Finite-time Singularity Research Articles

Yang-Mills flow on special-holonomy manifolds

This paper develops Yang-Mills flow on Riemannian manifolds with special holonomy. By analogy with the second-named author's thesis, we find that a supremum bound on a certain curvature component is sufficient to rule out finite-time singularities. Assuming such a bound, we prove that the infinite-time bubbling set is calibrated by the defining (n−4)-form.

Touchdown is the Only Finite Time Singularity in a Three-Dimensional MEMS Model

Touchdown is shown to be the only possible finite time singularity that may take place in a free boundary problem modeling a three-dimensional microelectromechanical system. The proof relies on the energy structure of the problem and uses smoothing effects of the semigroup generated in L 1 by the bi-Laplacian with clamped boundary conditions.

The effect of viscosity ratio on drop pinch-off dynamics in two-fluid flow

Drop pinch-off draws a lot of interest from scientists and engineers because of widely practical applications as well as the fascinating mechanism of finite-time singularities and self-similar behavior. This work experimentally investigates the effect of viscosity ratio on the local pinch-off mechanism of liquid drops in both air and viscous fluids, with a high-speed camera. The results show that for 56 ≤ λ ≤ 6.3×103 and 0.009 ≤ λ ≤ 0.061, drop pinch-off exhibits self-similar profile which is asymmetric and conical away from hmin. Drop pinch-off in air with low viscosity ratios shows either in inertial regime (I) or transition from inertial to viscous regime (I→V). But the kinetics undergoes the inertial regime (I) to inertial-viscous regime (IV) through an intermediate viscous regime (V) when the viscosity ratio becomes larger. Drop pinch-off in viscous fluids displays the transition from inertial to viscous regime (I→V). These results agree well with Eggers’s universal solution until it becomes unstable and the previous literature. The viscosity ratio indeed affects the drop pinch-off dynamics as well as interface deformation. Our experiments enrich the understanding of the interaction between the internal and external fluids on drop pinch-off behavior near the singularity point.

Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian

Abstract

A fundamental method for prediction of failure of strain hardening cementitious composites without prior information

Strain Hardening cementitious composites (SHCCs) have been known to produce excellent crack width control and very good tensile strain capacity. As a result, strategically introducing SHCCs would lead to significant carbon saving through the creation of structures that are durable and could potentially undergo autogenous self-healing. However, for wider acceptability of any new material requires long term empirical data. This may be circumvented by continually ensuring a structure made with this new material is serviceable i.e. material has not failed. We present an approach to predict failure time without any prior information on operating conditions and material information for SHCCs. This method is based on the modeling of ‘finite time singularity’ based on mathematical laws. The method produces fairly accurate results when applied to estimate failure for SHCCs with a wide range of failure times and to both linear and non-linear loading regimes. The practical applicability of this method was thus demonstrated.

A fluid mechanic's analysis of the teacup singularity.

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

Human wealth evolution: Trends and fluctuations

Is a causal description of human wealth history conceivable? To investigate the matter we introduce a simple causal albeit strongly aggregated model, assuming that the observed wealth growth is mainly driven by human collaborative efforts whose intensity itself increases with increasing wealth. As an empirical reference we use time series describing eight centuries of per capita annual gross domestic products (GDP) of three European countries, the UK, France and Sweden. The model requires a population large enough for disruptive events, e.g. famine, epidemics and wars, not to destroy the fundamental workings of society. Cultural interchanges between different geographical areas are not explicitly taken into account. The wealth evolution trend can then be described by an ordinary differential equation with three free parameters: one producing a short term exponential growth rate, one defining an additional minute constant growth rate and finally one that specifies the time scale below which human collaboration intensity can be treated as constant. Beyond that scale, wealth enhances the fundamental collaborative infrastructure of society. The solution features a finite time singularity, which implies infinite GDP per capita in finite time and thus suggests a lack of long term sustainability. The year at which the singularity occurs has a slight variation near 2020 AD from one country to another. GDP time series curtailed after 1900 AD produce similar values for the occurrence of the singularity, which thus could be predicted more than hundred years ago. Curtailing the GDP series from the early years up to 1700 AD also produces stable and consistent predictions for the singularity time. Power spectra are obtained for de-trended data spanning eight centuries, as well as for the first and last four centuries of the same period. All spectra have an overall signature where the power decays as the inverse frequency squared. The embedded peaks are reminiscent of the cycles described in the economic literature, but are also present in time series far predating industrialization. The background fluctuations in the GDP series is tentatively interpreted as societal response to disruptive stochastic events. e.g. new economic activities following epochal discoveries, as well as wars and epidemics.

The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

Anisotropic long-range interaction investigated with cold atoms

In two dimensions, a system of self-gravitating particles collapses and forms a singularity in finite time below a critical temperature $T_c$. We investigate experimentally a quasi two-dimensional cloud of cold neutral atoms in interaction with two pairs of perpendicular counter-propagating quasi-resonant laser beams, in order to look for a signature of this ideal phase transition: indeed, the radiation pressure forces exerted by the laser beams can be viewed as an anisotropic, and non-potential, generalization of two-dimensional self-gravity. We first show that our experiment operates in a parameter range which should be suitable to observe the collapse transition. However, the experiment unveils only a moderate compression instead of a phase transition between the two phases. A three-dimensional numerical simulation shows that both the finite small thickness of the cloud, which induces a competition between the effective gravity force and the repulsive force due to multiple scattering, and the atomic losses due to heating in the third dimension, contribute to smearing the transition.

Hermitian curvature flow on complex locally homogeneous surfaces

We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-K\"ahler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry.

On mean curvature flow of singular Riemannian foliations: Noncompact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and assure convergence of MCF of non-closed leaves of generalized isoparametric foliation on compact manifold.

Mixed Integral Norms for Ricci Flow

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.

Formation of finite-time singularities for nonlinear hyperbolic systems with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

Blow-Up Solutions to 3D Euler are Hydrodynamically Unstable

We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. This article explains why the prediction of such a blow-up, via direct numerical experiments, is so difficult. It is described how, in such a scenario, the solution becomes unstable as time approaches the blow-up time.

Scalar V-soliton equation and Kähler-Ricci flow on symplectic quotients

In this paper, we consider the V-soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein metrics, we can also reduce the V-soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We formulate some preliminary estimates for such a scalar equation on a compact Kähler manifold M.

A space time conservation element and solution element method for solving two-species chemotaxis model

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

Numerical study on the instability of localized buckling modes in the axially compressed strut on a distributed-spring elastic foundation with softening quadratic nonlinearity

Localized buckling modes in the axially compressed strut on a distributed-spring elastic foundation with a softening quadratic nonlinearity are numerically calculated based on a modified Petviashvili method in the spatial frequency domain. As the load decreases (increases), the maximum displacement of the corresponding localized buckling mode increases (decreases) and its width decreases (increases). Then, under the influence of longitudinal and transverse perturbations, stabilities of these localized buckling modes are numerically investigated. The adopted numerical method is the spatial Fourier transform in space and the finite difference method in time. For initial positive longitudinal perturbations, localized buckling modes are unstable showing focusing-type finite-time blowup singularities. For initial negative perturbations, localized buckling modes are unstable and become dispersed showing small-amplitude oscillations. For initial transverse perturbations, localized buckling modes are unstable and are transformed into three-dimensional locally confined modes, finally showing finite-time blowup singularities.

Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System

As a follow up to our work [27], we give examples of finite-energy and Lipschitz continuous velocity field and density $$(u_0,\rho _0)$$ which are $$C^\infty $$-smooth away from the origin and belong to a natural local well-posedness class for the Boussinesq equation whose corresponding local solution becomes singular in finite time. That is, while the sup norm of the gradient of the velocity field and the density remain finite on the time interval $$t\in [0,1)$$, both quantities become infinite as $$t\rightarrow 1$$. The key is to use scale-invariant solutions similar to those introduced in [27]. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of certain special infinite-energy solutions belonging to those critical spaces, and finally a cut-off argument to ensure finiteness of energy. All of this is done on spatial domains $$\{(x_1,x_2): x_1 \ge \gamma |x_2|\}$$ for any $$\gamma > 0$$ so that we can get arbitrarily close to the half-space case. We show that the 2D Euler equation is globally well-posed in all of the situations we look at, so that the singularity is not coming from the domain or the lack of smoothness on the data but from the vorticity amplification due to the presence of a density gradient. It is conceivable that our methods can be adapted to produce finite-energy $$C^\infty $$ solutions on $${\mathbb {R}}^2_+$$ which become singular in finite time.

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $\mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of $\mathcal{E}_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to $\mathcal{E}_0^{3/2}$ as $\mathcal{E}_0$ becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

On the energy of critical solutions of the binormal flow

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schrödinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity.