arXiv subjects

Hairong Liu

Publications and source records attributed to Hairong Liu.

At least 19 recordsLinked to original sources

arXiv2025

Well-posedness of the 3-D compressible Navier-Stokes equations with density-dependent viscosities in exterior domains with far-field vacuum

This paper investigates the local existence and uniqueness of strong solutions to the three-dimensional compressible Navier-Stokes equations with density-dependent viscosities in exterior domains. When both the shear and bulk viscosity coefficients depend on the density in a power law ($\rho^{\delta}$ with $0<\delta<1$) and Navier-slip boundary condition on the velocity is imposed, base on a reformulation of the problem using new variables to handle the degeneracy near vacuum, we establish the local well-posedness of regular solutions with far-field vacuum in inhomogeneous Sobolev spaces. Compared to the Cauchy problem, the initial-boundary value problem requires establishing non-standard weighted estimates and handling the unavailability of boundary conditions for higher-order terms. Our approach addresses these challenges via the conormal space technique.

math.AP↗

arXiv2025

Strong solutions to the 3-D compressible MHD equations with density-dependent viscosities in exterior domains with far-field vacuum

This paper investigates the existence and uniqueness of local strong solutions to the three-dimensional compressible magnetohydrodynamic (MHD) equations with density-dependent viscosities in an exterior domain. The system models the dynamics of electrically conducting fluids, such as plasmas, and incorporates the effects of magnetic fields on fluid motion. We focus on the case where the viscosity coefficients are proportional to the fluid density, and the far-field density approaches vacuum. By introducing a reformulation of the problem using new variables to handle the degeneracy near vacuum, we establish the local well-posedness of strong solutions for arbitrarily large initial data, even in the presence of far-field vacuum. Our analysis employs energy estimates, elliptic regularity theory, and a careful treatment of the Navier-slip boundary conditions for the velocity and perfect conductivity conditions for the magnetic field. To the best of our knowledge, such results are not available even for the Cauchy problem to the 3-D compressible MHD equations with degenerate viscosities.

math.AP↗

arXiv2024

Stability of steady states of the 3-D Navier-Stokes-Poisson equations with non-flat doping profile in exterior domains

This paper concerns an initial boundary value problem of compressible Navier-Stokes-Poisson equations with the non-flat doping profile in a 3-D exterior domain.The global existence of strong solutions near a steady state for compressible Navier-Stokes-Poisson equations with the general Navier-slip boundary conditions is established. For our setting, not only a steady state should be constructed in the exterior domain by the sub and super solution method, but also some new techniques should be adopted to obtain a priori estimates, especially some refined elliptic estimates and the estimates on the boundary.

math.AP↗

arXiv2024

On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains

In this paper, one of our aims is to investigate the instability of the distribution of the critical point set $\mathcal{C}(u)$ of a solution $u$ to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that $\mathcal{C}(u)$ in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set $\mathcal{C}(u)$ is unstable when any boundary of planar concentric circle annular domain $Ω$ has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets $u^{-1}_θ(0)(u_θ=\nabla u\cdot θ)$ and $u^{-1}(0)$, we prove that the solution $u$ on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that $\mathcal{C}(u)$ only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary $γ_E$ is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution $u$ by the moving plane method.

math.AP↗

arXiv2023

Measure upper bounds of nodal sets of solutions to Dirichlet problem of Schr\"{o}dinger equations

In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential function, and $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,\alpha}$ for any $0<\alpha<1$. By developing a delicate dividing iteration procedure, we show that upper bound of the $(n-1)$-dimensional Hausdorff measure of the nodal set of $u$ in $\Omega$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau.

math.AP↗

arXiv2023

Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation \begin{equation*} Δ^2_{X}u=Vu, \end{equation*} where $Δ_{X}=Δ_{x}+|x|^{2α}Δ_{y}$ ($0<α\leq1$), with $x\in\mathbb{R}^{m}, y\in\mathbb{R}^{n}$, denotes the Baouendi-Grushin type subelliptic operators, and the potential $V$ satisfies the strongly singular growth assumption $|V|\leq \frac{c_0}{ρ^4}$, where \begin{equation*} ρ=\left(|x|^{2(α+1)}+(α+1)^2|y|^2\right)^{\frac{1}{2(α+1)}} \end{equation*} is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

math.AP↗

arXiv2023

Quantitative unique continuation property for solutions to a bi-Laplacian equation with a potential

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} Δ^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.

math.AP↗

arXiv2021

Global Solutions to an initial boundary problem for the compressible 3-D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which is not available in literature, to the best of knowledge of the authors'.

math.AP↗

arXiv2020

Context-aware Stand-alone Neural Spelling Correction

Existing natural language processing systems are vulnerable to noisy inputs resulting from misspellings. On the contrary, humans can easily infer the corresponding correct words from their misspellings and surrounding context. Inspired by this, we address the stand-alone spelling correction problem, which only corrects the spelling of each token without additional token insertion or deletion, by utilizing both spelling information and global context representations. We present a simple yet powerful solution that jointly detects and corrects misspellings as a sequence labeling task by fine-turning a pre-trained language model. Our solution outperforms the previous state-of-the-art result by 12.8% absolute F0.5 score.

cs.CL↗

arXiv2020

Incremental Text-to-Speech Synthesis with Prefix-to-Prefix Framework

Text-to-speech synthesis (TTS) has witnessed rapid progress in recent years, where neural methods became capable of producing audios with high naturalness. However, these efforts still suffer from two types of latencies: (a) the {\em computational latency} (synthesizing time), which grows linearly with the sentence length even with parallel approaches, and (b) the {\em input latency} in scenarios where the input text is incrementally generated (such as in simultaneous translation, dialog generation, and assistive technologies). To reduce these latencies, we devise the first neural incremental TTS approach based on the recently proposed prefix-to-prefix framework. We synthesize speech in an online fashion, playing a segment of audio while generating the next, resulting in an $O(1)$ rather than $O(n)$ latency.

cs.CL↗

arXiv2020

Global Solutions to the initial boundary problem of 3-D compressible Navier-Stokes-Poisson on bounded domains

The initial boundary value problems for compressible Navier-Stokes-Poisson is considered on a bounded domain in $\mathbb{R}^3$ in this paper. The global existence of smooth solutions near a given steady state for compressible Navier-Stokes-Poisson with physical boundary conditions is established with the exponential stability. An important feature is that the steady state (except velocity) and the background profile are allowed to be of large variation.

math.AP↗

arXiv2020

Simultaneous Translation Policies: From Fixed to Adaptive

Adaptive policies are better than fixed policies for simultaneous translation, since they can flexibly balance the tradeoff between translation quality and latency based on the current context information. But previous methods on obtaining adaptive policies either rely on complicated training process, or underperform simple fixed policies. We design an algorithm to achieve adaptive policies via a simple heuristic composition of a set of fixed policies. Experiments on Chinese -> English and German -> English show that our adaptive policies can outperform fixed ones by up to 4 BLEU points for the same latency, and more surprisingly, it even surpasses the BLEU score of full-sentence translation in the greedy mode (and very close to beam mode), but with much lower latency.

cs.CL↗

arXiv2019

Machine Translation in Pronunciation Space

The research in machine translation community focus on translation in text space. However, humans are in fact also good at direct translation in pronunciation space. Some existing translation systems, such as simultaneous machine translation, are inherently more natural and thus potentially more robust by directly translating in pronunciation space. In this paper, we conduct large scale experiments on a self-built dataset with about $20$M En-Zh pairs of text sentences and corresponding pronunciation sentences. We proposed three new categories of translations: $1)$ translating a pronunciation sentence in source language into a pronunciation sentence in target language (P2P-Tran), $2)$ translating a text sentence in source language into a pronunciation sentence in target language (T2P-Tran), and $3)$ translating a pronunciation sentence in source language into a text sentence in target language (P2T-Tran), and compare them with traditional text translation (T2T-Tran). Our experiments clearly show that all $4$ categories of translations have comparable performances, with small and sometimes ignorable differences.

cs.CL↗

arXiv2019

STACL: Simultaneous Translation with Implicit Anticipation and Controllable Latency using Prefix-to-Prefix Framework

Simultaneous translation, which translates sentences before they are finished, is useful in many scenarios but is notoriously difficult due to word-order differences. While the conventional seq-to-seq framework is only suitable for full-sentence translation, we propose a novel prefix-to-prefix framework for simultaneous translation that implicitly learns to anticipate in a single translation model. Within this framework, we present a very simple yet surprisingly effective wait-k policy trained to generate the target sentence concurrently with the source sentence, but always k words behind. Experiments show our strategy achieves low latency and reasonable quality (compared to full-sentence translation) on 4 directions: zh<->en and de<->en.

cs.CL↗

arXiv2019

Robust Machine Translation with Domain Sensitive Pseudo-Sources: Baidu-OSU WMT19 MT Robustness Shared Task System Report

This paper describes the machine translation system developed jointly by Baidu Research and Oregon State University for WMT 2019 Machine Translation Robustness Shared Task. Translation of social media is a very challenging problem, since its style is very different from normal parallel corpora (e.g. News) and also include various types of noises. To make it worse, the amount of social media parallel corpora is extremely limited. In this paper, we use a domain sensitive training method which leverages a large amount of parallel data from popular domains together with a little amount of parallel data from social media. Furthermore, we generate a parallel dataset with pseudo noisy source sentences which are back-translated from monolingual data using a model trained by a similar domain sensitive way. We achieve more than 10 BLEU improvement in both En-Fr and Fr-En translation compared with the baseline methods.

cs.CL↗

arXiv2019

Robust Neural Machine Translation with Joint Textual and Phonetic Embedding

Neural machine translation (NMT) is notoriously sensitive to noises, but noises are almost inevitable in practice. One special kind of noise is the homophone noise, where words are replaced by other words with similar pronunciations. We propose to improve the robustness of NMT to homophone noises by 1) jointly embedding both textual and phonetic information of source sentences, and 2) augmenting the training dataset with homophone noises. Interestingly, to achieve better translation quality and more robustness, we found that most (though not all) weights should be put on the phonetic rather than textual information. Experiments show that our method not only significantly improves the robustness of NMT to homophone noises, but also surprisingly improves the translation quality on some clean test sets.

cs.CL↗

arXiv2018

Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

In this paper, we mainly study the critical points and critical zero points of solutions $u$ to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distributions of connected components of the super-level sets $\{x\in Ω: u(x)>t\}$ and sub-level sets $\{x\in Ω: u(x)<t\}$ for some $t$, we obtain the geometric structure of interior critical point sets of $u$. Precisely, let $Ω$ be a multiply connected domain with the interior boundary $γ_I$ and the external boundary $γ_E$, where $u|_{γ_I}=ψ_1(x),~u|_{γ_E}=ψ_2(x)$. When $ψ_1(x)$ and $ψ_2(x)$ have $N_1$ and $N_2$ local maximal points on $γ_I$ and $γ_E$ respectively, we deduce that $\sum_{i = 1}^k {m_i}\leq N_1+ N_2$, where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$. In addition, when $\min_{γ_E}ψ_2(x)\geq \max_{γ_I}ψ_1(x)$ and $u$ has only $N_1$ and $N_2$ equal local maxima relative to $\overlineΩ$ on $γ_I$ and $γ_E$ respectively, we develop a new method to show that one of the following three results holds $\sum_{i = 1}^k {m_i}=N_1+N_2$ or $\sum_{i = 1}^k {m_i}+1=N_1+N_2$ or $\sum_{i = 1}^k {m_i}+2=N_1+N_2$. Moreover, we investigate the geometric structure of interior critical zero points of $u.$ We obtain that the sum of multiplicities of the interior critical zero points of $u$ is less than or equal to the half of the number of its isolated zero points on the boundaries.

math.AP↗

arXiv2018

Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $Ω$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane.

math.AP↗