit in two-body interaction systems, which leads to Landau-type equations. Finally, we show that these limiting systems can also be formulated as GENERIC systems.

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arXiv2026

Multi-species kinetic models: GENERIC formulation and Fisher information

In this paper, we study the GENERIC structures of multi-species spatially inhomogeneous Boltzmann and Landau equations with Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics. In addition, under suitable assumptions on the collision kernels, we show that the Fisher information for the multi-species spatially homogeneous Boltzmann equation is non-increasing in time.

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arXiv2025

On a fuzzy Landau Equation: Part III. The grazing collision limit

In this paper, we study the grazing limit from the non-cutoff fuzzy Boltzmann equations to the fuzzy Landau equation, where particles interact through delocalised collisions. We show the grazing limit through variational formulations that correspond to the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) structure of the respective equations. We show that the variational formulation associated with a non-quadratic dual dissipation pair for the fuzzy Boltzmann equations converges to a variational formulation of the fuzzy Landau equation corresponding to a quadratic dissipation pair.

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arXiv2025

GENERIC formulation and small-angle limit for Kinetic wave equations

In this paper, we formulate the three-wave and four-wave kinetic equations into the GENERIC framework and formally derive a small-angle limit for the four-wave equation. This limit is akin to the well-known grazing limit from the kinetic Boltzmann equation to the kinetic Landau equation. We also show the GENERIC structure of the limiting system.

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arXiv2025

On a fuzzy Landau Equation: Part II. Solvability results

This article is the second in a series of our works on the fuzzy Landau equation, where particles interact via delocalised Coulomb collisions. In this work, we focus on the existence and propagation of regularity for solutions to the fuzzy Landau equation.

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arXiv2025

Passing to the limit in fuzzy Boltzmann equations

We study a fuzzy Boltzmann equation, where collisions are delocalised and modulated by a spatial kernel. We show that as the spatial kernel converges to a delta distribution, the solutions to these equations converge to renormalised solutions of the inhomogeneous Boltzmann equations.

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arXiv2025

On a fuzzy Landau Equation: Part I. A variational approach

This article is the first in a series of works on the fuzzy Landau equation, where particles interact through delocalised Coulomb collisions. Here, we establish a variational characterisation that recasts the fuzzy Landau equation within the framework of GENERIC systems (General Equations for Non-Equilibrium Reversible-Irreversible Coupling).

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arXiv2024

A variational approach to a fuzzy Boltzmann equation

We study a fuzzy Boltzmann equation, where particles interact via delocalised collisions, in contrast to classical Boltzmann equations. We discuss the existence and uniqueness of solutions and provide a natural variational characterisation by casting the fuzzy Boltzmann equation into the framework of GENERIC systems (General Equations for Non-Equilibrium Reversible-Irreversible Coupling).

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arXiv2021

Turbulent cascades for a family of damped Szeg\"o equations

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szeg\"o equations. The cubic Szeg\"o equation has been introduced as a toy model for a totally non-dispersive degenerate Hamiltonian equation. It is a completely integrable system which develops growth of high Sobolev norms, detecting transfer of energy and hence cascades phenomena. Here, we consider a two-parameter family of variants of the cubic Szeg\"o equation and prove that adding a damping term unexpectedly promotes the existence of turbulent cascades. Furthermore, we give a panorama of the dynamics for such equations on a six-dimensional submanifold.

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arXiv2020

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear elliptic equation for the stream function. The density function and the viscosity coefficient may have large variations. In addition, we formulate the solutions for the parallel, concentric and radial flows respectively, and as examples we calculate the solutions with piecewise-constant viscosity coefficients explicitly.

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