50 research outputs found
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 . 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 .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 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 solutions with a bounded domain, , where denotes the
volume and is a positive constant. In particular, we show that if the
cosmological constant , with the total mass , then the
non-trivial solutions in 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 in
We study the 2-dimensional Navier-Stokes-Poisson equations with
density-dependent viscosity 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 solution with compact support, to the pressureless
Euler-Poisson equations with attractive forces in . 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 is the mass of the solutions and is
the fixed volume of .
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}, Commun. Math. Sci.
\textbf{6} (2008), no. 3, 785--789.".
Key Words: Euler Equations, Euler-Poisson Equations, Blowup, Repulsive
Forces, Attractive Forces, 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 -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 ,
with compact support in , where is a positive constant and in the
sense which and for , under the initial
condition% blow up on or before the finite time
for pressureless fluids or
The main contribution of this article provides the blowup results of the
Euler or Euler-Poisson equations with repulsive
forces, and with pressure , as the previous blowup papers
(\cite{MUK} \cite{MP}, \cite{P} and \cite{CT}) cannot handle the systems with
the pressure term, for 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, 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