Blowup for the C^1 Solutions of the Euler-Poisson Equations of Gaseous Stars in R^N
Manwai Yuen
备注：
12 pages, We merged the result in "M.W. Yuen, Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces II, Pre-print, arXiv:1012.5143" in the updated version. Key Words: Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, $C^{1}$ Solutions, No-Slip Boundary Condition, Compact Support, Initial Value Problem, First Boundary Condition
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 $γ>1$, in radial symmetry. We could show that the $C^{1}$ non-trivial classical solutions $(ρ,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant with $ρ(t,r)=0$ 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 $γ>1.$ 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".
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…
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
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…
In this paper, we study the blowup phenomena for the regular solutions of the isentropic relativistic Euler-Poisson equations with a vacuum state in spherical symmetry. Using a general family of testing functions, we obtain new blowup conditions for the relativistic Euler-Poisson equations. We also show that the proposed blowup conditions are valid regardless of the speed requirement, which was one of the key constraints stated in "Y. Geng, Singularity Formation for Relativistic Euler and Euler-Poisson Equations with Repulsive Force, Commun. Pure Appl. Anal., 14 (2015), 549--564.".
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 Fin…
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 $(ρ,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant and in the sense which $ρ(t,r)=0$ 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 $γ>1.$ The main contribution of this article provides the blowup results of the Euler $(δ=0)$ or Euler-…
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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 $(ρ,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant wit…
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 $Λ<M/X_{0}$, with the total mass $M$, then the non-trivial $C^{2}$ solutions in $R^{N}$ with the irrotational i…
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We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-di…
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We study some particular solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure, in radial symmetry. With extension of the previous known blowup solutions for the Euler-Poisson equations / pressureless Navier-Stokes-Poisson with density-dependent viscosity, we constructed the corresponding analytical blowup solutions for the Navier-Stokes-Poisson Equations with density-dependent viscosity and with pressure.
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Stars are essentially gravitationally stabilised thermonuclear reactors in hydrostatic equilibrium. The fundamental differential equation for all Newtonian gaseous stars in equilibrium is \begin{align} \frac{dp(r)}{dr}=-\frac{\mathscr{G}\mathcal{M}(r)ρ(r)}{r^{2}}\nonumber \end{align} where $p(r),ρ(r)$ are the pressure, density at radius $r$ and $\mathcal{M}(r)$ is the mass contained within a shell of radius $r$ given by $\mathcal{M}(r)=\int_{0}^{r}4π\overline{r}^{2} ρ(\overline{r})d\overline{r}$, and $\mathscr{G}$ is Newton's constant. This simple but crucial differential equation for the pres…
