Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

流体力学方程组一类人为解构造方法

针对柱对称二维流体力学方程组,基于考虑方程右端附加源项的人为构造解方法,构造出一类统一形式的人为解.此类形式的人为解,对验证多维流体力学应用程序的正确性有重要的作用.同时将该类统一形式的人为解应用到PPM格式的程序,验证了构造的人为解的可行性.     

Vortical and self-similar flows of 2D compressible Euler equations

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino’s solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.     

On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity

The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of . It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

Convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations

We establish the convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations in this paper. The convergence is rigorously proved on the time interval where the smooth solution to the incompressible Euler equations exists. The proof relies on the compactness argument and the so-called relative-entropy method.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

Relaxation approximation of the Euler equations

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Regularity criteria for the Euler equations with nondecaying data

The local-in-time existence and uniqueness of strong solutions to the Euler equations in the whole space with nondecaying and certainly regular initial velocity are concerned. It is obtained that the spatial regularity of solutions coincides with that of initial velocity under the suitable setting of external forcing terms. Regularity criteria focusing into the vorticity are also discussed due to the similar arguments of Beale-Kato-Majda.     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

The stationary three‐dimensional Navier–Stokes equations with a non‐zero constant velocity at infinity

This paper is devoted to some mathematical questions related to the three‐dimensional stationary Navier–Stokes equations. Our approach is based on a combination of properties of Oseen problems in ?³. Copyright © 2008 John Wiley & Sons, Ltd.     

一类Euler方程解的渐近行为

给出了一类Ginzburg-Landau型泛函的极小元所满足的Euler方程的解的某些弱收敛性质。     

Strichartz estimates for the Euler equations in the rotational framework

We consider the initial value problems of the incompressible Euler equations in the rotational framework. We obtain the optimal range of the Strichartz estimate for the linear group associated with the Coriolis forces. As an application, we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high.     

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

Abstract

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t ^?(3/2). While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.     

有限初始能量的相对论欧拉方程组光滑解的爆破

在初始资料的某些限制下证明有限初始能量的相对论欧拉方程组柯西问题光滑解的爆破.该文的爆破条件不需要初始资料具有紧支集,部分补充了Pan和Smoller的经典爆破结果(2006).     

二维等熵可压欧拉方程古典解的存在性(英文)

研究二维等熵可压缩欧拉方程的古典解存在性.利用迭代技巧,得到解的局部存在性及唯一性,并且还证明了解在有限时间内爆破,即可压缩欧拉方程不存在全局古典解.     

Time-domain implementation of nonreflecting boundary-conditions for the nonlinear Euler equations

The present paper studies nonreflecting boundary-conditions for the 2-D unsteady nonlinear Euler equations, applied to the propagation of monochromatic pressure-waves in a uniform mean flow. Various boundary-conditions (1-D nonlinear, approximate linearized 2-D, and exact linearized 2-D) are compared for a wide range of both propagating and decaying waves. An original methodology, based on a moving-averages technique, is developed for the application of the exact linearized boundary-conditions, which requires the computation of 2-D (space–time) Fourier coefficients. It is shown that the exact linearized boundary-conditions yield very low reflection, and also that the approximate conditions may perform poorly in difficult cases. The reflection-coefficient shows some correlation with the group-velocity (direction and Mach-number) of the reflected waves, suggesting that proposed nonreflecting boundary-conditions should always be validated against the entire range of group-velocities.