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Manwai Yuen

Publications and source records attributed to Manwai Yuen.

32 records · Page 2Linked to original sources
arXiv2010

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 $γ>1$.

math.AP↗
arXiv2009

Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N

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}% ρ(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.

astro-ph.SR↗
arXiv2009

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_{Ω(t)}Kγ(γ-1)ρ^{γ-2}(\nablaρ)^{2}% dx+\int_{Ω(t)}Kγρ^{γ-1}Δρdx+ε\geq -δα(N)M, where $M$ is the mass of the solutions and $| Ω| $ is the fixed volume of $Ω(t)$. 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

math.AP↗
arXiv2009

Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars II

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

astro-ph.SR↗
arXiv2009

Stabilities for Euler-Poisson Equations in Some Special Dimensions

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 $2\int_Ω(ρ| u| ^{2}+2P)dx<gM^{2}-ε$, for any $γ$.

math.AP↗
arXiv2009

Analytically Periodic Solutions to the 3-dimensional Euler-Poisson Equations of Gaseous Stars with Negative Cosmological Constant

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

astro-ph.SR↗
arXiv2009

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 Symmetry

astro-ph.SR↗
arXiv2009

Analytical Solutions to the Navier-Stokes Equations

With the previous results for the analytical blowup solutions of the N-dimensional Euler-Poisson equations, we extend the similar structure to construct an analytical family of solutions for the isothermal Navier-Stokes equations and pressureless Navier-Stokes equations with density-dependent viscosity.

math-ph↗
arXiv2009

Analytical Solutions to the Navier-Stokes Equations with Density-dependent Viscosity and with Pressure

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 $γ$ and $θ$.

math-ph↗
arXiv2008

Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R^2. Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case, the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.

astro-ph↗
arXiv2008

Analytical Blowup Solutions to the 3-dimensional Pressureless Navier-Stokes-Poisson Equations with Density-dependent Viscosity

We study the pressureless Navier--Stokes-Poisson equations of describing the evolution of the gaseous star in astrophysics. The isothermal blowup solutions of Yuen, to the Euler-Poisson equations in R2, can be extended to the pressureless Navier-Stokes-Poisson equations with density-dependent viscosity in R3. Besides some remarks, about the meaning of the blowup solutions and the applicability of such solutions to the the drift-diffusion model in semiconductors, are discussed in the end.

astro-ph↗