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
We study the 4-dimensional pressureless Navier--Stokes-Poisson equations with density-dependent viscosity. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper.
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.".
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…
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-…
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 prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.
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…
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.
We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.
This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.
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 $γ>1$:%} [c]{c}% ρ=\max\{\frac{γ-1}{Kγ}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)] -\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{γ-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 …
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 $(c(t),b(t),ρ^{γ-1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c…
We study Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation several profiles of blowup solution have been noticed. Here we show the residual vanishing.
We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.
In this paper, we study the finite time blowup of the generalized Euler ODE in the matrix geometry. We can extend Sullivan's result which is about the finite time blowup result of initial invertible linear operators to singular linear operators. we can give a complete answer to the question of Sullivan in the case when the initial matrix A is symmetric in the finite dimensional vector space W. Some open questions are proposed in the last part of the paper.
We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding \emph{regular} solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. Th…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation …
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,α}$ initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,α}$ initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodyn…
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)(…
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.
In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure defined on the interval. Under Dirichlet boundary conditions we establish the exclusion of boundary blowup points, we show that the concentration mass does not have residual L1-terms and we determine the location of blowup points in terms of Kirchhoff's Hamiltonian. We …
Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations
University of Southern California（南加州大学）；Korea Institute for Advanced Study（韩国高等研究院）；New York University Abu Dhabi（阿布扎比纽约大学）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of pos…
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical s…
Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions i…
Compressible Euler-Poisson equations are the standard self-gravitating models for stellar dynamics in classical astrophysics. In this article, we construct periodic solutions to the isothermal ($γ=1$) Euler-Poisson equations in $R^{2}$ with possible applications to the formation of plate, spiral galaxies and the evolution of gas-rich, disk-like galaxies. The results complement Yuen's solutions without rotation (M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341(2008), 445--456.). Here, the periodic rotation …
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in \cite{luo2014potentially,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in \cite{luo2014potentially,luo2013potentially-2} share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical res…
This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Ampère system, where the fully non-linear Monge-Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.
In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_{3}\right)=-\,\fracμ{(1+t)^λ}\,ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), $$ where $x\in\mathbb R^3$, $μ>0$, $λ\geq 0$, and $\barρ>0$ are constants, $ρ_0,\, u_0\in C_0^{\infty}(\mathbb R^3)$, $(ρ_0, u_0)\not\equiv 0$, $ρ(0,\cdot)>0$, and $\varepsilon>0$ is sufficiently small. For $0\leqλ\leq1$, we show that …
We investigate rescaling transformations for the Vlasov-Poisson and Euler-Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behaviour of the solutions and can be applied to other models in kinetic theory (2-dimensional symmetric Vlasov-Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger-Poisson system, nonlinear Schrödinger equation).
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 $γ$.
The dynamics of gaseous stars can be described by the Euler-Poisson system. Inspired by Rein's stability result for $γ>{4/3}$, we prove the nonlinear instability of steady states for the adiabatic exponent $γ={6/5}$ in spherically symmetric and isentropic motion.
A fundamental two-fluid model for describing dynamics of a plasma is the Euler-Poisson system, in which compressible ion and electron fluids interact with their self-consistent electrostatic force. Global smooth electron dynamics were constructed in Guo due to dispersive effect of the electric field. In this paper, we construct global smooth irrotational solutions with small amplitude for ion dynamics in the Euler-Poisson system.
We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced…
This article concerns the global-in-time existence of smooth solutions with small amplitude to two space dimensional Euler-Poisson system. The main difficulty lies in the slow time decay $(1+t)^{-1}$ of the linear system. Inspired by Ozawa, Tsutaya, and Tsutsumi's work, we show that such smooth solutions exist for radially symmetric flows.
We prove that the one-dimensional Euler-Poisson system driven by the Poisson forcing together with the usual γ-law pressure, γ ≥ 1, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the 2x2 p-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.
This paper is concerned with the 1-D compressible Euler-Poisson equations with moving physical vacuum boundary condition. It is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local well-posedness of classical solutions is established in the case of the adiabatic index $1<γ<3$.
The motion of a continuum of matter subject to gravitational interaction is classically described by the Euler-Poisson system. Prescribing the density of matter at initial and final times, we are able to obtain weak solutions for this equation by minimizing the action of the Lagrangian which is a convex functional. Then we see that such minimizing solutions are consistent with smooth solutions of the Euler-Poisson system and enjoy some special regularity properties. Meanwhile some intersting links with with Hamilton-Jacobi equations are found.
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t ω+ {\mathcal L}_u ω&= 0\\ u &= δ\tilde η^{-1} Δ^{-1} ω\end{align*} where $ω$ is the vorticity $2$-form, ${\mathcal L}_u$ denotes the Lie derivative with respect to the velocity field $u$, $Δ$ is the Hodge Laplacian, $δ$ is the codifferential (the negative of the divergence operator), and $\tilde η^{-1}$ is the canonical map from $2$-forms to $2$-vector fields induced by the Euclidean metric $η$. In this paper we consider a generali…
We give a necessary and sufficient condition for the global existence of the classical solution to the Cauchy problem of the compressible Euler-Poisson equations with radial symmetry. We introduce a new quantity which describes the balance between the initial velocity of the flow and the strength of the force governed by Poisson equation.
