Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Existence of standing and traveling waves in quantum hydrodynamics with viscosity

We prove existence of standing waves for two quantum hydrodynamics systems with linear and nonlinear viscosity. Moreover, global existence of traveling waves is proved for the former without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result in \cite{LMZ2020}, where it was required that the viscosity is sufficiently strong.

Traveling waves for quantum hydrodynamics with nonlinear viscosity

In this paper we study existence of traveling waves for 1-D compressible Euler system with dispersion (which models quantum effects through the Bohm potential) and nonlinear viscosity in the context of quantum hydrodynamic models for superfluidity. The existence of profiles is proved for appropriate (super- or sub- sonic) end states defining Lax shocks for the underlying Euler system formulated in terms of density and velocity without restrictions for the viscosity and dispersion parameters. On the other hand, the interplay of the dispersion and the viscosity plays a crucial role in proving th…

带粘性量子流体力学中驻波的数值谱分析

Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity

本文利用Evans函数技术研究了带线性和非线性粘性的量子流体力学系统在驻波附近的线性化谱，数值发现存在实不稳定特征值，为谱不稳定性提供了数值证据。

粘性-分散型冲击波轮廓在等熵可压缩流体中的存在性及谱稳定性分析

Existence and spectral stability analysis of viscous-dispersive shock profiles for isentropic compressible fluids of Korteweg type

本文研究了等熵可压缩流体中粘性-分散型冲击波轮廓的存在性及稳定性，证明了其唯一性及任意幅值特性，并分析了其谱稳定性。

On the vanishing electron-mass limit in plasma hydrodynamics in unbounded media

We consider the zero-electron-mass limit for the Navier-Stokes-Poisson system in unbounded spatial domains. Assuming smallness of the viscosity coefficient and ill-prepared initial data, we show that the asymptotic limit is represented by the incompressible Navier-Stokes system, with a Brinkman damping, in the case when viscosity is proportional to the electron-mass, and by the incompressible Euler system provided the viscosity is dominated by the electron mass. The proof is based on the RAGE theorem and dispersive estimates for acoustic waves, and on the concept of suitable weak solutions for…

具有Bohm势和线性黏性的量子流体动力学系统的适定性与衰减结构

Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity

本文研究一维量子流体动力学中的可压缩黏性-色散Euler系统，证明其局部适定性，并利用傅里叶空间补偿矩阵符号建立线性衰减估计，进而证明亚音速平衡态小扰动的全局存在与时间衰减。

$L^2$-收缩性与KdV-伯格斯方程的激波

$L^2$-contraction of Shock Waves for KdV-Burgers Equation

本文研究了KdV-伯格斯方程中粘性色散激波的$L^2$收缩性，证明了其时间渐进稳定性，并探讨了振荡激波的稳定性及结构性质。

On the Boundary-Domain Integral Equation Method for the Dirichlet Problem of the Compressible Stokes System with Variable Viscosity

We derive two systems of boundary-domain integral equations (BDIEs) equivalent to the Dirichlet problem for the compressible Stokes system using the potential method with an explicit parametrix (Levi function). The BDIEs are given in terms of surface and volume hydrodynamic potentials. We study the mapping properties of these integral potential operators and used them to prove existence and uniqueness of solution of the two systems of BDIEs obtained taking into account the non-trivial kernels of the single layer and hypersingular hydrodynamic surface potentials.

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.

Global Regularity for A Radiation Hydrodynamics Model with Viscosity and Thermal Conductivity

In this paper, we study the global wellposedness of a radiation hydrodynamics model with viscosity and thermal conductivity. It is now well-understood that, unlike the compressible Euler equations whose smooth solutions must blow up in finite time no matter how small and how smooth the initial data is, the dissipative structure of such a radiation hydrodynamics model can indeed guarantee that its one-dimensional Cauchy problem admits a unique global smooth solution provided that the initial data is sufficiently small, while for large initial data, even if the heat conductivity is taken into ac…

Analysis of acoustic oscillations for a class of hydrodynamic systems describing quantum fluids

Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relation is given by the Bogoliubov dispersion relation for weakly interacting Bose gases. We provide refined Strichartz estimates allowing for an accurate control of acoustic oscillations. Applications to the low Mach number limit for the quantum Navier-St…