arXiv subjects

Yachun Li

Publications and source records attributed to Yachun Li.

At least 19 recordsLinked to original sources

arXiv2025

Global well-posedness and asymptotic behavior of large strong solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent coefficients

It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this paper, we study the 3D Navier-Stokes equations with temperature-dependent coefficients in the whole space. When the initial density and the initial temperature are linearly equivalent to some large constant states, we establish the first result on the global existence of large strong solution. Moreover, the optimal decay rates of the solution to its associated equilibrium are established when the initial data belong to $L^{p_0}(\mathbb{R}^3)$ for some $p_0\in[1,2]$.

math.AP↗

arXiv2025

Initial boundary value problem for one-dimensional hyperbolic compressible Navier-Stokes equations

An initial boundary value problem for one-dimensional hyperbolic compressible Navier-Stokes equations is investigated. After transforming the system into Lagrangian coordinate, the resulting system possesses a structure with uniform characteristic boundary. By constructing an approximate system with non-characteristic boundary, we get a uniform global smooth solutions and obtain a global solution of the original problem by passing to a limit. Moreover, the global relaxation limit is also obtained.

math.AP↗

arXiv2024

$L^{p}$ Mild Solution to Stochastic Incompressible inhomogeneous Navier-Stokes Equations

In this paper, we establish the global $L^{p}$ mild solution of inhomogeneous incompressible Navier-Stokes equations in the torus $\mathbb{T}^{N}$ with $N<p<6$, $ 1 \leqslant N \leqslant 3$, driven by the Wiener Process. We introduce a new iteration scheme coupled the density $\rho$ and the velocity $\mathbf{u}$ to linearize the system, which defines a semigroup. Notably, unlike semigroups dependent solely on $x$, the generators of this semigroup depend on both time $t$ and space $x$. After demonstrating the properties of this time- and space-dependent semigroup, we prove the local existence and uniqueness of mild solution, employing the semigroup theory and Banach's fixed point theorem. Finally, we show the global existence of mild solutions by Zorn's lemma. Moreover, for the stochastic case, we need to use the operator splitting method to do some estimates separately.

math.AP↗

arXiv2024

Local classical solutions to Navier-Stokes equations with degenerate viscosities and vacuum

We consider the 3D isentropic compressible Navier-Stokes equations with degenerate viscousities and vacuum. The degenerate viscosities $\mu(\rho)$ and $\lambda(\rho)$ are proportional to some power of density, while the powers of density in $\mu(\rho)$ and $\lambda(\rho)$ are different(i.e., $\delta_1\neq \delta_2$). The local well-posedness of classical solution is established by introducing a ``quasi-symmetric hyperbolic''--``degenerate elliptic'' coupled structure to control the behavior of the velocity of the fluid near the vacuum and give some uniform estimates. In particular, the initial data allows vacuum in an open set and we do not need any initial compatibility conditions.

math.AP↗

arXiv2024

Existence and non-uniqueness of probabilistically strong solutions to 3D stochastic magnetohydrodynamic equations

We are concerned with the 3D stochastic magnetohydrodynamic (MHD) equations driven by additive noise on torus. For arbitrarily prescribed divergence-free initial data in $L^{2}_x$, we construct infinitely many probabilistically strong and analitically weak solutions in the class $L^{r}_{\Omega}L_{t}^{\gamma}W_{x}^{s,p}$, where $r>1$ and $(s, \gamma, p)$ lie in a supercritical regime with respect to the the Lady\v{z}henskaya-Prodi-Serrin (LPS) criteria. In particular, we get the non-uniqueness of probabilistically strong solutions, which is sharp at one LPS endpoint space. Our proof utilizes intermittent flows which are different from those of Navier-Stokes equations and derives the non-uniqueness even in the high viscous and resistive regime beyond the Lions exponent 5/4. Furthermore, we prove that as the noise intensity tends to zero, the accumulation points of stochastic MHD solutions contain all deterministic solutions to MHD solutions, which include the recently constructed solutions in [28, 29] to deterministic MHD systems.

math.AP↗

arXiv2024

Global Strong Solutions to the Cauchy Problem of Three-dimensional Isentropic Magnetohydrodynamics Equations with Large Initial Data

We consider the Cauchy problem to the three-dimensional isentropic compressible Magnetohydrodynamics (MHD) system with density-dependent viscosities. When the initial density is linearly equivalent to a large constant state, we prove that strong solutions exist globally in time, and there is no restriction on the size of the initial velocity and initial magnetic field.

math.AP↗

arXiv2024

Global Existence and Asymptotic Behavior of Large Strong Solutions to the 3D Full Compressible Navier-Stokes Equations with Density-dependent Viscosities

The purpose of this work is to investigate the Cauchy problem of global-in-time existence of large strong solutions to the Navier-Stokes equations for compressible viscous and heat conducting fluids. A class of density-dependent viscosity is considered. By introducing the modified effective viscous flux and using the bootstrap argument, we establish the global existence of large strong solution when the initial density is linearly equivalent to a large constant state. It is worthy of mentioning that, different from the work of Matsumura and Nishida (J. Math. Kyoto Univ., 1980) with small initial perturbation and the work of Huang and Li (Arch. Ration. Mech. Anal., 2018) with small energy but possibly large oscillations, our global large strong solution is uniform-in-time in $H^2$ Sobolev space and the uniform-in-time bounds of both density and temperature are obtained without any restrictions on the size of initial velocity and initial temperature. In addition, when the initial data belongs to $L^{p_0}\cap H^2$ with $p_0\in[1,2]$, we establish the convergence of the solution to its associated equilibrium with an explicit decay rate whether the initial data close to or far away from the equilibrium in the whole space. As a result, we give a specific large strong solution in Sobolev space satisfying the global existence assumptions proposed by Villani (Mem. Amer. Math. Soc., 2009), He, Huang, and Wang (Arch. Ration. Mech. Anal., 2019), Zhang and Zi (Ann. Inst. H. Poincare Anal. Non Lineaire, 2020) in studying the asymptotic behavior of solution, and extend the second above result to the nonisentropic case. This paper considers for the first time the application of Fourier splitting method to Navier-Stokes equations with variable viscosity.

math.AP↗

arXiv2024

Stationary solution to Stochastically Forced Euler-Poisson Equations in Bounded Domain: Part 2. 1-D Ohmic Contact Boundary

In this paper, we establish the asymptotic stability of the steady-state for a 1-D stochastic Euler-Poisson equations with Ohmic contact boundary conditions forced by the Wiener process. We utilize Banach's fixed point theorem and the a priori energy estimates uniformly in time to ensure the global existence of solutions around the steady state. In contrast to the deterministic case, the presence of stochastic forces lead to the lack of temporal derivatives of momentum, posing challenges for energy estimates. Furthermore, Ohmic contact boundary conditions pose greater challenges for energy estimates compared to systems with insulating boundary conditions. To address this issue, we establish asymptotic stability concerning the spatial derivatives through weighted energy estimates for the estimates of stochastic integrals, employing a technique distinct from that of the deterministic case. Furthermore, we demonstrate the existence of an invariant measure based on the a priori energy estimates. This invariant measure precisely corresponds to the Dirac measure generated by the steady state, due to the exponential decay of perturbed solutions around the steady state.

math.AP↗

arXiv2024

Stationary solution to Stochastically Forced Euler-Poisson Equations in Bounded Domain: Part 1. 3-D Insulating Boundary

This paper is concerned with $3$-D stochastic Euler-Poisson equations with insulating boundary conditions forced by the Wiener process. We first establish the global existence and uniqueness of the solution to the system, then we prove that the solution converges to its steady-state time-asymptotically. To obtain the converging rate, we need to develop weighted energy estimates, which are not required for the deterministic counterpart of the problem. Moreover, we observe that the invariant measure is just the Dirac measure generated by the steady-state, in which the time-exponential convergence rate to the steady-state plays an essential role.

math.AP↗

arXiv2024

Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations

This paper concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. These results in particular yield that the Lions exponent is the sharp viscosity threshold for the uniqueness/non-uniqueness in law of Leray solutions. Our proof utilizes the self-similarity and instability programme developed by Jia Šverák [42,43] and Albritton-Brué-Colombo [1], together with the theory of martingale solutions including stability for non-metric spaces and gluing procedure.

math.PR↗

arXiv2024

On blow-up to the one-dimensional Navier-Stokes equations with degenerate viscosity and vacuum

In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law(i.e., $μ(ρ)=ρ^δ, δ>1$). We first obtain the local existence of the regular solution, then show that the regular solution will blow-up in finite time if initial data has an isolated mass group, no matter how small and smooth the initial data are. It is worth mentioning that based on the transport structure of some intrinsic variables, we obtain the $L^\infty$ bound of the density, which helps to remove the restriction $δ\leq γ$ in Li-Pan-Zhu[21] and Huang-Wang-Zhu[13].

math.AP↗

arXiv2023

Approximations of Euler-Maxwell systems by drift-diffusion equations through zero-relaxation limits near non-constant equilibrium

Due to extreme difficulties in numerical simulations of Euler-Maxwell equations, which are caused by the highly complicated structures of the equations, this paper concerns the simplification of Euler-Maxwell system through the zero-relaxation limit towards the drift-diffusion equations with non-constant doping functions. We carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near non-constant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness arguments. Furthermore, stream function methods are carefully generalized to the non-constant equilibrium case, with which as well as the anti-symmetric structure of the error system and an induction argument, we establish global-in-time error estimates between smooth solutions to the Euler-Maxwell system and those to drift-diffusion system, which are bounded by some power of relaxation parameter.

math.AP↗

arXiv2023

Single Domain Dynamic Generalization for Iris Presentation Attack Detection

Iris presentation attack detection (PAD) has achieved great success under intra-domain settings but easily degrades on unseen domains. Conventional domain generalization methods mitigate the gap by learning domain-invariant features. However, they ignore the discriminative information in the domain-specific features. Moreover, we usually face a more realistic scenario with only one single domain available for training. To tackle the above issues, we propose a Single Domain Dynamic Generalization (SDDG) framework, which simultaneously exploits domain-invariant and domain-specific features on a per-sample basis and learns to generalize to various unseen domains with numerous natural images. Specifically, a dynamic block is designed to adaptively adjust the network with a dynamic adaptor. And an information maximization loss is further combined to increase diversity. The whole network is integrated into the meta-learning paradigm. We generate amplitude perturbed images and cover diverse domains with natural images. Therefore, the network can learn to generalize to the perturbed domains in the meta-test phase. Extensive experiments show the proposed method is effective and outperforms the state-of-the-art on LivDet-Iris 2017 dataset.

cs.CV↗

arXiv2022

Non-uniqueness for the hypo-viscous compressible Navier-Stokes equations

We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-Δ)^α$ with $α\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations (INS). In particular, in view of the Ladyženskaja-Prodi-Serrin criteria, the obtained non-uniqueness of $L^2_tC_x$ weak solutions to the hypo-viscous INS is sharp, and reveals that $α=1$ is the sharp viscosity threshold for the well-posedness in $L^2_tC_x$. Furthermore, we prove that the Hölder continuous weak solutions to the compressible Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of weak solutions to the hypo-viscous CNS.

math.AP↗

arXiv2022

Global Convergence and Error Estimates in Infinity-ion-mass Limits for Bipolar Euler-Poisson System

This paper is concerned with the global-in-time convergence from bipolar Euler-Poisson system (BEP) to unipolar one (UEP) through the infinity-ion-mass limit by letting the ratio of the mass of ion $m_i$ over that of electron $m_e$ goes to infinity. The global convergence of the limit is obtained for smooth solutions sufficiently close to constant equilibrium states. Furthermore, by applying the stream function method and taking advantage of the anti-symmetric structure of the error system, one obtains the corresponding global-in-time error estimates between smooth solutions of (BEP) and (UEP). It is worth mentioning that due to the strong coupling through the Poisson equation in bipolar system, stream functions for ions and electrons equations should be constructed separately based on asymptotic expansions of solutions, which is very different from the case of unipolar system.

math.AP↗

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↗