arXiv subjects

Zhilei Liang

Publications and source records attributed to Zhilei Liang.

13 recordsLinked to original sources

arXiv2026

On weak solutions for the stationary Cahn-Hillard-Navier-Stokes equations with singular potential

The stationary Navier--Stokes--Cahn--Hilliard equations are considered, governing the motion of a compressible, two-phase fluid mixture with a diffuse interface. The free energy density in this paper has a singular logarithmic (Flory-uggins) form, ensuring that the mass fraction remains in the physical range and allowing for vacuum states. We prove the existence of weak solutions in a three-dimensional bounded domain under structural assumptions on the adiabatic exponent. The stationary setting poses two main mathematical challenges: the absence of an energy inequality driven by the evolution process to control the singular potential, and the degeneracy of the density near the vacuum. To address these issues, we introduce a specialized regularization of the logarithmic term that eliminates the quadratic growth induced by anti-diffusion, thereby restoring compactness. Uniform estimates are obtained through a special choice of artificial pressure and an interpolation argument that controls the desired norm of the density. A two-level limiting process then yields a weak solution that satisfies the physical bounds almost everywhere on the support of the density.To our knowledge, this is the first existence result for the steady compressible Navier--Stokes--Cahn--Hilliard system that incorporates both a singular free energy and vacuum regions.

math.AP↗

arXiv2025

Existence of traveling waves for vector valued gradient flows

Allen-Cahn equation is a fundamental continuum model that describes phase transitions in multi-component mixtures. We prove the existence of traveling waves for vector valued Allen-Cahn equations in the context of Ginzburg-Landau theories; in addition, we find the largest wave speed and provide its bounds from upper and below. Our method is based on a variation technique and can be applied to system of equations with a gradient flow structure.

math.AP↗

arXiv2022

Weak solutions to the equations of stationary compressible flows in active liquid crystals

The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the strong coupling of active particles and fluids, and some weighted estimates on the energy functions are achieved so that the weak solutions can be constructed for all values of the adiabatic exponent $γ>1$.

math.AP↗

arXiv2022

Long-time behavior of weak solutions for compressible Navier-Stokes equations with degenerate viscosity

The long-time regularity and asymptotic of weak solutions are studied for compressible Navier-Stokes equations with degenerate viscosity in a bounded periodic domain in two and three dimensions. It is shown that the density keeps strictly positive from below and above after a finite period of time. Moreover, higher velocity regularity is obtained via a parabolic type iteration technique. Since then the weak solution conserves its energy equality, and decays exponentially to the equilibrium in $L^{2}$-norm as time goes to infinity. In addition, assume that the initial momentum is zero, the exponential decay rate is derived for the derivative functions, and the weak solution becomes a strong one in two dimensional space.

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arXiv2021

Weak solutions to the stationary Cahn-Hillard/Navier-Stokes equations for compressible fluids

We are concerned with the Cahn-Hilliard/Navier-Stokes equations for the stationary compressible flows in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations describing the compressible fluid flows and the stationary Cahn-Hilliard type diffuse equation for the mass concentration difference. We prove the existence of weak solutions when the adiabatic exponent $γ$ satisfies $γ>\frac{4}{3}$. The proof is based on the weighted total energy estimates and the new techniques developed to overcome the difficulties from the capillary stress.

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arXiv2020

Stationary Cahn-Hilliard-Navier-Stokes equations for the diffuse interface model of compressible flows

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations for compressible fluids and a stationary Cahn-Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

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arXiv2020

A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity

We consider a vanishing viscosity sequence of weak solutions of the three-dimensional Navier--Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for Hölder continuous solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification.

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arXiv2019

Local strong solution for the viscous compressible and heat-conductive fluids with vacuum in 2D space

This paper considers the Cauchy problem of equations for the viscous compressible and heat-conductive fluids in the two-dimensional(2D) space. We establish the local existence theory of unique strong solution under some initial layer compatibility conditions. The initial data can be arbitrarily large, the initial density is allowed to vanish in any set and the far field state is assumed to be vacuum.

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arXiv2018

Energy equality in compressible fluids with physical boundaries

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrtρ \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.

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arXiv2017

Vanishing pressure limit for compressible Navier-Stokes equations with degenerate viscosities

In this paper we study a vanishing pressure process for highly compressible Navier-Stokes equations as the Mach number tends to infinity. We first prove the global existence of weak solutions for the pressureless system in the framework [Li-Xin, arXiv:1504.06826v2], where the weak solutions are established for compressible Navier-Stokes equations with degenerate viscous coefficients. Furthermore, a rate of convergence of the density in $L^{\infty}\left(0,T;L^{2}(\rr)\right)$ is obtained, in case when the velocity corresponds to the gradient of density at initial time.

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arXiv2017

Large-time behavior for spherically symmetric flow of viscous polytropic gas in exterior unbounded domain with large initial data

This paper deals with the spherically symmetric flow of compressible viscous and polytropic ideal fluid in unbounded domain exterior to a ball in $\mathbb{R}^n$ with $n\ge2$. We show that the global solutions are convergent as time goes to infinity. The critical step is obtaining the point-wise bound of the specific volume $v(x,t)$ and the absolute temperature $θ(x,t)$ from up and below both for $x$ and $t$. Note that the initial data can be arbitrarily large and, compared with \cite{nn}, our method applies to the spatial dimension $n=2.$ The proof is based on the elementary energy methods.

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arXiv2014

Some Uniform Estimates and Large-Time Behavior for One-Dimensional Compressible Navier-Stokes System in Unbounded Domains with Large Data

This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, the global solution is showed to be asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

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arXiv2013

On Classical Solutions to the Cauchy Problem of the Two-Dimensional Barotropic Compressible Navier-Stokes Equations with Vacuum

This paper concerns the Cauchy problem of the barotropic compressible Navier-Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.

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