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Yanlin Liu

Publications and source records attributed to Yanlin Liu.

33 records · Page 2Linked to original sources
arXiv2023

On the existence and structures of almost axisymmetric solutions to 3-D Navier-Stokes equations

In this paper, we consider 3-D Navier-Stokes equations with almost axisymmetric initial data, which means that by writing $u_0 =u^r_0 e_r+u^θ_0 e_θ+u^z_0 e_z$ in the cylindrical coordinates, then $\partial_θu^r_0,\,\partial_θu^θ_0$ and $\partial_θu^z_0$ are small in some sense (recall axisymmetric means these three quantities vanish). Then with additional smallness assumption on $u^θ_0$, we prove the global existence of a unique strong solution $u$, and this solution keeps close to some axisymmetric vector field. We also establish some refined estimates for the integral average in $θ$ variable for $u$. Moreover, as $u^r_0,\,u^θ_0$ and $u^z_0$ here depend on $θ$, it is natural to expand them into Fourier series in $θ$ variable. And we shall consider one special form of $u_0$, with some small parameter $\varepsilon$ to measure its swirl part and oscillating part. We study the asymptotic expansion of the corresponding solution, and the influences between different profiles in the asymptotic expansion. In particular, we give some special symmetric structures that will persist for all time. These phenomena reflect some features of the nonlinear terms in Navier-Stokes equations.

math.AP↗
arXiv2021

The influence of viscous coefficients on the lifespan of 3-D anisotropic Navier-Stokes system

The anisotropic Navier-Stokes system arises in geophysical fluid dynamics, which is derived by changing $-ν\D$ in the classical Navier-Stokes system to $-(ν_1\partial^2_1+ν_2\partial^2_2+ν_3\partial^2_3)$. Here $ν_1,\,ν_2\,,ν_3$ are the viscous coefficients, which can be different from each other. This reflects that the fluid can behave differently in each direction. The purpose of this paper is to derive some lower bound estimates on the lifespan to such anisotropic Navier-Stokes system. We not only investigate the case when $ν_1,\,ν_2\,,ν_3$ are all positive, but also the more sophisticated cases when one or two of them vanish. We find that in these lower bound estimates, the weights of $ν_1,\,ν_2\,,ν_3$ are not equal. A detailed study of this problem can also help us to have a better understanding of the nonlinear structure in the classical Navier-Stokes system.

math.AP↗
arXiv2021

Latest Studies of the SM Higgs Boson Couplings to Fermions at ATLAS

This proceeding presents the latest studies on the Yukawa couplings of the Standard Model Higgs boson with 139 fb$^{-1}$ data collected using the ATLAS detector at a center-of-mass energy of 13 TeV. A first direct probe of $CP$ violation in the top-quark Yukawa coupling using events where the Higgs boson is produced in association with top quarks ($t\bar{t}H$ and $tH$), and decays into two photons ($H \rightarrow γγ$ ) is discussed. The latest results on the Higgs boson production in association with a $W$ or $Z$ boson ($VH$) in the $H \rightarrow b \bar{b}$ channel are depicted as well. Finally, the searches for Higgs boson decays into two muons ($H \rightarrow μ^{+} μ^{-}$) and two electrons ($H \rightarrow e^{+} e^{-}$) are presented.

hep-ex↗
arXiv2020

Relative Equivariant Coarse Index Theorem and Relative $L^2$-Index Theorem

In this paper, we give a definition of the relative equivariant coarse index for proper actions and derive a relative equivariant coarse index theorem connecting this index with the localized equivariant coarse indices. This is an equivariant version of Roe's relative coarse index theorem in arXiv:arch-ive/1210.6100. Furthermore, we present a definition of the relative $L^2$-index and prove a relative $L^2$-index theorem which is a relative version of Atiyah's $L^2$-index theorem.

math.OA↗
arXiv2020

Elastic Cherenkov effects in transversely isotropic soft materials-I: Theoretical analysis, simulations and inverse method

A body force concentrated at a point and moving at a high speed can induce shear-wave Mach cones in dusty-plasma crystals or soft materials, as observed experimentally and named the elastic Cherenkov effect (ECE). The ECE in soft materials forms the basis of the supersonic shear imaging (SSI) technique, an ultrasound-based dynamic elastography method applied in clinics in recent years. Previous studies on the ECE in soft materials have focused on isotropic material models. In this paper, we investigate the existence and key features of the ECE in anisotropic soft media, by using both theoretical analysis and finite element (FE) simulations, and we apply the results to the non=invasive and non-destructive characterization of biological soft tissues. We also theoretically study the characteristics of the shear waves induced in a deformed hyperelastic anisotropic soft material by a source moving with high speed, considering that contact between the ultrasound probe and the soft tissue may lead to finite deformation. On the basis of our theoretical analysis and numerical simulations, we propose an inverse approach to infer both the anisotropic and hyperelastic parameters of incompressible transversely isotropic (TI) soft materials. Finally, we investigate the properties of the solutions to the inverse problem by deriving the condition numbers in analytical form and performing numerical experiments. In Part II of the paper, both ex vivo and in vivo experiments are conducted to demonstrate the applicability of the inverse method in practical use.

cond-mat.soft↗
arXiv2019

Global well-posedness of $3$-D anisotropic Navier-Stokes system with small unidirectional derivative

In \cite{LZ4}, the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier-Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier-Stokes system $(ANS)$ with only horizontal dissipation. More precisely, given initial data $u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12},$ $(ANS)$ has a unique global solution provided that $|D_\h|^{-1}\pa_3u_0$ is sufficiently small in the scaling invariant space $\cB^{0,\f12}.$

math.AP↗
arXiv2019

Semi-Supervised Brain Lesion Segmentation with an Adapted Mean Teacher Model

Automated brain lesion segmentation provides valuable information for the analysis and intervention of patients. In particular, methods based on convolutional neural networks (CNNs) have achieved state-of-the-art segmentation performance. However, CNNs usually require a decent amount of annotated data, which may be costly and time-consuming to obtain. Since unannotated data is generally abundant, it is desirable to use unannotated data to improve the segmentation performance for CNNs when limited annotated data is available. In this work, we propose a semi-supervised learning (SSL) approach to brain lesion segmentation, where unannotated data is incorporated into the training of CNNs. We adapt the mean teacher model, which is originally developed for SSL-based image classification, for brain lesion segmentation. Assuming that the network should produce consistent outputs for similar inputs, a loss of segmentation consistency is designed and integrated into a self-ensembling framework. Specifically, we build a student model and a teacher model, which share the same CNN architecture for segmentation. The student and teacher models are updated alternately. At each step, the student model learns from the teacher model by minimizing the weighted sum of the segmentation loss computed from annotated data and the segmentation consistency loss between the teacher and student models computed from unannotated data. Then, the teacher model is updated by combining the updated student model with the historical information of teacher models using an exponential moving average strategy. For demonstration, the proposed approach was evaluated on ischemic stroke lesion segmentation, where it improves stroke lesion segmentation with the incorporation of unannotated data.

cs.CV↗
arXiv2018

Global solutions of $3$-D Navier-Stokes system with small unidirectional derivative

Given initial data $u_0=(u_0^\h,u_0^3)\in H^{-\d,0}\cap H^{\f12}(\R^3)$ with both~$\uh_0$ and~$\nabla_{\rm h}\uh_0$ belonging to ~$L^2(\R^3)\cap L^\infty(\R_\v;L^2(\R^2_\h))$ and $u_0^\h\in L^\infty(\R_\v, H^{-\d}(\R^2_\h))$ for some $δ\in ]0,1[,$ if in addition $\pa_3u_0$ belongs to $H^{-\frac12,0}\cap H^{\frac12,0}(\R^3),$ we prove that the classical $3$-D Navier-Stokes system has a unique global Fujita-Kato solution provided that $\|\pa_3u_0\|_{H^{-\f12,0}}$ is sufficiently small compared to a constant which depends only on the norms of the initial data. In particular, this result provides some classes of large initial data which generate unique global solutions to 3-D Navier-Stokes system.

math.AP↗
arXiv2018

Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient

In this paper, we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. Then we shall construct a family of initial data, $u_{0,ν},$ which vary fast enough in the vertical variable and which are not small in the space, $BMO^{-1}.$ Yet $u_{0,ν}$ generates a unique global solution to the classical 3-D Navier-Stokes system provided that $ν$ is sufficiently large.

math.AP↗
arXiv2017

Critical one component anisotropic regularity for 3-D Navier-Stokes system

Let us consider an initial data $v_0$ for the classical 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p\in]4,\infty[,~q_1\in[1,2[,~μ>0, ~q_2\in\bigl[2,\bigl(1/p+μ\bigr)^{-1}\bigr[,~κ\in ]1,\infty[$, and any unit vector $e$, the $L^p$ estimate in time of $\bigl\|(v(t)|e)_{\mathbb{R}^3}\bigr\|_{L^{\frac{3p}{p-2}}}^p +\bigl\|(v(t)|e)_{\mathbb{R}^3}\bigr\|^p_{ \bigl(\dot{B}^{μ+\frac2p+\frac2{q_1}-1}_{q_1,κ}\bigr)_{\rm h} \bigl(\dot{B}^{\frac1{q_2}-μ}_{q_2,κ}\bigr)_{\rm v}}$ blows up at $T^\star$.

math.AP↗
arXiv2017

On the global well-posedness of 3-D axi-symmetric Navier-Stokes system with small swirl component

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier-Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.

math.AP↗
arXiv2015

On the critical one-component velocity regularity criteria to 3-D incompressible MHD system

Let $(u,b)$ be a smooth enough solution of 3-D incompressible MHD system. We prove that if $(u,b)$ blows up at a finite time $T^*$, then for any $p\in]4,\infty[$, there holds $\int_0^{T^*}\bigl(\|u^3(t')\|^p_{\dH^{\frac 12+\frac 2p}}+\|b(t')\|^p_{\dH^{\frac 12+\frac 2p}}\bigr)dt'=\infty$. We remark that all these quantities are in the critical regularity of the MHD system.

math.AP↗