We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem is globally in state space. Furthermore, non-increasing energy at the junction and a maximum principle are proven. A numerical example is given in which the new conditions are the only known conditions leading to the physically correct wave types. The approach generalizes to full gas dynamics.
Consider the linear stability of the three dimensional isentropic compressible Navier-Stokes equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We prove the enhanced dissipation phenomenon for the linearized isentropic compressible Navier-Stokes equations around the Couette flow $(y, 0, 0)^\top$. Moreover, the lift-up phenomenon is also shown in this paper. Compared with the 3D incompressible Navier-Stokes equations [Ann. of Math.,185(2017), 541--608], the lift-up effect here is stronger due to the loss of the incompressible condition.
This note is devoted to the linear stability of the Couette flow for the non-isentropic compressible Euler equations in a domain $\mathbb{T}\times \mathbb{R}$. Exploiting the several conservation laws originated from the special structure of the linear system, we obtain a Lyapunov type instability for the density, the temperature, the compressible part of the velocity field, and also obtain an inviscid damping for the incompressible part of the velocity field.
In this paper, we consider the initial-boundary value problem of three-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms in an exterior domain with Navier-slip boundary condition and with far-field vacuum. This problem is related to the motion of the compressible viscous flow past a rotating obstacle. We establish the global existence and uniqueness of classical solutions, provided that the initial mass is small. The initial data and the angular velocity of the obstacle are allowed to have large oscillations.
We are concerned with the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain $\mathbb{T}\times \mathbb{R}$. For a general initial data settled in Sobolev spaces, we obtain a Lyapunov type instability of the density, the temperature, the compressible part of the velocity field, and also obtain an inviscid damping for the incompressible part of the velocity field. Moreover, if the initial density, the initial temperature and the incompressible part of the initial velocity field satisfy some quality relation, …
We are concerned with global weak solutions to the isentropic compressible Euler equations with cylindrically symmetric rotating structure, in which the origin is included. Due to the presence of the singularity at the origin, only the case excluding the origin $|\vec{x}|\geq1$ has been considered by Chen-Glimm \cite{Chen3}. The convergence and consistency of the approximate solutions are proved by using $L^{\infty}$ compensated compactness framework and vanishing viscosity method. We observe that if the blast wave initially moves outwards and if initial density and velocity decay to zero at c…
In this paper, we study the global existence and asymptotic behavior of classical solutions near vacuum for the initial-boundary value problem modeling isentropic supersonic flows through divergent ducts. The governing equations are the compressible Euler equations with a small parameter, which can be written as a hyperbolic system in terms of the Riemann invariants with a non-dissipative source. We provide a new result for the global existence of classical solutions to initial-boundary value problems of non-dissipative hyperbolic balance laws without the assumption of small data. The work is …
We are concerned with isentropic gas flow in the Laval nozzle with a friction due to viscosity. It is well known that the flow attains the sonic state at the throat, where the cross section is minimum in the Laval nozzle. However, the present friction changes the position of the sonic state into downstream, which is called chooking. Our goal in this paper is to investigate this phenomena mathematically. From the mathematical point of view, the friction is different from normal frictions and difficult to treat with. In spite of its physical importance, the friction has not received much attenti…
In this paper, we study the initial-boundary value problem of one-dimensional isentropic compressible Euler equations with the source term $βρ|u|^αu$. By means of wave decomposition and the uniform a-priori estimates, we prove the global existence of smooth solutions under small perturbations around the supersonic Fanno flow. Then, by Gronwall's inequality, we get the smooth solution is time-periodic.
Global Solutions to the Gas-Vacuum Interface Problem of Isentropic Compressible Inviscid Flows with Damping in Spherically Symmetric Motions and Physical Vacuum
Decomposition of Three-Dimensional Steady Non-isentropic Compressible Euler System and Stability of Spherically Symmetric Subsonic Flows and Transonic Shocks under Multidimensional Perturbations
Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions
本文研究了喷管内等熵气体的运动。这是流体动力学的一个主要课题。事实上，喷嘴是用来增加火箭发动机的推力。此外，喷嘴流与天体物理学密切相关。这些现象都是由可压缩欧拉方程控制的，而欧拉方程是非齐次守恒律中的关键方程之一。本文考虑其非定常流动，致力于证明一般喷管Cauchy问题解的整体存在性和稳定性。该定理已在（Tsuge in Arch. Ration.机械肛门209：365-400（2013））。然而，这一结果仅限于小数据。我们在本文中的目的是消除这种限制，也就是说，我们考虑大数据。虽然这门学科在数学、物理和工程学中很重要，但它仍然开放了很长一段时间。问题似乎在于近似解的有界估计，因为我们只有方法来研究关于时间变量的行为。为了解决这个问题，我们首先引入一个广义不变区域。与已有的上界和下界相比，它的上界和下界是空间变量函数的推广常数。然而，我们不能将新的不变区域应用于传统的差分法。因此，我们发明了改进的Goddom方案。近似解由对应于不变区域上界和下界的函数组成。这些方法使我们能够研究近似解关于空间变量的行为。这些思想也适用于其他涉及类似困难的非线性问题。
Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions i…
It is well-known that a Leray-Hopf weak solution in $L^4 (0,T; L^4(Ω))$ for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations. This is a…
Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum
本文考虑三维等熵可压缩Navier-斯托克斯方程的Cauchy问题。当粘性系数是密度幂的常数倍数时（$ρ^δ$ with $0<δ<1$），在对该系统的非线性结构进行分析的基础上，通过引入新的变量和初始相容条件，确定了在非齐次索伯列夫空间中存在具有远场真空和有限能量的局部正则解的初始数据类，该方法解决了Bresh-Desjardins-Metivier [3]、Jiu-Wang-Xin [11]等部分提到的退化粘性流的一个公开问题。我们证明了当时间$t$趋于无穷大时，不可能得到任何整体正则解的$u$的$L^\infty$范数衰减到零。
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1…
The interaction of elementary waves for isentropic flow in a variable cross-section duct is obtained (\cite{ShengZhang}). The authors have discussed rarefaction wave or shock wave interacts with stationary wave. In this paper, we extend their results to the nonisentropic flow. It can be proved that if one changes $(u,ρ)$ plane in isentropic flow with $(u,p)$ plane in nonisentropic case, the interaction results of rarefaction wave or shock wave with stationary wave can be moved parallel from the isentropic case when contact discontinuity is involved. Thus we mainly focus on the interactions bet…
In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $L^2$ norm grows as $t^{1/2}$ and this confirms previous observations in the physics literature. Instead, the solenoidal component of the velocity field experience inviscid damping, meaning that it decays to zero even in the absence of…
Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
