arXiv subjects

Yinghui Zhang

Publications and source records attributed to Yinghui Zhang.

26 records · Page 2Linked to original sources

arXiv2022

On instability of a generic compressible two-fluid model in $\mathbb R^3$

We are concerned with the instability of a generic compressible two-fluid model in the whole space $\mathbb{R}^3$, where the capillary pressure $f(α^-ρ^-)=P^+-P^-\neq 0$ is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, $f'(1)<0$, Evje-Wang-Wen established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu-Yao-Zhang proved global stability of the constant equilibrium state for the case $P^+=P^-$ (corresponding to $f'(1)=0$). In this work, we investigate the instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, $f'(1)>0$. First, by employing Hodge decomposition technique and making detailed analysis of the Green's function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space $H^k$, thus leading to a global instability result for the linearized problem. Moreover, with the help of the global linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can then show the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup. Therefore, our result shows that for the case $f'(1)>0$, the constant equilibrium state of the two-fluid model is linearly globally unstable and nonlinearly locally unstable in the sense of Hadamard, which is in contrast to the cases $f'(1)<0$ and $P^+=P^-$ (corresponding to $f'(1)=0$) where the constant equilibrium state of the two--fluid model is nonlinearly globally stable.

math.AP↗

arXiv2022

Global well--posedness and large time behavior of classical solutions to a generic compressible two-fluid model

In this paper, we investigate a generic compressible two-fluid model with common pressure ($P^+=P^-$) in $\mathbb{R}^3$. Under some smallness assumptions, Evje-Wang-Wen [Arch Rational Mech Anal 221:1285--1316, 2016] obtained the global solution and its optimal decay rate for the 3D compressible two-fluid model with unequal pressures $P^+\neq P^-$. More precisely, the capillary pressure $f(α^-ρ^-)=P^+-P^-\neq 0$ is taken into account, and is assumed to be a strictly decreasing function near the equilibrium. As indicated by Evje-Wang-Wen, this assumption played an key role in their analysis and appeared to have an essential stabilization effect on the model. However, global well-posedness of the 3D compressible two-fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we establish global existence and large time behavior of classical solutions to the 3D compressible two-fluid model with common pressure. One of key observations here is that to closure the higher-order energy estimates of non-dissipative variables (i.e, fraction densities $α_{\pm}ρ_\pm$), we will introduce the linear combination of two velocities ($u^\pm$): $v=(2μ^++λ^+)u^+-(2μ^-+λ^-)u^-$ and explore its good regularity, which is particularly better than ones of two velocities themselves.

math.AP↗

arXiv2021

Global existence and decay rates for a generic compressible two-fluid model

We investigate global existence and optimal decay rates of a generic non-conservative compressible two--fluid model with general constant viscosities and capillary coefficients.The main novelty of this work is three--fold: First, for any integer $\ell\geq3$, we show that the densities and velocities converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and the $k$($\in [1, \ell]$)--order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, which are the same as ones of the compressible Navier--Stokes system, Navier--Stokes--Korteweg system and heat equation. Second, the linear combination of the fraction densities ($β^+α^+ρ^++β^-α^-ρ^-$) converges to its corresponding equilibrium state at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and its $k$($\in [1, \ell]$)--order spatial derivative converges to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, but the fraction densities ($α^\pmρ^\pm$) themselves converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{1}{4}}$, and the $k$($\in [1, \ell]$)--order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{1}{4}-\frac{k}{2}}$, which are slower than ones of their linear combination ($β^+α^+ρ^++β^-α^-ρ^-$) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well--chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two--fluid model.

math.AP↗

arXiv2021

Optimal large time behavior of the compressible Bipolar Navier--Stokes--Poisson system with unequal viscosities

This paper is concerned with the Cauchy problem of the 3D compressible bipolar Navier--Stokes--Poisson (BNSP) system with unequal viscosities, and our main purpose is three--fold: First, under the assumption that $H^l\cap L^1$($l\geq 3$)--norm of the initial data is small, we prove the optimal time decay rates of the solution as well as its all--order spatial derivatives from one--order to the highest--order, which are the same as those of the compressible Navier--Stokes equations and the heat equation. Second, for well--chosen initial data, we also show the lower bounds on the decay rates. Therefore, our time decay rates are optimal. Third, we give the explicit influences of the electric field on the qualitative behaviors of solutions, which are totally new as compared to the results for the compressible unipolar Navier--Stokes--Poisson(UNSP) system [Li et al., in Arch. Ration. Mech. Anal., {196} (2010), 681--713; Wang, J. Differ. Equ., { 253} (2012), 273--297]. More precisely, we show that the densities of the BNSP system converge to their corresponding equilibriums at the same $L^2$--rate $(1+t)^{-\frac{3}{4}}$ as the compressible Navier--Stokes equations, but the momentums of the BNSP system and the difference between two densities decay at the $L^2$--rate $(1+t)^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})}$ and $(1+t)^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}}$ with $1\leq p\leq \frac{3}{2}$, respectively, which depend directly on the initial low frequency assumption of electric field, namely, the smallness of $\|\nabla ϕ_0\|_{L^p}$.

math.AP↗

arXiv2020

Learning Order Parameters from Videos of Dynamical Phases for Skyrmions with Neural Networks

The ability to recognize dynamical phenomena (e.g., dynamical phases) and dynamical processes in physical events from videos, then to abstract physical concepts and reveal physical laws, lies at the core of human intelligence. The main purposes of this paper are to use neural networks for classifying the dynamical phases of some videos and to demonstrate that neural networks can learn physical concepts from them. To this end, we employ multiple neural networks to recognize the static phases (image format) and dynamical phases (video format) of a particle-based skyrmion model. Our results show that neural networks, without any prior knowledge, can not only correctly classify these phases, but also predict the phase boundaries which agree with those obtained by simulation. We further propose a parameter visualization scheme to interpret what neural networks have learned. We show that neural networks can learn two order parameters from videos of dynamical phases and predict the critical values of two order parameters. Finally, we demonstrate that only two order parameters are needed to identify videos of skyrmion dynamical phases. It shows that this parameter visualization scheme can be used to determine how many order parameters are needed to fully recognize the input phases. Our work sheds light on the future use of neural networks in discovering new physical concepts and revealing unknown yet physical laws from videos.

cs.LG↗

arXiv2020

Optimal decay rates of a non-conservative compressible two-phase fluid model

We are concerned with the time decay rates of strong solutions to a non-conservative compressible viscous two-phase fluid model in the whole space R3. Compared to the previous related works, the main novelty of this paper lies in the fact that it provides a general framework that can be used to extract the optimal decay rates of the solution as well as its all-order spatial derivatives from one-order to the highest-order, which are the same as those of the heat equation. Furthermore, for well-chosen initial data, we also show the lower bounds on the decay rates. Our methods mainly consist of Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis and energy method based on finite induction.

math.AP↗

arXiv2019

Restricted Linearized Augmented Lagrangian Method for Euler's Elastica Model

Euler's elastica model has been extensively studied and applied to image processing tasks. However, due to the high nonlinearity and nonconvexity of the involved curvature term, conventional algorithms suffer from slow convergence and high computational cost. Various fast algorithms have been proposed, among which, the augmented Lagrangian based ones are very popular in the community. However, parameter tuning might be very challenging for these methods. In this paper, a simple cutting-off strategy is introduced into the augmented Lagrangian based algorithms for minimizing the Euler's elastica energy, which leads to easy parameter tuning and fast convergence. The cutting-off strategy is based on an observation of inconsistency inside the augmented Lagrangian based algorithms. When the weighting parameter of the curvature term goes to zero, the energy functional boils down to the ROF model. So, a natural requirement is that its augmented Lagrangian based algorithms should also approach the augmented Lagrangian based algorithms formulated directly for solving the ROF model from the very beginning. Unfortunately, this is not the case for certain existing augmented Lagrangian based algorithms. The proposed cutting-off strategy helps to decouple the tricky dependence between the auxiliary splitting variables, so as to remove the observed inconsistency. Numerical experiments suggest that the proposed algorithm enjoys easier parameter-tuning, faster convergence and even higher quality of image restorations.

eess.IV↗

arXiv2011

Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Let $λ^{*}>0$ denote the largest possible value of $λ$ such that $$ \{{array}{lllllll} Δ^{2}u=λ(1+u)^{p} & {in}\ \ \B, %0 \frac{n+4}{n-4}$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the unit ball.

math.AP↗