arXiv subjects

Zheng-an Yao

Publications and source records attributed to Zheng-an Yao.

At least 19 recordsLinked to original sources

arXiv2026

Global uniform regularity and vanishing vertical viscosity limit for the compressible Navier--Stokes equations in the half-space

In geophysical flows such as large-scale ocean dynamics, the vertical viscosity is often much smaller than the horizontal viscosity. This anisotropy makes it natural to ask whether solutions of the full anisotropic compressible Navier--Stokes equations converge, as the vertical viscosity coefficient $\varepsilon \to 0$, to solutions of a horizontally dissipative limit system, and whether this limit can be justified globally in time. Prior work has answered this question locally in time or in the incompressible setting. We resolve this problem for the three-dimensional compressible Navier--Stokes equations in the upper half-space with the Navier slip boundary condition. This paper establishes two main results for small perturbations of a constant equilibrium state. First, we prove the existence of a unique global-in-time solution whose conormal Sobolev norm remains uniformly bounded for all $t \ge 0$ and all $\varepsilon \in (0,1)$. Second, we justify the global vanishing vertical viscosity limit. More precisely, we show that the solutions converge to a global solution of the horizontally dissipative compressible Navier--Stokes system. This provides the first rigorous justification of the anisotropic viscosity limit for compressible flows on an infinite time interval.

math.AP↗

arXiv2026

The perfect divisibility and chromatic number of some odd hole-free graphs

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least one edge admits a partition of $V(H)$ into sets $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. $G$ is short-holed if every hole in $G$ has length 4. A hammer is the graph obtained by identifying one vertex of a $K_3$ and one end vertex of a $P_3$. In this paper, we prove that (i) (odd hole, hammer, $K_{2,3}$)-free graphs are perfectly divisible, (ii) $χ(G)\le 4ω(G)(ω(G)-1)$ if $G$ is short-holed and $(K_1+C_4)$-free, (iii) $χ(G)\le 2ω(G)-1$ if $G$ is short-holed and $(K_1\cup K_3)$-free, and (iv) $χ(G)\le 16ω(G)-24$ if $G$ is short-holed and $(K_1+(K_1\cup K_3))$-free.

math.CO↗

arXiv2025

Stability and large-time behavior for the N-Dimensional Euler-FENE dumbbell model near an equilibrium

This paper studies the N-dimensional FENE dumbbell model without velocity dissipation, focusing on the stability and decay of perturbations near the steady solution $(0,\pin)$. Due to the lack of velocity dissipation, the above problems are highly challenging. In fact, without coupling, the corresponding N-dimensional Euler equation near u=0 is well known to be unstable. To overcome this difficulty, we analyze the wave structure arising in the system governing perturbations around the steady state, which originates from the equilibrium configuration and the coupling effects. This wave structure enables us to establish the global stability in the $H^s$-type Sobolev norms. Also, we highlight the critical role of wave structure in the decay estimates of the Euler-FENE dumbbell model. By combining this property with the Fourier splitting method, we derive the decay rate, which is identical to that of the general FENE dumbbell with velocity dissipation.

math.AP↗

arXiv2025

Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near a background magnetic field

This paper resolves the global regularity problem for the three-dimensional incompressible magnetohydrodynamics (MHD) equations in the upper half-space with slip boundary conditions, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of conormal derivatives with the decay of tangential derivatives. These global conormal regularity estimates, together with sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of a magnetic field, the global-in-time vanishing viscosity limit for the 3D incompressible Navier-Stokes equations with anisotropic dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the vanishing viscosity limit.

math.AP↗

arXiv2025

Incompressible limit for the 3D compressible FENE dumbbell model

In this work, we study the global-in-time incompressible limit of the compressible FENE dumbbell model on the three-dimensional torus T^3, where the incompressible limit is driven by large volume viscosity. To establish this limit, we develop time-weighted a priori estimates that yield decay rates for strong solutions. A key challenge arises from the fact that increasing the volume viscosity suppresses the decay of high-frequency components, thereby weakening the dissipation of the density and complicating the derivation of uniform-in-time decay estimates. To overcome this difficulty, we introduce a novel momentum-based estimate and show that the incompressible component of the momentum decays faster in time than the velocity itself. Exploiting this enhanced decay, we successfully close the a priori estimates and establish a time-decreasing convergence rate toward the incompressible limit.

math.AP↗

arXiv2025

Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near an equilibrium

This paper solves the global conormal regularity problem for the three-dimensional incompressible MHD equations with slip boundary condition near a background magnetic field. Motivated by applications in geophysics, the MHD system considered here is anisotropic with small vertical dissipation and small horizontal magnetic diffusion. By exploiting the enhanced dissipation due to the background magnetic field and introducing three layers of energy functionals, we are able to establish global-in-time uniform bounds that are independent of vertical viscosity and horizontal resistivity. These global conormal regularity estimates allow us to pass to the limit and obtain the convergence to the MHD system with no vertical dissipation and horizontal magnetic diffusion. In the special case of the 3D incompressible Navier-Stokes, explicit long-time rates are also extracted in the zero vertical viscosity limit.

math.AP↗

arXiv2024

Decay Rates for Viscous Surface Waves of Isotropic Micropolar Fluids With or Without Surface Tension

In this paper, we consider a layer of viscous incompressible isotropic micropolar fluid in a uniform gravitational field of finite depth, lying above a flat rigid bottom and below the atmosphere in a three-dimensional horizontally periodic setting. The fluid dynamics are governed by gravity-driven incompressible micropolar equations. We investigate the global well-posedness for both the cases with and without surface tension. On one hand, in the case with surface tension (i.e. σ > 0), we show that the global solution decays to the equilibrium exponentially. On the other hand, in the case without surface tension (i.e. σ = 0), the solution decays to the equilibrium at an almost exponential rate. Comparing the two different cases for σ > 0 and σ = 0 reveals that the surface tension can enhance the decay rate.

math.AP↗

arXiv2024

Local existence and uniqueness of solution to the two-dimensional inhomogeneous Prandtl equations by energy method

In this paper, we consider the local existence and uniqueness result for the inhomogeneous Prandtl equations in dimension two by energy method. First of all, for the homogeneous case, the local-in-time well-posedness theory of unsteady Prandtl equations was obtained by [Alexandre, Wang, Xu, Yang, J. Am. Math. Soc., 28 (3), 745-784 (2015)] and [Masmoudi, Wong, Comm. Pure Appl. Math., 68 (10), 1683-1741 (2015)] independently by energy method without any transformation. However, for the inhomogeneous case, the appearance of density will create some new difficulties for us to overcome the loss of tangential derivative of horizontal velocity. Thus, our first result is to overcome the loss of tangential derivative such that one can establish the local-in-time well-posedness result for the inhomogeneous Prandtl equations by energy method. Secondly, for the homogeneous case, the local-in-x well-posedness in higher regular space for the steady Prandtl equations was obtained by [Guo, Iyer, Comm. Math. Phys., 382 (3), 1403-447 (2021)] by energy method since they firstly found the good quantity(called `quotient'). With the help of this quotient, our second result is to establish the local-in-x well-posedness in higher regular Sobolev space for the steady inhomogeneous Prandtl equations.

math.AP↗

arXiv2023

Well-posedness and exponential stability of the inhomogeneous anisotropic incompressible Navier-Stokes equation with far-field vacuum in two-dimensional whole space

In this paper, we investigate the well-posedness theory and exponential stability for the inhomogeneous incompressible Navier-Stokes equation with only horizontal dissipative structure. Due to the lack of the vertical dissipative term and appearance of vacuum, it is a highly challenging tricky problem for us to study the well-posedness, stability and large-time behavior problems in two-dimensional whole space. The local-in-time well-posedness theory is successfully established at first because we develop some good estimates for the density and vorticity to control the nonlinear term. Finally, these good estimates of density and vorticity help us to establish the global-in-time well-posedness and exponential stability if the initial velocity is suitable small.

math.AP↗

arXiv2023

Global Stability of a PDE-ODE model for acid-mediated tumor invasion

In this paper, we study the global dynamics of a general reaction-diffusion model based on acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. A key feature of this model is the density-limited tumor diffusion term for tumor cells, which might give rise to the degeneracy of the parabolic equation. Our theoretical results characterize the effects of acid resistance and mutual competition of healthy cells and tumor cells on tumor progression in the long term, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. The approach relies on the construction of suitable Lyapunov functionals and upper/lower solutions.

math.AP↗

arXiv2022

Optimal decay of full compressible Navier-Stokes equations with potential force

In this paper, we aim to investigate the optimal decay rate for the higher order spatial derivative of global solution to the full compressible Navier-Stokes (CNS) equations with potential force in $\mathbb{R}^3$. We establish the optimal decay rate of the solution itself and its spatial derivatives (including the highest order spatial derivative) for global small solution of the full CNS equations with potential force. With the presence of potential force in the considered full CNS equations, the difficulty in the analysis comes from the appearance of non-trivial ststionary solutions. These decay rates are really optimal in the sense that it coincides with the rate of the solution of the linerized system. In addition, the proof is accomplished by virtue of time weighted energy estimate, spectral analysis, and high-low frequency decomposition.

math.AP↗

arXiv2021

Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime

In this paper, we investigate the higher regularity and asymptotic behavior for the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Due to the degeneracy of horizontal velocity near boundary, the higher regularity of solution is a tricky problem. By constructing suitable approximated system and establishing closed energy estimate for a good quantity(called "quotient" in \cite{Guo-Iyer-2021}), our first result is to solve this higher regularity problem. Furthermore, we show the global well-posedness and global-in-$x$ asymptotic behavior when the initial data are small perturbation of the classical Hartmann layer in Sobolev space. By using the energy method to establish closed estimate for the quotient, we overcome the difficulty arising from the degeneracy of horizontal velocity near boundary. Due to the damping effect, we also point out that this global solution will converge to the equilibrium state(called Hartmann layer) with exponent decay rate.

math.AP↗

arXiv2021

Optimal decay of compressible Navier-Stokes equations with or without potential force

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, it has been shown in \cite{guo2012} that the $N$-th order spatial derivative of global small solution of the CNS equations without potential force tends to zero with the $L^2-$rate $(1+t)^{-(s+N-1)}$ when the initial perturbation around the constant equilibrium state belongs to $H^N(\mathbb{R}^3)\cap \dot H^{-s}(\mathbb{R}^3)(N \ge 3 \text{~and~} s\in [0, \frac32))$. Thus, our first result improves this decay rate to $(1+t)^{-(s+N)}$. Secondly, we establish the optimal decay rate for the global small solution of the CNS equations with potential force as time tends to infinity. These decay rates for the solution itself and its spatial derivatives are really optimal since the upper bounds of decay rates coincide with the lower ones.

math.AP↗

arXiv2021

The Optimal Decay Rate of Strong Solution for the Compressible Nematic Liquid Crystal Equations with Large Initial Data

This paper is devoted to establishing the optimal decay rate of the global large solution to compressible nematic liquid crystal equations when the initial perturbation is large and belongs to $L^1(\mathbb R^3)\cap H^2(\mathbb R^3)$. More precisely, we show that the first and second order spatial derivatives of large solution $(ρ-1, u, \nabla d)(t)$ converges to zero at the $L^2-$rate $(1+t)^{-\frac54}$ and $L^2-$rate $(1+t)^{-\frac74}$ respectively, which are optimal in the sense that they coincide with the decay rates of solution to the heat equation. Thus, we establish optimal decay rate for the second order derivative of global large solution studied in [12,18] since the compressible nematic liquid crystal flow becomes the compressible Navier-Stokes equations when the director is a constant vector. It is worth noticing that there is no decay loss for the highest-order spatial derivative of solution although the associated initial perturbation is large. Moreover, we also establish the lower bound of decay rates of $(ρ-1, u, \nabla d)(t)$ itself and its spatial derivative, which coincide with the upper one. Therefore, the decay rates of global large solution $\nabla^2(ρ-1,u,\nabla d)(t)$ $(k=0,1,2)$ are actually optimal.

math.AP↗

arXiv2021

Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.

math.AP↗

arXiv2020

Decay of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data

In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier-Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the first order spatial derivative of large solution of the full compressible Navier-Stokes equations converging to zero in $L^2-$norm is $(1+t)^{-5/4}$, which coincides with the heat equation. This extends the previous decay rate $(1+t)^{-3/4}$ obtained in \cite{he-huang-wang2}.

math.AP↗

arXiv2020

Optimal decay for the full compressible Navier-Stokes system in critical $L^p$ Besov spaces

Danchin and He (Math. Ann. 64: 1-38, 2016) recently established the global existence in critical $L^p$-type regularity framework for the $N$-dimensional $(N\geq 3)$ non-isentropic compressible Navier-Stokes equations. The purpose of this paper is to further investigate the large time behavior of solutions constructed by them. More precisely, we prove that if the initial data at the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-σ_1}$ with $σ_1\in (2-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^s$ norm of the critical global solutions exhibits the optimal decay $(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{s+σ_1}{2}}$ for suitable $p$ and $s$. The main tool we use is the pure energy argument without the spectral analysis, which enables us to \emph{remove the smallness assumption} of initial data at the low-frequency.

math.AP↗

arXiv2019

The Optimal Decay Rate of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data

In recent paper 5, it is shown that the upper decay rate of global solution of compressible Navier-Stokes(CNS) equations converging to constant equilibrium state $(1, 0)$ in $H^1-$norm is $(1+t)^{\frac34(\frac{2}{p}-1)}$ when the initial data is large and belongs to $H^2(\mathbb{R}^3) \cap L^p(\mathbb{R}^3) (p\in[1,2))$. Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in $H^1-$norm is $(1+t)^{-\frac32(\frac1p-\frac12)-\frac12}$. For the case of $p=1$, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state $(1, 0)$ in $L^2-$norm is $(1+t)^{-\frac{3}{4}}$ if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in $L^2-$norm is $(1+t)^{-\frac{3}{4}}$ although the associated initial data is large.

math.AP↗