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.Comment: 10 page

Stabilities for Euler-Poisson Equations with Repulsive Forces in R^N

This article extends the previous paper in "M.W. Yuen, \textit{Stabilities for Euler-Poisson Equations in Some Special Dimensions}, J. Math. Anal. Appl. \textbf{344} (2008), no. 1, 145--156.", from the Euler-Poisson equations for attractive forces to the repulsive ones in $R^{N}$ $(N\geq2)$. The similar stabilities of the system are studied. Additionally, we explain that it is impossible to have the density collapsing solutions with compact support to the system with repulsive forces for $\gamma>1$.Comment: Key Words: Euler-Poisson Equations, Repulsive Forces, Stabilities, Frictional Damping, Second Inertia Function, Non-collapsing Solutions; 6 Page

Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R^{N}

Based on Makino's solutions with radially symmetry, we extend the corresponding ones with elliptic symmetry for the compressible Euler and Navier-Stokes equations in R^{N} (N\geq2). By the separation method, we reduce the Euler and Navier-Stokes equations into 1+N differential functional equations. In detail, the velocity is constructed by the novel Emden dynamical system: {| a_{i}(t)=({\xi}/(a_{i}(t)({\Pi}a_{k}(t))^{{\gamma}-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} with arbitrary constants {\xi}, a_{i0} and a_{i1}. Some blowup phenomena or global existences of the solutions obtained could be shown.Comment: 6 pages, Key Words: Euler Equations, Navier-Stokes Equations, Analytical Solutions, Elliptic Symmetry, Makino's Solutions, Self-Similar, Drift Phenomena, Emden Equation, Blowup, Global Solution

Blowup for C2C^{2} Solutions of the N-dimensional Euler-Poisson Equations in Newtonian Cosmology

Pressureless Euler-Poisson equations with attractive forces are standard models in Newtonian cosmology. In this article, we further develop the spectral dynamics method and apply a novel spectral-dynamics-integration method to study the blowup conditions for $C^{2}$ solutions with a bounded domain, $\left\Vert X(t)\right\Vert \leq X_{0}$, where $\left\Vert\cdot\right\Vert$ denotes the volume and $X_{0}$ is a positive constant. In particular, we show that if the cosmological constant $\Lambda<M/X_{0}$, with the total mass $M$, then the non-trivial $C^{2}$ solutions in $R^{N}$ with the irrotational initial condition blow up at a finite time.Comment: 10 Page

Analytical Collapsing Solutions to Pressureless Navier-Stokes-Poisson Equations with Density-dependent Viscosity θ=1/2\theta=1/2 in R2R^{2}

We study the 2-dimensional Navier-Stokes-Poisson equations with density-dependent viscosity $\theta=1/2$ without pressure of gaseous stars in astrophysics. The analytical solutions with collapsing in radial symmetry, are constructed in this paper.Comment: 8 page

Blowup of C^2 Solutions for the Euler Equations and Euler-Poisson Equations in R^N

In this paper, we use integration method to show that there is no existence of global $C^{2}$ solution with compact support, to the pressureless Euler-Poisson equations with attractive forces in $R^{N}$. And the similar result can be shown, provided that the uniformly bounded functional:% \int_{\Omega(t)}K\gamma(\gamma-1)\rho^{\gamma-2}(\nabla\rho)^{2}% dx+\int_{\Omega(t)}K\gamma\rho^{\gamma-1}\Delta\rho dx+\epsilon\geq -\delta\alpha(N)M, where $M$ is the mass of the solutions and $| \Omega|$ is the fixed volume of $\Omega(t)$. On the other hand, our differentiation method provides a simpler proof to show the blowup result in "D. H. Chae and E. Tadmor, \textit{On the Finite Time Blow-up of the Euler-Poisson Equations in}$R^{N}$, Commun. Math. Sci. \textbf{6} (2008), no. 3, 785--789.". Key Words: Euler Equations, Euler-Poisson Equations, Blowup, Repulsive Forces, Attractive Forces, $C^{2}$ SolutionsComment: 7 page

Rotational and Self-similar Solutions for the Compressible Euler Equations in R^3

In this paper, we present rotational and self-similar solutions for the compressible Euler equations in R^3 using the separation method. These solutions partly complement Yuen's irrotational and elliptic solutions in R^3 [Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4524-4528] as well as rotational and radial solutions in R^2 [Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2172-2180]. A newly deduced Emden dynamical system is obtained. Some blowup phenomena and global existences of the responding solutions can be determined. The 3D rotational solutions provide concrete reference examples for vortices in computational fluid dynamics.Comment: 9 pages; Key Words: Compressible Euler Equations, Rotational Solutions, Self-similar Solutions, Symmetry Reduction, Vortices, 3-dimension, Navier-Stokes Equation

Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces

In this paper, we study the blowup of the $N$-dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant and in the sense which $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition% $H_{0}=\int_{0}^{R}rV_{0}dr>0$ blow up on or before the finite time $T=R^{3}/(2H_{0})$ for pressureless fluids or $\gamma>1.$ The main contribution of this article provides the blowup results of the Euler $(\delta=0)$ or Euler-Poisson $(\delta=1)$ equations with repulsive forces, and with pressure $(\gamma>1)$, as the previous blowup papers (\cite{MUK} \cite{MP}, \cite{P} and \cite{CT}) cannot handle the systems with the pressure term, for $C^{1}$ solutions.Comment: Accepted by Nonlinear Analysis Series A: Theory, Methods & Applications Key Words: Euler Equations, Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, $C^{1}$ Solutions, No-Slip Conditio

Analytical Blowup Solutions to the Isothermal Euler-Poisson Equations of Gaseous Stars in R^N

This article is the continued version of the analytical blowup solutions for 2-dimensional Euler-Poisson equations in "M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341 (1)(2008), 445-456." and "M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars II. arXiv:0906.0176v1". With the extension of the blowup solutions with radial symmetry for the isothermal Euler-Poisson equations in R^2, other special blowup solutions in R^N with non-radial symmetry are constructed by the separation method. Key words: Analytical Solutions, Euler-Poisson Equations, Isothermal, Blowup, Special Solutions, Non-radial SymmetryComment: 13 page

Blow-up Phenomena for Compressible Euler Equations with Non-vacuum Initial Data

In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.Comment: 19 pages, Key Words: Euler Equations, Integration Method, Blowup, Radial Symmetry, Non-vacuum, Initial Value Problem