Searcharxiv⌕ Search

arXiv subjects

Zhi-Min Chen

Publications and source records attributed to Zhi-Min Chen.

18 recordsLinked to original sources
arXiv2026

Stability of purely convective steady-states of fractional Boussinesq equations in an exterior domain

A thermal convection flow in the three-dimensional unbounded fluid domain exterior to a sphere is considered. The viscosity force is determined by a fractional power of the Stokes operator. A purely conductive steady state arises due to the fluid heated from the sphere. A weak solution of the fluid motion problem is obtained and global stability of the steady-state solution in $L^2$ is provided.

math.AP↗
arXiv2025

Strong solutions of fractional Boussinesq equations in an exterior domain

A thermal convection fluid motion in the three-dimensional domain exterior to a sphere is considered. A purely conductive steady state arises due to the fluid heated from the sphere. A fractional equation system is introduced by using spectral presentation. The existence of small strong solutions in a Hilbert space is obtained. The strong solution existence implies the local stability of the steady state, which attracts asymptotically the flows evolving initially from the vector fields close to the steady state.

math.AP↗
arXiv2024

Enhanced and unenhanced dampings of the Kolmogorov flow

In the present study, Kolmogorov flow represents the stationary sinusoidal solution $(\sin y,0)$ to a two-dimensional spatially periodic Navier-Stokes system, driven by an external force. This system admits the additional non-stationary solution $(\sin y,0)+e^{-νt} (\sin y,0)$, which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity $ν$. Enhanced damping or enhanced dissipation of the problem is obtained by presenting higher decay rate for the difference between a solution and the non-stationary basic solution. Moreover, for the understanding of the metastability problem in an explicit manner, a variety of exact solutions are presented to show enhanced and unenhanced dampings.

math.AP↗
arXiv2024

Hopf bifurcation of a non-parallel Navier-Stokes flow

A plane non-parallel flow in a square fluid domain exhibits an odd number of vortices. A spectral structure is found to have a non-real solution of the spectral problem linearized around the flow. With the use of this structure, Hopf bifurcation or secondary time periodic flows branching of a basic square eddy flow is found. In contrast to a square eddy flow involving an even number of vortices in earlier analytical and experimental investigations, instability of the flow leads to steady-state bifurcations.

math-ph↗
arXiv2021

Bifurcating steady-state flows involving energy dissipation over a Hartmann boundary layer

A plane non-parallel vortex flow in a square fluid domain is examined. The energy dissipation of the flow is dominated by viscosity and linear friction effect of a Hartmann layer. This is a traditional Navier-Stokes flow when the linear friction effect is not involved, whereas it is a magnetohydrodynamic flow when the energy dissipation is fundamentally dominated by the friction. It is proved that linear critical values of a spectral problem are nonlinear thresholds leading to the onset of secondary steady-state flows, the nonlinear phenomenon observed in laboratory experiments.

math.AP↗
arXiv2021

Quasi-stationary solutions of the surface quasi-geostrophic equation

In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $κ(-Δ)^α$ ($κ>0$, $0\le α<1$), involved in the equation. When the equation is supercritical ($0\le α<\frac12$), the problem on the existence of large global regular solutions remains open. This study, however, provides explicit sample solutions for the understanding of the uncertain problem.

math.AP↗
arXiv2021

Energy stability of the Charney-DeVore quasi-geostrophic equation for atmospheric blocking

Charney and DeVore [J. Atmos. Sci. 36 (1979), 1205-1216] found multiple equilibrium states as a consequence of bottom topography in their pioneering work on the quasi-geostrophic barotropic flow over topography in a $β$-plane channel. In the present paper, we prove that the basic flow is asymptotically stable in a parameter region, including the flat topography situation, which excludes the existence of multiple equilibrium states therein. Moreover, we show that an additional condition on the average zonal force or the average zonal velocity is indispensable to the well-posedness of the Charney-DeVore quasi-geostrophic equation. Coexistence of at least three equilibrium states is confirmed by a pseudo-arclength continuation method for different topographic amplitudes. The stabilities of the equilibrium states are examined by high-resolution direct numerical simulations.

math.AP↗
arXiv2020

New formulation of the finite depth free surface Green function

For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [F. John, On the motion of floating bodies II. Simple harmonic motions, Communs. Pure Appl. Math. 3 (1950) 45-101] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain $(0, \infty)$. When the source point together with a field point is on the free surface, John's integral and its gradient are not convergent since the integration $\int^\infty_κ$ of the corresponding integrands does not tend to zero in a uniform manner as $κ$ tends to $\infty$. Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice.

physics.flu-dyn↗
arXiv2020

Global wellposedness to the $n$-dimensional compressible Oldroyd-B model without damping mechanism

The Cauchy problem of the compressible Oldroyd-B model without damping mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in density and stress tensor in the model is compensated by exploiting an intrinsic structure and introducing new quantities between density, velocity and stress tensor. Therefore, global solutions to the system with small initial data in critical Besov spaces are obtained. As a byproduct, optimal time decay rates of the solutions are derived by using an energy estimation argument. The results remain valid for the compressible viscoelastic system without the `div-curl structure assumption and thus improve those given by Hu and Wang [ J. Differential Equations, {\bf 250}, 1200--1231, 2011] and Qian and Zhang [Arch. Ration. Mech. Anal., {\bf 198}, 835--868, 2010].

math.AP↗
arXiv2019

Long-time behavior for three dimensional compressible viscous and heat-conductive gases

We study the large-time behavior of solutions to the compressible Navier-Stokes equations for a viscous and heat-conductive gases in $\mathbb{R}^3$. More precisely, under a suitable additional condition involving only the low frequencies of the initial data, we exhibit the optimal time decay rates for the constructed global solutions. The proof relies on some new energy arguments developed by Xin and Xu [39] for the compressible Navier-Stokes equations.

math.AP↗
arXiv2019

Global large solutions and incompressible limit for the compressible Navier-Stokes equations

The present paper is dedicated to the global large solutions and incompressible limit for the compressible Navier-Stokes system in $\mathbb{R}^d$ with $d\ge 2$. We aim at extending the work by Danchin and Mucha (Adv. Math., 320, 904--925, 2017) in $L^2$ structure to that in a critical $L^p$ framework. The result implies the existence of global large solutions initially from large highly oscillating velocity fields.

math.AP↗
arXiv2019

Instability of the Kolmogorov flow in a wall-bounded domain

In the magnetohydrodynamics (MHD) experiment performed by Bondarenko and his co-workers in 1979, the Kolmogorov flow loses stability and transits into a secondary steady state flow at the Reynolds number $R=O(10^3)$. This problem is modelled as a MHD flow bounded between lateral walls under slip wall boundary condition. The existence of the secondary steady state flow is now proved. The theoretical solution has a very good agreement with the flow measured in laboratory experiment at $R=O(10^3)$. Further transition of the secondary flow is observed numerically. Especially, well developed turbulence arises at $R=O(10^4)$.

math.AP↗
arXiv2018

Straightforward integration for free surface Green function and body wave motions

An alternative manner is provided for solving the classical linearised problem of the radiation and diffraction of regular water waves caused by oscillation of a floating body in deep water. It is shown that the singular wave integrals of the three-dimensional free surface Green function $G$ and its gradient $\nabla G$ can be regarded as regular wave integrals and are integrated directly. The method is validated by comparing with benchmark data for a floating or submerged body undergoing oscillatory wave motions. The comparison shows that the evaluation is sufficiently accurate for practical purposes. As the significance of the method, the numerical approximation stability for the gradient $\nabla G$ is shown to be the same with that for $G$.

physics.flu-dyn↗
arXiv2018

Ill-posedness of waterline integral of time domain free surface Green function for surface piercing body advancing at dynamic speed

In the linear time domain computation of a floating body advancing at a dynamic speed, the source formulation for the velocity potential of the hydrodynamic problem is commonly used so that the velocity potential is expressed as the integral of time domain free surface sources distributed on the two-dimensional wetted body surface and the one-dimensional waterline, which is the intersection of the wetted body surface and the mean free water surface. A time domain free surface source is corresponding to the time domain free surface Green function associated with a suitable source strength, which is to be solved from body boundary condition and normal velocity boundary integral equation of the source formulation. The normal velocity boundary integral equation contains an integral of the normal derivative of the time domain free surface Green function on the waterline. It is shown that the waterline integral is ill-posed. Thus the source strength of velocity potential is not obtainable.

physics.flu-dyn↗
arXiv2017

Steady-state bifurcation analysis of a strong nonlinear atmospheric vorticity equation

The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $κ>0$. It is proved that the steady-state solution is unique for $κ>1$ while multiple steady-state solutions exist for $κ<κ_{crit}$ with respect to critical value $κ_{crit}<1$. Without involvement of viscosity, the equation has strong nonlinearity as its nonlinear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semi-linear equations such as the Navier-Stokes equations but is not available for the quasi-geostrophic equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.

math.AP↗
arXiv2014

A Direct Inspection of the Displacement Current Using the Phase Measurement

After J. C. Maxwell brought forward the concept of displacement currents, H. R. Hertz and other scholars verified the existence of electromagnetic waves in experimental, and then confirmed indirectly the conceptive correctness of displacement currents. During the recent years, along with the evolution of electronic measurement technologies, the researchers are attempting to validate directly the amplitude and orientation of displacement currents in experimental. The paper proposes and achieves one phase measurement experiment to scrutinize the orientation of displacement currents. The study indicates that the existing measurement technology is capable of inspecting directly the amplitude and orientation of displacement currents. The test results do not locate on the predicted range of classical electromagnetic theory presently. The displacement current may not be treatable similar to the conductive current to a certain extent. This conclusion enriches the understanding to the property of displacement currents.

physics.gen-ph↗