三维空间中带非线性阻尼项的等熵欧拉方程组轴对称解的爆破

研究了三维空间中带非线性阻尼项的可压缩等熵欧拉方程组Dirichlet初边值问题.采用泛函方法,定义几种不同的泛函,当初始速度足够大时分别得到了经典解在某一时间内必定爆破的结论.由于出现了非线性阻尼项,较之线性阻尼的情形,经典解爆破的难度随之增加.     

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.     

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.     

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.     

Blowup for the compressible isothermal Euler equations with non-vacuum initial data

ABSTRACT

In this paper, we study the compressible isothermal Euler equations with non-vacuum initial data. First, we prove the property of finite propagation to this Cauchy problem by using local energy estimates. Second, we establish the blowup results of the multi-dimensional case in radial symmetry and the one-dimensional case in non-radial symmetry by making assumptions on the initial velocity. Third, we present the blowup results of the three-dimensional case in non-radial symmetry by making assumptions on the initial momentum.     

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.     

带自由边界的可压缩欧拉与欧拉-泊松方程组径向对称解的爆破北大核心CSCD

本文在R^(N)(N=2,3)中研究描述流向外部真空的可压缩流体的欧拉与欧拉-泊松方程组径向对称解的爆破.在分离流体与真空的连续自由边界条件下考虑其自由边值问题.对于径向对称的欧拉方程组,证明若初始流平均向外流动,则其光滑解将在有限时刻爆破.对于带有斥力与弛豫项的单极与双极径向对称欧拉-泊松方程组,证明若某个与初始动量有关的加权泛函适当大,则其光滑解将在有限时刻爆破。     

Blow up of incompressible Euler solutions

We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L ₂-residuals of computed solutions under decreasing mesh size. AMS subject classification (2000)  35Q30, 65M60     

Global existence of classical solutions for the 3D generalized compressible Oldroyd-B model

In this paper, we prove the global existence of small classical solutions to the 3D generalized compressible Oldroyd-B system. It can be seen as compressible Euler equations coupling the evolution of stress tensor τ. The result mainly shows that singularity of solutions to compressible Euler equations can be prevented by the coupling of viscoelastic stress tensor. Moreover, unlike most complex fluids containing compressible Euler equations, the irrotational condition ∇×u=0 would not be proposed here to achieve the global well-posedness.     

三维可压缩Euler方程球对称解的生命区间

对于可压缩的三维Ｅｕｌｅｒ方程，当其初值为振幅ε的小扰动且具有球对称性质时，我们研究了经典解的生命区间．并证明无论初值的扰动多么小，经典解都在有限时间内爆破．     

Blowup phenomena of solutions to the Euler equations for compressible fluid flow

The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form c(t)x are constructed to show the expanding and blowup properties. The solution with velocity of the form for γ?1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form c(t)|x|^α-1x for α≠1 and multi-space dimensions.     

Note for “Some Exact Blowup Solutions to the pressureless Euler equations in R” [Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2993-2998]

In this supplementary note, we can generalize the exact solutions for the pressureless Euler equations in [Yuen MW. Some exact blowup solutions to the pressureless Euler equations in R^N, Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2993-8]. Here, the solutions are constructed in implicit or explicit forms.     

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.     

Existence and asymptotic behavior of C solutions to the multi-dimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of C¹$C^1$ solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.     

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.     

On the finite-time singularities of the 3D incompressible Euler equations

We prove the finite-time vorticity blowup, in the pointwise sense, for solutions of the 3D incompressible Euler equations assuming some conditions on the initial data and its corresponding solutions near initial time. These conditions are represented by the relation between the deformation tensor and the Hessian of pressure, both coupled with the vorticity directions associated with the initial data and solutions near initial time. We also study the possibility of the enstrophy blowup for the 3D Euler and the 3D Navier-Stokes equations, and prove the finite-time enstrophy blowup for initial data satisfying suitable conditions. Finally, we obtain a new blowup criterion that controls the blowup by a quantity containing the Hessian of the pressure. © 2006 Wiley Periodicals, Inc.     

A Priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

In this paper, we prove a new type of energy estimate for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed k. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin to when you have a free boundary. In the incompressible case our energies reduce to those in Christodoulou and Lindblad, and our proof in particular gives a simplified proof of their estimates with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long‐time existence also for slightly compressible liquids with a free surface.© 2017 Wiley Periodicals, Inc.     

Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices

In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L^∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case. Received: November 18, 2003     

Analytical solutions to the Navier-Stokes equations for non-Newtonian fluid

The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional （N ≥ 2） Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term （μ（ρ）｜ u｜^α u）.     

The Asymptotic Behavior and the Quasineutral Limit forthe Bipolar Euler-Poisson System withBoundary Effects and a Vacuum

In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy’s law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.