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Xueke Pu

Publications and source records attributed to Xueke Pu.

At least 19 recordsLinked to original sources
arXiv2025

Modulation approximation for the non-isentropic Euler-Poisson system

As a formal approximation, the nonlinear Schrödinger (NLS) equation can be derived to describe the evolution of the envelopes of small oscillating wave packets-like solutions to the Euler-Poisson system. In this paper we rigorously justify that the wave packets for the non-isentropic Euler-Poisson system can be approximated by solutions of the NLS equation over a physically relevant $\mathcal{O}(ε^{-2})$ time scale. Besides the difficulties such as resonances at $k=0$ and $k=\pm k_0$ and loss of derivatives arising in the modulation approximation problem in the isentropic Euler-Poisson system, new difficulties arise in the non-isentropic case. In the non-isentropic Euler-Poisson system, new resonances at wave number $k=\pm 2k_0$ appear which necessitate rescaling the correction to the modulation approximation differently for different wave numbers. In addition, it is more difficult to obtain the uniform estimates for the error $(R_{0},R_{1},R_{-1})$ between the real solutions and the approximate solutions, due to the extra interactions with the temperature. To overcome the difficulties aroused by resonances and loss of derivatives, we find several important structural identities between the diagonalized unknowns and apply a series of normal-form transforms, to obtain uniform estimates for the error over the desired $\mathcal{O}(ε^{-2})$ long time scale.

math.AP↗
arXiv2025

The bidirectional NLS approximation for the one-dimensional Euler-Poisson system

The nonlinear Schrödinger (NLS) equation is known as a universal equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet in various dispersive systems. In this paper, we prove that under a certain multiple scale transformation, solutions to the Euler-Poisson system can be approximated by the sums of two counter-propagating waves solving the NLS equations. It extends the earlier results [Liu and Pu, Comm. Math. Phys., 371(2), (2019)357-398], which justify the unidirectional NLS approximation to the Euler-Poisson system for the ion-coustic wave. We demonstrate that the solutions could be convergent to two counter-propagating wave packets, where each wave packet involves independently as a solution of the NLS equation. We rigorously prove the validity of the NLS approximation for the one-dimensional Euler-Poisson system by obtaining uniform error estimates in Sobolev spaces. The NLS dynamics can be observed at a physically relevant timespan of order $\mathcal{O}(ε^{-2})$. As far as we know, this result is the first construction and valid proof of the bidirectional NLS approximation.

math.AP↗
arXiv2025

Linear stability analysis of the Couette flow for 2D compressible Navier-Stokes-Poisson system

In this paper, we study the linear stability of Couette flow for 2D compressible Navier-Stokes-Poisson system at high Reynolds number in the domain $\mathbb{T}\times\mathbb{R}$ with initial perturbation in Sobolev spaces. We establish the upper bounds for the solutions of linearized system near Couette flow. In particular, we show that the irrotational component of the perturbation may have a transient growth, after which it decays exponentially.

math.AP↗
arXiv2024

Linear stability analysis of the Couette flow for the 2D Euler-Poisson system

This paper is concerned with the linear stability analysis for the Couette flow of the Euler-Poisson system for both ionic fluid and electronic fluid in the domain $\bb{T}\times\bb{R}$. We establish the upper and lower bounds of the linearized solutions of the Euler-Poisson system near Couette flow. In particular, the inviscid damping for the solenoidal component of the velocity is obtained.

math.AP↗
arXiv2022

Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation

The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by the hydrostatic approximation. On the other hand, the mathematically rigorous derivation of the primitive equations without coupling with the temperature is also known. In this paper, we generalize the above result from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergence rate $O(\varepsilon)$, as the aspect ratio $\varepsilon$ goes to zero.

math.AP↗
arXiv2022

Stability threshold for 2D shear flows near Couette of the Navier-Stokes equation

In this paper, we consider the stability threshold of the 2D shear flow $(U(y),0)^{\top}$ of the Navier-Stokes equation at high Reynolds number $Re$. When the shear flow is near in Sobolev norm to the Couette flow $(y,0)^{\top}$ in some sense, we prove that if the initial data $u_0$ satisfies $\|u_0-(U(y),0)^{\top}\|\leq εRe^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to the Couette flow for $t\gg Re^{1/3}$, as $t\to\infty$.

math.AP↗
arXiv2022

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is obtained under the assumption of initial data $(v_0,T_0) \in H^1$ with $\partial_z v_0 \in L^4$. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, in the cases of $H^1$ initial data, $H^1$ initial data with additional regularity $\partial_z v_0 \in L^4$ and $H^2$ initial data, respectively, as the aspect ration parameter $λ$ goes to zero, and the rate of convergence is of the order $O(λ^{η/2})$ with $η=\min\{2,β-2,γ-2\}(2<β,γ<\infty)$. The convergence result implies a rigorous justification of the hydrostatic approximation.

math.AP↗
arXiv2022

On the rigorous mathematical derivation for the viscous primitive equations with density stratification

In this paper, we rigorously derive the governed equations describing the motion of stable stratified fluid, from the mathematical point of view. Specially, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ration parameter goes to zero, and the rate of convergence is of the same order as the aspect ratio parameter. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

math.AP↗
arXiv2020

Stability threshold for 2D shear flows of the Boussinesq system near Couette

In this paper, we consider the stability threshold for the shear flows of the Boussinesq system in a domain $\mathbb{T} \times \mathbb{R}$. The main goal is to prove the nonlinear stability of the shear flow $(U^S,Θ^S)=((e^{νt\partial_{yy}}U(y),0)^{\top},αy)$ with $U(y)$ close to $y$ and $α\geq0$. We separate two cases: one is $α\geq 0$ small scaling with the viscosity coefficients and the case without smallness of $α$ and fixed heat diffusion coefficient. The novelty here is that we don't require $μ=ν$ and only need to assume that $μ$ is scaled with $ν$ or fixed, where $μ$ is the inverse of the Reynolds number and $ν$ is the heat diffusion coefficient.

math.AP↗
arXiv2019

Justification of the NLS approximation for ion Euler-Poisson equation

The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.

math.AP↗
arXiv2018

Partial regularity to the Landau-Lifshitz equation with spin accumulation

In this paper, we consider a model for the spin-magnetization system that takes into account the diffusion process of the spin accumulation. This model consists of the Landau-Lifshitz equation describing the precession of the magnetization, coupled with a quasi-linear parabolic equation describing the diffusion of the spin accumulation. This paper establishes the global existence and uniqueness of weak solutions for large initial data in $\Bbb R^2$. Moreover, partial regularity is shown. In particular, the solution is regular on $\Bbb R^2\times(0,\infty)$ with the exception of at most finite singular points.

math.AP↗
arXiv2017

The QKP limit of the quantum Euler-Poisson equation

In this paper, we consider the derivation of the Kadomtsev-Petviashvili (KP) equation for cold ion-acoustic wave in the long wavelength limit of the two-dimensional quantum Euler-Poisson system, under different scalings for varying directions in the Gardner-Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter $H>0$. The QKP-I is derived for $H>2$, QKP-II for $0<H<2$ and the dispersive-less KP (dKP) equation for the critical case $H=2$. The rigorous proof for these limits is given in the well-prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner-Morikawa transform. The results can be generalized in several directions.

math.AP↗
arXiv2016

Quasineutral limit for the quantum Navier-Stokes-Poisson equation

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.

math-ph↗
arXiv2015

Long wavelength limit for the quantum Euler-Poisson equation

In this paper, we consider the long wavelength limit for the quantum Euler-Poisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(ε^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.

math.AP↗
arXiv2015

Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.

math-ph↗
arXiv2013

Quasineutral limit of the Euler-Poisson equation for a cold, ion-acoustic plasma

In this paper, we consider the quasineutral limit of the Euler-Poisson equation for a clod, ion-acoustic plasma when the Debye length tends to zero. When the ion-acoustic plasma is cold, the Euler-Poisson equation is pressureless and hence fails to be Friedrich symmetrisable, which excludes the application of the classical energy estimates method. This brings new difficulties in proving uniform estimates independent of $\varepsilon$. The main novelty in this article is to introduce new $\varepsilon$-weighted norms of the unknowns and to combine energy estimates in different levels with weights depending on $\varepsilon$. Finally, that the quasineutral regimes are the incompressible Euler equations is proven for well prepared initial data.

math-ph↗
arXiv2012

Dispersive Limit of the Euler-Poisson System in Higher Dimensions

In this paper, we consider the dispersive limit of the Euler-Poisson system for ion-acoustic waves. We establish that under the Gardner-Morikawa type transformations, the solutions of the Euler-Poisson system converge globally to the Kadomtsev-Petviashvili II equation in $\Bbb R^2$ and the Zakharov-Kuznetsov equation in $\Bbb R^3$ for well-prepared initial data, under different scalings. This justifies rigorously the KP-II limit and the ZKE limit of the Euler-Poisson equation.

math-ph↗