Publications (26)0 Total impact
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Manwai Yuen
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ABSTRACT: 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.
01/2013;
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Manwai Yuen
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ABSTRACT: In this paper, we construct a new class of blowup solutions with elementary
functions to the 3-dimensional compressible or incompressible Euler and
Navier-Stokes equations. In detail, we obtain a class of global rotational
exact solutions for the compressible fluids with $\gamma>1$:%} [c]{c}%
\rho=\max\{\frac{\gamma-1}{K\gamma}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)]
-\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{\gamma-1}} u_{1}=a(t)+C(y-z)
u_{2}=a(t)+C(-x+z) u_{3}=a(t)+C(x-y). where a(t)=c_{0}+c_{1}t and
b(t)=3c_{0}c_{1}t+{3/2}c_{1}^{2}t^{2}+c_{2}% with $C$, $c_{0}$, $c_{1}$ and
$c_{2}$ are arbitrary constants; And the corresponding blowup or global
solutions for the incompressible Euler equations are also given. Our
constructed solutions are similar to the famous Arnold-Beltrami-Childress (ABC)
flow. The solutions with infinite energy can exhibit the interesting behaviors
locally. Besides, the corresponding global solutions are also given for the
compressible Euler equations. Furthermore, due to $\operatorname{div}\vec{u}=0$
for the solutions, the solutions also work for the 3-dimnsional incompressible
Euler and Navier-Stokes equations.
05/2011;
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Manwai Yuen
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ABSTRACT: Based on Yuen's solutions with radially symmetry of the pressureless
density-dependent Navier-Stokes in $R^{N}$, the corresponding ones with
elliptic symmetry are constructed by the separation method. In detail, we
successfully reduce the pressureless Navier-Stokes equations with
density-dependent viscosity into $1+N$ differential functional equations. In
particular for $\kappa_{1}>0$ and $\kappa_{2}=0$, the velocity is built by the
new Emden dynamical system with force-force interaction:%\{{array} [c]{c}%
\ddot{a}_{i}(t)=\frac{-\xi(\sum_{k=1}^{N}\frac{\dot{a}_{k}(t)}%
{a_{k}(t)})}{a_{i}(t)(\underset{k=1}{\overset{N}{\Pi}}% a_{k}(t))
^{\theta-1}}\text{for}i=1,2,...,N\
a_{i}(0)=a_{i0}>0,\text{}\dot{a}_{i}(0)=a_{i1}% {array}. with arbitrary
constants $\xi$, $a_{i0}$ and $a_{i1}$. We can show some blowup phenomena or
global existences for the obtained solutions. Based on the complication of the
deduced Emden dynamical systems, the author conjectures there exist limit
cycles or chaos for this kind of flows. Numerical simulation or mathematical
proofs for the Emden dynamical systems are expected in the future.
05/2011;
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Manwai Yuen
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ABSTRACT: 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.
04/2011;
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Manwai Yuen
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ABSTRACT: The Newtonian Euler-Poisson equations with attractive forces are the
classical models for the evolution of gaseous stars and galaxies in
astrophysics. In this paper, we use the integration method to study the blowup
problem of the $N$-dimensional system with adiabatic exponent $\gamma>1$, in
radial symmetry. We could show that the $C^{1}$ non-trivial classical solutions
$(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant
with $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition
\begin{equation}
H_{0}=\int_{0}^{R}r^{n}V_{0}dr>\sqrt{\frac{2R^{2n-N+4}M}{n(n+1)(n-N+2)}}%
\end{equation} with an arbitrary constant $n>\max(N-2,0),$\newline blow up
before a finite time $T$ for pressureless fluids or $\gamma>1.$ Our results
could fill some gaps about the blowup phenomena to the classical $C^{1}$
solutions of that attractive system with pressure under the first boundary
condition.\newline In addition, the corresponding result for the repulsive
systems is also provided. Here our result fully covers the previous case for
$n=1$ in "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations
with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods &
Applications 74 (2011), 1465--1470".
12/2010;
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Manwai Yuen
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ABSTRACT: In this paper, we continue to study the blowup problem of the $N$-dimensional compressible Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. In details, we extend the recent result of "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods & Applications \textbf{74} (2011), 1465--1470.". We could further apply the integration method to obtain the more general results which the non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant with $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition% \begin{equation} H_{0}=\int_{0}^{R}r^{n}V_{0}dr>0 \end{equation} where an arbitrary constant $n>0$, blow up on or before the finite time $T=2R^{n+2}/(n(n+1)H_{0})$ for pressureless fluids or $\gamma>1.$ The results obtained here fully cover the previous known case for $n=1$. Comment: 9 pages; Key Words: Euler Equations, Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, C? Solutions, No-Slip Condition
12/2010;
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Manwai Yuen
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ABSTRACT: We study the construction of analytical non-radially solutions for the
1-dimensional compressible adiabatic Euler equations in this article. We could
design the perturbational method to construct a new class of analytical
solutions. In details, we perturb the linear velocity:% \begin{equation}
u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler
equations. By comparing the coefficients of the polynomial, we could deduce the
corresponding functional differential system of
$(c(t),b(t),\rho^{\gamma-1}(0,t)).$ Then by skillfully applying the Hubble's
transformation: \begin{equation} c(t)=\frac{\dot{a}(t)}{a(t)}, \end{equation}
the functional differential equations can be simplified to be the system of
$(a(t),b(t),\rho^{\gamma-1}(0,t))$. After proving the existence of the
corresponding ordinary differential equations, a new class of blowup or global
solutions can be shown. Here, our results fully cover the previous known ones
by choosing $b(t)=0$.
12/2010;
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Manwai Yuen
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ABSTRACT: In this article, we study the perturbational method to construct the
non-radially symmetric solutions of the compressible 2-component Camassa-Holm
equations. In detail, we first combine the substitutional method and the
separation method to construct a new class of analytical solutions for that
system. In fact, we perturb the linear velocity: u=c(t)x+b(t), and substitute
it into the system. Then, by comparing the coefficients of the polynomial, we
can deduce the functional differential equations involving
$(c(t),b(t),\rho^{2}(0,t)).$ Additionally, we could apply the Hubble's
transformation c(t)={\dot{a}(3t)}/{a(3t)}, to simplify the ordinary
differential system involving $(a(3t),b(t),\rho ^{2}(0,t))$. After proving the
global or local existences of the corresponding dynamical system, a new class
of analytical solutions is shown. And the corresponding solutions in radial
symmetry are also given. To determine that the solutions exist globally or blow
up, we just use the qualitative properties about the well-known Emden equation:
{array} [c]{c} {d^{2}/{dt^{2}}}a(3t)= {\xi}{a^{1/3}(3t)}, a(0)=a_{0}>0
,\dot{a}(0)=a_{1} {array} . Our solutions obtained by the perturbational
method, fully cover the previous known results in "M.W. Yuen,
\textit{Self-Similar Blowup Solutions to the 2-Component Camassa-Holm
Equations,}J. Math. Phys., \textbf{51} (2010) 093524, 14pp." by the separation
method.
12/2010;
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Manwai Yuen
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ABSTRACT: In this article, we study the self-similar solutions of the 2-component
Degasperis-Procesi water system:% [c]{c}%
\rho_{t}+k_{2}u\rho_{x}+(k_{1}+k_{2})\rho u_{x}=0
u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\rho\rho_{x}=0. By the
separation method, we can obtain a class of self-similar solutions,% [c]{c}%
\rho(t,x)=\max(\frac{f(\eta)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x
\overset{\cdot\cdot}{a}(s)-\frac{\xi}{4a(s)^{\kappa}}=0,\text{}a(0)=a_{0}%
\neq0,\text{}\overset{\cdot}{a}(0)=a_{1}
f(\eta)=\frac{k_{3}}{\xi}\sqrt{-\frac{\xi}{k_{3}}\eta^{2}+(\frac{\xi}{k_{3}}\alpha)
^{2}}% where $\eta=\frac{x}{a(s)^{1/4}}$ with $s=4t;$ $\kappa=\frac{k_{1}}{2}%
+k_{2}-1,$ $\alpha\geq0,$ $\xi<0$, $a_{0}$ and $a_{1}$ are constants. which the
local or global behavior can be determined by the corresponding Emden equation.
The results are very similar to the one obtained for the 2-component
Camassa-Holm equations. Our analytical solutions could provide concrete
examples for testing the validation and stabilities of numerical methods for
the systems. With the characteristic line method, blowup phenomenon for
$k_{3}\geq0$ is also studied.
08/2010;
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Manwai Yuen
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ABSTRACT: In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems. Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundary
07/2010;
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ABSTRACT: In this paper, we study the Euler and Euler-Poisson equations in $R^{N}$,
with multiple $\gamma$-law for pressure function: \begin{equation}
P(\rho)=e^{s}\sum_{j=1}^{m}\rho^{\gamma_{j}}, \end{equation} where all
$\gamma_{i+1}>\gamma_{i}\geq1$, is the constants. The analytical line solutions
are constructed for the systems. It is novel to discover the analytical
solutions to handle the systems with mixed pressure function. And our solutions
can be extended to the systems with the generalized multiple damping and
pressure function.
05/2010;
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Manwai Yuen
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ABSTRACT: 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 Pages
01/2010;
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Manwai Yuen
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ABSTRACT: 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 Condition
01/2010;
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Manwai Yuen
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ABSTRACT: The pressureless Euler equations can be used as simple models of cosmology or
plasma physics. In this paper, we construct the exact solutions in non-radial
symmetry to the pressureless Euler equations in $R^{N}:$% [c]{c}%
\rho(t,\vec{x})=\frac{f(\frac{1}{a(t)^{s}}\underset{i=1}{\overset
{N}{\sum}}x_{i}^{s})}{a(t)^{N}}\text{,}\vec{u}(t,\vec{x}%
)=\frac{\overset{\cdot}{a}(t)}{a(t)}\vec{x}, a(t)=a_{1}+a_{2}t. \label{eq234}%
where the arbitrary function $f\geq0$ and $f\in C^{1};$ $s\geq1$, $a_{1}>0$ and
$a_{2}$ are constants$.$\newline In particular, for $a_{2}<0$, the solutions
blow up on the finite time $T=-a_{1}/a_{2}$.
Moreover, the functions (\ref{eq234}) are also the solutions to the
pressureless Navier-Stokes equations.
10/2009;
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Manwai Yuen
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ABSTRACT: We study the stabilities and classical solutions of Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. In fact, we extend the study the stabilities of Euler-Poisson equations with or without frictional damping term to some special dimensional spaces. Besides, by using the second inertia function in 2 dimension of Euler-Poisson equations, we prove the non-global existence of classical solutions with $2\int_{\Omega}(\rho| u| ^{2}+2P)dx<gM^{2}-\epsilon$, for any $\gamma$. Comment: 15 pages
07/2009;
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Manwai Yuen
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ABSTRACT: 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, $C^{2}$ Solutions Comment: 7 pages
07/2009;
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Manwai Yuen
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ABSTRACT: 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 Symmetry Comment: 13 pages
05/2009;
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Manwai Yuen
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ABSTRACT: This article is the continued version of the analytical blowup solutions for
2-dimensional Euler-Poisson equations \cite{Y1}. With extension of the blowup
solutions with radial symmetry for the isothermal Euler-Poisson equations in
$R^{2}$, other special blowup solutions in $R^{2}$ with non-radial symmetry are
constructed by the separation method. We notice that the results are the first
evolutionary solutions with non-radial symmetry for the system.
Key words: Analytical Solutions, Euler-Poisson Equations, Isothermal, Blowup,
Non-radial Symmetry, Line Source or Sink
05/2009;
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ABSTRACT: This article is the continued version of the analytical solutions for the pressureless Navier-Stokes equations with density-dependent viscosity in "M.W. Yuen, Analyitcal Solutions to the Navier-Stokes Equations, J. Math. Phys., 49 (2008) No. 11, 113102, 10pp". We are able to extend the similar solutions structure to the case with pressure under some restriction for $\gamma$ and $\theta$. Comment: 7 pages, Typos are corrected
02/2009;
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Manwai Yuen
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ABSTRACT: By the extension of the 3-dimensional analytical solutions of Goldreich and Weber "P. Goldreich and S. Weber, Homologously Collapsing Stellar Cores, Astrophys, J. 238, 991 (1980)" with adiabatic exponent gamma=4/3, to the (classical) Euler-Poisson equations without cosmological constant, the self-similar (almost re-collapsing) time-periodic solutions with negative cosmological constant (lambda<0) are constructed. The solutions with time-periodicity are novel. On basing these solutions, the time-periodic and almost re-collapsing model is conjectured, for some gaseous stars. Key Words: Analytically Periodic Solutions, Re-collapsing, Cosmological Constant, Euler-Poisson Equations, Collapsing Comment: 9 pages
01/2009;
