Symmetric Self-similar Solutions Research Articles

Movable Singularity of Some Hamiltonian Systems Associated with Blowup Phenomena

The motivation for this paper comes from a blowup problem for a semi-linear wave equation and a heat equation. Following an idea of J. Leray, we study a radially symmetric self-similar solution which has singularities on the characteristic cone. This naturally leads to the study of a Hamiltonian system called a profile equation. The novelty of this paper is that we focus on the movable singularity of the Hamiltonian system and we use Borel summability in constructing a singular solution. By “movable singularity” we mean that the singularity does not appear in the coefficients of the equation and depends on the respective solution. In the proof of our theorem we reduce the Hamiltonian system to a simpler form by a method similar to the so-called Birkhoff reduction. We obtain the parametrization of a singular solution by an elementary function. We also give applications to a semi-linear wave equation and a heat equation.

Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

In the supercritical range of the polytropic indices gamma in (1,frac{4}{3}) we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case gamma =1. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.

On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles

In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension $d\ge 2$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic $\mathcal{C}^\infty$ self-similar solutions with suitable decay at infinity. The $\mathcal{C}^\infty$ regularity is used in a fundamental way in our companion paper (part II) in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions $d=2,3$.

On the Properties of a Radially Symmetric Self-Similar Solution of a Nonlinear Heat Conduction Equation with a Source

In this work, new properties of a radially symmetric self-similar solution of the nonlinear heat equation with source were obtained. A self-similar solution was derived using the method of standard equations. These properties were proved by regulation the solution of the self-similar equation relative to the parameter of the nonlinear source by adding an additional parameter. The obtained properties were verified by numerical experiments.

Asymptotic behavior of solutions of a k-Hessian evolution equation

We study the long-time behavior of solutions of the k-Hessian evolution equation ut=Sk(D2u), posed on a bounded domain of the n-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of u is precisely described by this special solution. Further, we initiate the study of that equation on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call k-Barenblatt solutions. These solutions present some common properties as those of well-known Barenblatt solutions for the porous media equation and the p-Laplacian equation.

Movable singularity of semi linear Heun equation and application to blowup phenomenon

In this paper we shall study the movable singularity of semi linear Heun equation and its application to the blowup of a semi linear wave equation. In fact, the semi linear Heun equation appears if we consider a radially symmetric self-similar solution of the semi linear wave equation. By the movable singularity we mean the singularity which does not appear in the coefficients of the equation and that depends on the respective solution. We focus on movable singularity when we construct a singular solution of the semi linear wave equation with singularities on the characteristic cone. In the proof of our theorem we reduce our equation to a simpler form by the method similar to the so-called Birkhoff reduction, then we analyze the reduced equation. The latter part is closely related with the parametrization of a solution in terms of the Jacobi elliptic function.

Critical collapse of ultrarelativistic fluids: Damping or growth of aspherical deformations

We perform fully nonlinear numerical simulations to study aspherical deformations of the critical self-similar solution in the gravitational collapse of ultra-relativistic fluids. Adopting a perturbative calculation, Gundlach predicted that these perturbations behave like damped or growing oscillations, with the frequency and damping (or growth) rates depending on the equation of state. We consider a number of different equations of state and degrees of asphericity and find very good agreement with the findings of Gundlach for polar $\ell = 2$ modes. For sufficiently soft equations of state, the modes are damped, meaning that, in the limit of perfect fine-tuning, the spherically symmetric critical solution is recovered. We find that the degree of asphericity has at most a small effect on the frequency and damping parameter, or on the critical exponents in the power-law scalings. Our findings also confirm, for the first time, Gundlach's prediction that the $\ell = 2$ modes become unstable for sufficiently stiff equations of state. In this regime the spherically symmetric self-similar solution can no longer be recovered by fine-tuning to the black-hole threshold, and one can no longer expect power-law scaling to hold to arbitrarily small scales.

A homotopy continuation approach for analysing finite-time singularities in thin liquid films

We consider self-similar solutions related to rupture of thin liquid films on a solid substrate that evolve solely under the stabilizing influence of surface tension and the destabilizing influence of effective van der Waals interactions between the free surface of the film and the substrate. Such solutions have been previously analysed in the literature and various numerical approaches to obtain such solutions have been proposed. Such approaches are based either on shooting or finite-difference schemes and require well-chosen initial guesses for solutions. We propose an alternative numerical method, which is based on a homotopy approach and continuation techniques and allows one to reach self-similar solutions from analytically known small-amplitude steady solutions of the related thin-film equation. We argue that this method is more robust than previously proposed methods and does not require initial guesses to obtain solutions. Although the present study focuses on the particular case of self-similar solutions related to planar rupture that have square-root far-field behaviour, our approach can also be used to obtain planar solutions having a different far-field behaviour and radially symmetric self-similar solutions for the considered thin-film equation. We expect the approach to be also valid for other equations of similar type that show a subcritical primary bifurcation and finite-time singularities.

Convergence to self-similar solutions for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$

We consider the Cauchy problem for a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$, modeling chemotaxis and self-attracting particles, with $L^1$-initial data. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, by using similarity arguments, it is shown that the nonnegative solution converges to a radially symmetric self-similar solution at rate $o(t^{-1+1/p})$ in the $L^p$-norm $(1\le p\le\infty)$ as time goes to infinity.

Parallel and Orthogonal Cylindrically Symmetric Self-Similar Solutions

In this paper, we evaluate kinematic self-similar perfect fluid and dust solutions for the most general cylindrically symmetric spacetime. We explore kinematic self-similar solutions of the first, second, zeroth and infinite kinds for parallel and orthogonal cases. It is found that the parallel case gives solutions for both perfect fluid and dust cases in all kinds except the zeroth kind of the dust case where there exists no solution. The orthogonal perfect fluid case gives stiff fluid solution only in the first kind and vacuum solution for the dust case. We obtain a total of thirteen solutions out of which eleven are independent. The correspondence of these solutions with those already available in the literature is also given.

Self-similar cosmological solutions with dark energy. I. Formulation and asymptotic analysis

Based on the asymptotic analysis of ordinary differential equations, we classify all spherically symmetric self-similar solutions to the Einstein equations which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(\ensuremath{\gamma}\ensuremath{-}1)\ensuremath{\mu}$ with $0<\ensuremath{\gamma}<2/3$. This corresponds to a ``dark energy'' fluid and the Friedmann solution is accelerated in this case due to antigravity. This extends the previous analysis of spherically symmetric self-similar solutions for fluids with positive pressure ($\ensuremath{\gamma}>1$). However, in the latter case there is an additional parameter associated with the weak discontinuity at the sonic point and the solutions are only asymptotically ``quasi-Friedmann,'' in the sense that they exhibit an angle deficit at large distances. In the $0<\ensuremath{\gamma}<2/3$ case, there is no sonic point and there exists a one-parameter family of solutions which are genuinely asymptotically Friedmann at large distances. We find eight classes of asymptotic behavior: Friedmann or quasi-Friedmann or quasistatic or constant-velocity at large distances, quasi-Friedmann or positive-mass singular or negative-mass singular at small distances, and quasi-Kantowski-Sachs at intermediate distances. The self-similar asymptotically quasistatic and quasi-Kantowski-Sachs solutions are analytically extendible and of great cosmological interest. We also investigate their conformal diagrams. The results of the present analysis are utilized in an accompanying paper to obtain and physically interpret numerical solutions.

Self-similar cosmological solutions with dark energy. II. Black holes, naked singularities, and wormholes

We use a combination of numerical and analytical methods, exploiting the equations derived in a preceding paper, to classify all spherically symmetric self-similar solutions which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(\ensuremath{\gamma}\ensuremath{-}1)\ensuremath{\mu}$ with $0<\ensuremath{\gamma}<2/3$. The expansion of the Friedmann universe is accelerated in this case. We find a one-parameter family of self-similar solutions representing a black hole embedded in a Friedmann background. This suggests that, in contrast to the positive pressure case, black holes in a universe with dark energy can grow as fast as the Hubble horizon if they are not too large. There are also self-similar solutions which contain a central naked singularity with negative mass and solutions which represent a Friedmann universe connected to either another Friedmann universe or some other cosmological model. The latter are interpreted as self-similar cosmological white hole or wormhole solutions. The throats of these wormholes are defined as two-dimensional spheres with minimal area on a spacelike hypersurface and they are all nontraversable because of the absence of a past null infinity.

Self-similar solutions for a semilinear heat equation with critical Sobolev exponent

The Cauchy problem for a semilinear heat equation with singular initial data (w t =Δw+w p in R N x(0,∞) w(x,0) =λa(x/|x|)|x| -2/(p-1) in R N \{0} is studied, where N > 2, p = (N+2)/(N-2), A > 0 is a parameter, and a > 0, a ≢ 0. We show that there exists a constant λ* > 0 such that the problem has at least two positive self-similar solutions for λ∈ (0,λ*) when N = 3,4,5, and that, when N ≥ 6 and a ≡ 1, the problem has a unique positive radially symmetric self-similar solution for λ∈ (0, λ * ) with some λ * ∈ (0, λ*). Our proofs are based on the variational methods and Pohozaev type arguments to the elliptic problem related to the profiles of self-similar solutions.

Self-similar solutions of semilinear wave equations with a focusing nonlinearity

We prove that in three space dimensions a nonlinear wave equation utt − Δu = up, with p⩾ 7 being an odd integer, has a countable family of regular spherically symmetric self-similar solutions.

Unstable sixth-order thin film equation: I. Blow-up similarity solutions

We study blow-up behaviour of solutions of the sixth-order thin film equation containing an unstable (backward parabolic) second-order term. By a formal matched expansion technique, we show that, for the first critical exponent where N is the space dimension, the free-boundary problem (FBP) with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions where T > 0 is the blow-up time. We also study the Cauchy problem (CP) in RN × R+ and show that the corresponding self-similar family {uk(x, t)} is countable and consists of solutions of maximal regularity, which are oscillatory near the interfaces. Actually, we show that compactly supported oscillatory blow-up profiles for the CP exist for all n ∊ (0, nh), where is a heteroclinic bifurcation point for the ordinary differential equation involved. The FBP ceases to exist before, at .

Unstable sixth-order thin film equation: II. Global similarity patterns

We continue the study begun in part I (Evans J D, Galaktionov V A and King J R 2007 Unstable sixth-order thin film equation: I. Blow-up of similarity solutions Nonlinearity 20 1799–841) of asymptotic large time behaviour of global solutions of the sixth-order thin film equation with bounded integrable initial data. We show that for the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, We also study the Cauchy problem, for which we construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n ∊ (0, nh), where is a ‘heteroclinic bifurcation’ point for a related nonlinear ordinary differential equation. We use various concepts based on the branching of sufficiently small solutions from the known eigenfunctions of the linear rescaled operator corresponding to n = 0.

X-ray emission by a shocked fast wind from the central stars of planetary nebulae

We calculate the X-ray emission from the shocked fast wind blown by the central stars of planetary nebulae (PNe) and compare with observations. Using spherically symmetric self-similar solutions, we calculate the flow structure and X-ray temperature for a fast wind slamming into a previously ejected slow wind. We find that the observed X-ray emission of six PNe can be accounted for by shocked wind segments that were expelled during the early-PN phase, if the fast wind speed is moderate, v2∼ 400–600 km s−1, and the mass-loss rate is a few times 10−7 M⊙ yr−1. We find, as proposed previously, that the morphology of the X-ray emission is in the form of a narrow ring inner to the optical bright part of the nebula. The bipolar X-ray morphology of several observed PNe, which indicates an important role of jets, rather than a spherical fast wind, cannot be explained by the flow studied here.

Classification of radially symmetric self-similar solutions of $u_t=\Delta log u$ in higher dimensions

We give a complete classification of radially symmetric self-similar solutions of the equation $u_t=\Delta\log u$, $u>0$, in higher dimensions. For any $n\ge 2$, $\eta>0$, $\alpha$, $\beta\in \mathbb {R}$, we prove that there exists a radially symmetric solution for the corresponding elliptic equation $\Delta\log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $ \mathbb {R}^n$, $v(0)=\eta$, if and only if either $\alpha\ge 0$ or $\beta>0$. For $n\ge 3$, we prove that $\lim_{r\to\infty}r^2v(r) =2(n-2)/(\alpha -2\beta)$ if $\alpha>\max (2\beta,0)$ and $\lim_{r\to\infty}r^2v(r)/\log r=2(n-2)/\beta$ if $\alpha=2\beta>0$. For $n\ge 2$ and $2\beta>\max (\alpha ,0)$, we prove that $\lim_{r\to\infty}r^{\alpha/\beta}v(r)=A$ for some constant $A>0$.

Kink instability and stabilization of the Friedmann universe with scalar fields

The evolution of weak discontinuity is investigated in the flat FRW universe with a single scalar field and with multiple scalar fields. We consider both massless scalar fields and scalar fields with exponential potentials. Then we find that a new type of instability, i.e., kink instability develops in the flat FRW universe with massless scalar fields. The kink instability develops with scalar fields with considerably steep exponential potentials, while less steep exponential potentials do not suffer from kink instability. In particular, assisted inflation with multiple scalar fields does not suffer from kink instability. The stability of general spherically symmetric self-similar solutions is also discussed.

Letter: Self-Similar Collapse of Scalar Field with Plane Symmetry

Plane symmetric self-similar solutions to Einstein's four-dimensional theory of gravity are studied and all such solutions are given analytically in closed form. The local and global properties of these solutions are investigated and it is shown that some of the solutions can be interpreted as representing gravitational collapse of the scalar field. During the collapse, trapped surfaces are never developed. As a result, no black hole is formed. Although the collapse always ends with spacetime singularities, it is found that these singularities are spacelike and not naked.