arXiv subjects

Yachun Li

Publications and source records attributed to Yachun Li.

At least 37 records · Page 2Linked to original sources

arXiv2022

A Boundary Value Problem for a Class of Anisotropic Degenerate Parabolic-Hyperbolic Equations

We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for $L^2$-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove existence and uniqueness of the entropy solution. This is a revised corrected version of the paper published in Arch.\ Ration.\ Mech.\ Anal.\ {\bf 226} (2017), no. 3, 975--1008.

math.AP↗

arXiv2022

Asymptotic Decay of Besicovitch Almost Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

We prove the well-posedness and the asymptotic decay to the mean value of Besicovitch almost periodic entropy solutions to nonlinear aniso\-tropic degenerate parabolic-hyperbolic equations. After setting up the problem and its kinetic formulation on the Bohr compact, the main result, that is, the decay property, ia achieved by devising a suitable adaptation of the technique introduced by Chen and Perthame (2009) in their proof of the decay of periodic entropy solutions to the same equations.

math.AP↗

arXiv2022

Sharp non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^γ_tW^{s,p}_x$, where the exponents $(s,γ,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Ladyženskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regularity outside a small fractal singular set in time with zero $\mathcal{H}^{η_*}$-Hausdorff dimension, where $η_*$ can be any given small positive constant. Furthermore, we prove the strong vanishing viscosity and resistivity result, which yields the failure of Taylor's conjecture along some subsequence of weak solutions to the hyper viscous and resistive MHD beyond the Lions exponent.

math.AP↗

arXiv2022

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: above the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent $α$ can be larger than the Lions exponent $5/4$. It is well-known that, due to Lions [55], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $α\geq 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L^γ_tW^{s,p}_x$, in view of the generalized Ladyženskaja-Prodi-Serrin condition. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints $(3/p+1-2α, \infty, p)$ and $(2α/γ+1-2α, γ, \infty)$. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff $\mathcal{H}^{η_*}$ measure, where $η_*>0$ is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, the strong vanishing viscosity result is obtained for the hyperdissipative Navier-Stokes equations.

math.AP↗

arXiv2022

Unimodal-Concentrated Loss: Fully Adaptive Label Distribution Learning for Ordinal Regression

Learning from a label distribution has achieved promising results on ordinal regression tasks such as facial age and head pose estimation wherein, the concept of adaptive label distribution learning (ALDL) has drawn lots of attention recently for its superiority in theory. However, compared with the methods assuming fixed form label distribution, ALDL methods have not achieved better performance. We argue that existing ALDL algorithms do not fully exploit the intrinsic properties of ordinal regression. In this paper, we emphatically summarize that learning an adaptive label distribution on ordinal regression tasks should follow three principles. First, the probability corresponding to the ground-truth should be the highest in label distribution. Second, the probabilities of neighboring labels should decrease with the increase of distance away from the ground-truth, i.e., the distribution is unimodal. Third, the label distribution should vary with samples changing, and even be distinct for different instances with the same label, due to the different levels of difficulty and ambiguity. Under the premise of these principles, we propose a novel loss function for fully adaptive label distribution learning, namely unimodal-concentrated loss. Specifically, the unimodal loss derived from the learning to rank strategy constrains the distribution to be unimodal. Furthermore, the estimation error and the variance of the predicted distribution for a specific sample are integrated into the proposed concentrated loss to make the predicted distribution maximize at the ground-truth and vary according to the predicting uncertainty. Extensive experimental results on typical ordinal regression tasks including age and head pose estimation, show the superiority of our proposed unimodal-concentrated loss compared with existing loss functions.

cs.CV↗

arXiv2022

Few-shot One-class Domain Adaptation Based on Frequency for Iris Presentation Attack Detection

Iris presentation attack detection (PAD) has achieved remarkable success to ensure the reliability and security of iris recognition systems. Most existing methods exploit discriminative features in the spatial domain and report outstanding performance under intra-dataset settings. However, the degradation of performance is inevitable under cross-dataset settings, suffering from domain shift. In consideration of real-world applications, a small number of bonafide samples are easily accessible. We thus define a new domain adaptation setting called Few-shot One-class Domain Adaptation (FODA), where adaptation only relies on a limited number of target bonafide samples. To address this problem, we propose a novel FODA framework based on the expressive power of frequency information. Specifically, our method integrates frequency-related information through two proposed modules. Frequency-based Attention Module (FAM) aggregates frequency information into spatial attention and explicitly emphasizes high-frequency fine-grained features. Frequency Mixing Module (FMM) mixes certain frequency components to generate large-scale target-style samples for adaptation with limited target bonafide samples. Extensive experiments on LivDet-Iris 2017 dataset demonstrate the proposed method achieves state-of-the-art or competitive performance under both cross-dataset and intra-dataset settings.

cs.CV↗

arXiv2022

Non-uniqueness of weak solutions to 3D magnetohydrodynamic equations

We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, we prove that the weak solutions constructed by Beekie-Buckmaster-Vicol [2] for the ideal MHD can be obtained as a strong vanishing viscosity and resistivity limit of a sequence of weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishing viscosity and resistivity limits. Unlike in the context of the NSE [13] and the ideal MHD [2], new types of velocity and magnetic flows, featuring both the refined spatial and temporal intermittency, are constructed to respect the geometry of MHD and to control the strong viscosity and resistivity. Compatible algebraic structure is derived in the convex integration scheme. More interestingly, the new intermittent flows indeed enable us to prove the aforementioned results for the hyper-viscous and hyper-resistive MHD equations up to the sharp exponent $5/4$, which coincides exactly with the Lions exponent for 3D hyper-viscous NSE.

math.AP↗

arXiv2022

A Boundary Value Problem for a Class of Anisotropic Stochastic Degenerate Parabolic-Hyperbolic Equations

We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain $\cO=\cO'\X\cO''$ where a Neumann boundary condition is imposed on $\po\cO'\X\cO"$, the hyperbolic boundary, and a Dirichlet condition is imposed on $\cO'\X\po\cO"$, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem.

math.AP↗

arXiv2020

The Strong Trace Property and the Neumann Problem for Stochastic Conservation Laws

We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a major step for establishing the uniqueness of the kinetic solution to the referred problem we establish the new strong trace property for stochastic conservation laws. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.

math.AP↗

arXiv2019

On the global-in-time inviscid limit of the 3D isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

In the recent paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power ($(ρ^ε)^δ$ with $δ>1$), for regular solutions to the corresponding Cauchy problem, via introducing one "quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure to control the behavior of the velocity near the vacuum, we establish the uniform energy estimates for the local sound speed in $H^3$ and $(ρ^ε)^{\frac{δ-1}{2}}$ in $H^2$ with respect to the viscosity coefficients for arbitrarily large time under some smallness assumption on the initial density. Second, by making full use of this structure's quasi-symmetric property and the weak smooth effect on solutions, we prove the strong convergence of the regular solutions of the degenerate viscous flow to that of the inviscid flow with vacuum in $H^2$ for arbitrarily large time. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the B-D relation for viscosities, and seems to be the first on the global-in-time inviscid limit of smooth solutions which have large velocities and contain vacuum state for compressible flow in three space dimensions without any symmetric assumption.

math.AP↗

arXiv2019

Well-posedness of nonisentropic Euler equations with physical vacuum

We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum boundary. Our proof is based on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term. Then we construct the solutions to this degenerate parabolic problem and establish the estimates that are uniform with respect to the artificial viscosity parameter. Solutions to the compressible Euler equations are obtained as the limit of the vanishing artificial viscosity. Different from the isentropic case \cite{Coutand4, Lei}, our momentum equation of conservation laws has an extra term $p_{S}S_x$ that leads to some extra terms in the energy function and causes more difficulties even for the case of $γ=2$. Moreover, we deal with this free boundary problem starting from the general cases of $2\leqγ<3$ and $1<γ<2 $ instead of only emphasizing the isentropic case of $γ=2$ in \cite{Coutand4, jang1, Lei}.

math.AP↗

arXiv2019

Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible Navier-Stokes equations with density-dependent viscosities, arbitrarily large initial data and vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention that, via introducing a "quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure, we can also give some uniformly bounded estimates of $\displaystyle\Big(ρ^{\frac{γ-1}{2}}, u\Big)$ in $H^3$ space and $ρ^{\frac{δ-1}{2}}$ in $H^2$ space (adiabatic exponent $γ>1$ and $1<δ\leq \min\{3, γ\}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ (for any $s'\in [2, 3)$) space with the rate of $ε^{2(1-s'/3)}$. Further more, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications. This paper is based on our early preprint in 2015.

math.AP↗

arXiv2019

Re-Identification Supervised Texture Generation

The estimation of 3D human body pose and shape from a single image has been extensively studied in recent years. However, the texture generation problem has not been fully discussed. In this paper, we propose an end-to-end learning strategy to generate textures of human bodies under the supervision of person re-identification. We render the synthetic images with textures extracted from the inputs and maximize the similarity between the rendered and input images by using the re-identification network as the perceptual metrics. Experiment results on pedestrian images show that our model can generate the texture from a single image and demonstrate that our textures are of higher quality than those generated by other available methods. Furthermore, we extend the application scope to other categories and explore the possible utilization of our generated textures.

cs.CV↗

arXiv2018

Global existence of weak solutions for quantum MHD equations

In this paper, we consider the quantum MHD equations with both the viscosity coefficient and the magnetic diffusion coefficient are depend on the density. we prove the global existence of weak solutions and perform the lower planck limit in a 3-dimensional torus for large initial data. The global existence is shown by using Faedo-Galerkin method and weak compactness techniques for the adiabatic exponent $γ>1$.

math.AP↗

arXiv2015

On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids with viscosities depending on density in a power law and allowing initial vacuum. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by vacuum. Moreover, for certain classes of initial data with local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

math.AP↗

arXiv2015

On classical solutions to 2D Shallow water equations with degenerate viscosities

In this paper, the $2$-D isentropic Navier-Stokes systems for compressible fluids with density-dependent viscosity coefficients are considered. In particular, we assume that the viscosity coefficients are proportional to density. These equations, including several models in $2$-D shallow water theory, are degenerate when vacuum appears. We introduce the notion of regular solutions and prove the local existence of solutions in this class allowing the initial vacuum in the far field. This solution is further shown to be stable with respect to initial data in $H^2$ sense. A Beal-Kato-Majda type blow-up criterion is also established.

math.AP↗

arXiv2014

Existence results for compressible radiation hydrodynamics equations with vacuum

In this paper, we consider the 3-D compressible isentropic radiation hydrodynamics (RHD) equations. The local existence of strong solutions with vacuum is firstly established when the initial data is arbitrarily large, contains vacuum and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero, it may vanish in some open set or decay at infinity. We also prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we prove a blow-up criterion for the local strong solution. The similar result also holds for the general barotropic flow with pressure law $p_m=p_m(ρ)\in C^1(\mathbb{\overline{R}}^+)$.

math.AP↗

arXiv2014

On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum

In this paper, we discuss the Cauchy Problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we establish the local existence of regular solutions by the fundamental methods in the theory of quasi-linear symmetric hyperbolic systems under some physical assumptions. Then we give the non-global existence of regular solutions caused by the effect of vacuum for $1<γ\leq 3$. Finally, we extend our result to the initial-boundary value problem under some suitable boundary conditions. These blow-up results tell us that the radiation cannot prevent the formation of singularities caused by the appearance of vacuum.

math-ph↗