arXiv subjects

Guangyi Hong

Publications and source records attributed to Guangyi Hong.

8 recordsLinked to original sources

arXiv2026

Global stability and anisotropic large-time behavior of the three-dimensional compressible Navier--Stokes equations with eddy diffusion

We study the Cauchy problem for the three-dimensional compressible Navier--Stokes equations with eddy diffusion, an anisotropic dissipative mechanism that arises naturally in geophysical fluid dynamics (cf.~\cite{Jabin-Bresch-2018,Temam-Ziane-2004}). In contrast to the classical compressible Navier--Stokes system, the momentum equation here carries no full vertical Laplacian: the velocity is diffused only in the horizontal directions, and the sole vertical regularization it receives is the partial one transmitted through the compressible mode $\operatorname{div}\mathbf{u}$. This degeneracy invalidates the standard parabolic energy framework as well as the classical high--low frequency Green-function bounds. We prove that the constant non-vacuum equilibrium $(\bar{\rho},0)$ is globally nonlinearly stable against small Sobolev perturbations: global classical solutions exist in $H^{N}(\mathbb{R}^{3})$ for every $N\ge 3$, and the density and velocity relax to equilibrium with explicit, genuinely anisotropic decay rates. The mechanism behind the result is a hidden dissipation produced by the pressure--divergence coupling between $\nabla\rho$ and $\operatorname{div}\mathbf{u}$, which compensates for the missing vertical smoothing of the density and the compressible part of the velocity; the solenoidal part of the velocity, by contrast, is governed by a purely horizontal heat flow and therefore decays only at the two-dimensional rate. The analysis rests on a refined anisotropic spectral decomposition of the Green matrix, a div--curl treatment of the velocity, and time-weighted nonlinear energy estimates tailored to the degenerate dissipation. To the best of our knowledge, this is the first global stability and large-time behavior result for the three-dimensional compressible Navier--Stokes equations with eddy diffusion in the whole space.

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arXiv2026

Stability of vertically charged steady magnetic field in 3D incompressible magneto-micropolar fluids without magnetic and angular viscosity in a strip domain

This paper intends to understand the regularity and stability problem on the 3D incompressible magneto-micropolar equations with zero magnetic and angular viscosities in a strip domain. The magneto-micropolar system models the electrically conducting micropolar fluid in the presence of a magnetic field. The lack of magnetic diffusion and angular dissipation makes it impossible to prove even small data global well-posedness result, let alone general large data global regularity. This paper presents a steady-state setup around which any perturbations can be shown to be globally regular and stable. More precisely, any small perturbation near a steady magnetic field perpendicular to the horizontal boundary leads to a unique global classical solution. In addition, the solution is shown to converge to the steady state at an almost exponential rate as time goes to infinity. These appear to be the very first rigorous global results on the magneto-micropolar equations concerned here.

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arXiv2024

Convergence of boundary layers of chemotaxis models with physical boundary conditions~II: Non-degenerate

This paper establishes the convergence of boundary-layer solutions of the consumption type Keller-Segel model with non-degenerate initial data subject to physical boundary conditions, which is a sequel of \cite{Corrillo-Hong-Wang-vanishing} on the case of degenerate initial data. Specifically, we justify that the solution with positive chemical diffusion rate $\varepsilon>0 $ converges to the solution with zero diffusion $\varepsilon=0 $ (outer-layer solution) plus the boundary-layer profiles (inner-layer solution) for any time $t>0$ as $ \varepsilon \rightarrow 0 $. Compared to \cite{Corrillo-Hong-Wang-vanishing}, the main difficulty in the analysis is the lack of regularity of the outer- and boundary-layer profiles since only the zero-order compatibility conditions for the leading-order boundary-layer profiles can be fulfilled with non-degenerate initial data. Our new strategy is to regularize the boundary-layer profiles with carefully designed corner-corrector functions and approximate the low-regularity leading-order boundary-layer profiles by higher-regularity profiles with regularized boundary conditions. By using delicate weight functions involving boundary-layer profiles to cancel the multi-scaled linear terms in the perturbed equations, we manage to obtain the requisite uniform-in-$ \varepsilon $ estimates for the convergence analysis. This cancellation technique enables us to prove the convergence to boundary-layer solutions for any time $ t >0 $, which is different from the convergence result in \cite{Corrillo-Hong-Wang-vanishing} which holds true only for some finite time depending on the Dirichlet boundary value.

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arXiv2024

Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data

The celebrated experiment of Tuval et al. \cite{tuval2005bacterial} showed that the bacteria living a water drop can form a thin layer near the air-water interface, where a so-called chemotaxis-fluid system with physical boundary conditions was proposed to interpret the mechanism underlying the pattern formation alongside numerical simulations. However, the rigorous proof for the existence and convergence of the boundary layer solutions to the proposed model still remains open. This paper shows that the model with physical boundary conditions proposed in \cite{tuval2005bacterial} in one dimension can generate boundary layer solution as the oxygen diffusion rate $\varepsilon>0$ is small. Specifically, we show that the solution of the model with $\varepsilon>0$ will converge to the solution with $\varepsilon=0$ (outer-layer solution) plus the boundary layer profiles (inner-layer solution) with a sharp transition near the boundary as $ \varepsilon \rightarrow 0$. There are two major difficulties in our analysis. First, the global well-posedness of the model is hard to prove since the Dirichlet boundary condition can not contribute to the gradient estimates needed for the cross-diffusion structure in the model. Resorting to the technique of taking anti-derivative, we remove the cross-diffusion structure such that the Dirichlet boundary condition can facilitate the needed estimates. Second, the outer-layer profile of bacterial density is required to be degenerate at the boundary as $ t \rightarrow 0 ^{+}$, which makes the traditional cancellation technique incapable. Here we employ the Hardy inequality and delicate weighted energy estimates to overcome this obstacle and derive the requisite uniform-in-$\varepsilon$ estimates allowing us to pass the limit $\varepsilon \to 0$ to achieve our results.

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arXiv2023

On the Splash Singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in $ \mathbb{R}^{3}$, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension $d\ge 2$.

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arXiv2021

Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions

In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize to the unique stationary solution as time tends to infinity against some suitable small perturbations. To the best of our knowledge, this is the first result obtained on the global well-posedness and asymptotic behavior of solutions to the exogenous chemotaxis system with physical boundary conditions.

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arXiv2020

Nonlinear stability of phase transition steady states to a hyperbolic-parabolic system modelling vascular networks

This paper is concerned with the existence and stability of phase transition steady states to a quasi-linear hyperbolic-parabolic system of chemotactic aggregation, which was proposed in \cite{ambrosi2005review, gamba2003percolation} to describe the coherent vascular network formation observed {\it in vitro} experiment. Considering the system in the half line $ \mathbb{R}_{+}=(0,\infty)$ with Dirichlet boundary conditions, we first prove the existence \textcolor{black}{and uniqueness of non-constant phase transition steady states} under some structure conditions on the pressure function. Then we prove that this unique phase transition steady state is nonlinearly asymptotically stable against a small perturbation. We prove our results by the method of energy estimates, the technique of {\it a priori} assumption and a weighted Hardy-type inequality.

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arXiv2015

Global classical solution to 3D compressible magnetohydrodynamic equations with large initial data and vacuum

In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $(γ-1)^{\frac{1}{6}}E_{0}^{\frac{1}{2}}$, together with the $\|H_{0}\|_{L^{2}}$, is suitably small, a result on the existence of global classical solutions is obtained. It should be pointed out that the initial energy $E_{0}$ except the $L^{2}$- norm of $H_{0}$ can be large as $γ$ goes to 1, and that throughout the proof of the theorem in the present paper, we make no restriction upon the initial data $(ρ_{0},u_{0})$. Our result improves the one established by Li-Xu-Zhang in \cite{H.L. L}, where, with small initial engergy, the existence of classical solution was proved.

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