Finite-time Singularity Formation Research Articles

On singularity formation of a 3D model for incompressible Navier–Stokes equations

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.

Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a Singular Nonlinearity Modeling a MEMS Capacitor

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric 2D case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

Nonsingular exponential gravity: A simple theory for early- and late-time accelerated expansion

A theory of exponential modified gravity which explains both early-time inflation and late-time acceleration, in a unified way, is proposed. The theory successfully passes the local tests and fulfills the cosmological bounds and, remarkably, the corresponding inflationary era is proven to be unstable. Numerical investigation of its late-time evolution leads to the conclusion that the corresponding dark energy epoch is not distinguishable from the one for the $\Lambda$CDM model. Several versions of this exponential gravity, sharing similar properties, are formulated. It is also shown that this theory is non-singular, being protected against the formation of finite-time future singularities. As a result, the corresponding future universe evolution asymptotically tends, in a smooth way, to de Sitter space, which turns out to be the final attractor of the system.

Singularity formations for a surface wave model

In this paper we study the Burgers equation with a nonlocal term of the form Hu where H is the Hilbert transform. This system has been considered as a quadratic approximation for the dynamics of a free boundary of a vortex patch (see Biello and Hunter 2010 Commun. Pure Appl. Math. LXIII 0303–36; Marsden and Weinstein 1983 Physica D 7 305–23). We prove blowup in finite time for a large class of initial data with finite energy. Considering a more general nonlocal term, of the form ΛαHu for 0 < α < 1, finite time singularity formation is also shown.

Wave breaking for the periodic weakly dissipative Dullin–Gottwald–Holm equation

In this paper, we consider the periodic weakly dissipative Dullin–Gottwald–Holm equation. The present work is mainly concerned with blow-up phenomena for the Cauchy problem for this new kind of equation. We apply the optimal constant to give sufficient conditions via an appropriate integral form of the initial data, which guarantee the finite-time singularity formation for the corresponding solution.

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.

Critical thresholds in hyperbolic relaxation systems

Critical threshold phenomena in one-dimensional 2 × 2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.

Wave Breaking and Persistence Properties for the Dispersive Rod Equation

This paper is concerned with some aspects of blow up of solutions and persistence properties. Firstly, we will try to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Then a particular class of initial data for the periodic case is also considered in this paper. Finally, we investigate the persistence properties of the strong solutions.

Blow up and propagation speed of solutions to the DGH equation

A wave-breaking mechanism for solutionswith certain initial profiles and propagation speed are discussed inthis paper. Firstly, we apply the best constant to give sufficientcondition via an appropriate integral form of the initial data,which guarantees finite time singularity formation for thecorresponding solution, then we establish blow up criteria via theconserved quantities. Finally, persistence properties of the strongsolutions are presented and infinite propagation speed is alsoinvestigated in the sense that the corresponding solution $u(x,t)$does not have compact spatial support for $t>0$ though $u_0 \inC_0^{\infty}(\mathbb{R})$.

Critical thresholds in a quasilinear hyperbolic model of blood flow

Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.

Blow-up phenomenon for a periodic rod equation

In this paper, firstly we find the optimal constant for a kind of Sobolev inequality on the unit circle via Fourier series. Then we apply the optimal constant on a nonlinear rod equation to give sufficient conditions on the initial datum, which guarantee finite-time singularity formation for the corresponding solution.

Critical thresholds in a relaxation model for traffic flows

In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traYc flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual sub-characteristic con- dition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formation of solutions si- multaneously by showing the critical threshold phenomena associated with the underlying relaxation system. We identify five upper thresh- olds for finite time singularity in solutions and three lower thresholds for global existence of smooth solutions. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope. Our results show that the shorter the drivers' respond- ing time to the traYc, the larger the set of initial conditions leading to global smooth solutions which correctly predicts the empirical findings for traY cfl ows.

Current sheet formation at a magnetic neutral line in Hall magnetohydrodynamics

The dynamics of a plasma in the vicinity of a neutral line of the magnetic field is considered in the framework of incompressible Hall magnetohydrodynamics (MHD). A self-similar solution for the collapse to a current sheet is obtained. Numerical and analytical results are presented. In contrast to the standard incompressible MHD model in two dimensions, the Hall effect leads to the formation of finite-time singularities of the velocity derivatives and the electric current density at the neutral line. Analytical expressions are derived for the collapse time and the form of the solution near the singularity, which agree closely with the numerical results. If the ion skin depth is set to zero, the singularity formation time becomes infinite, corresponding to the standard MHD result. Implications of the results for Hall MHD models of fast magnetic reconnection are discussed.

Blow-up of solutions to the DGH equation

In this paper, firstly we find best constants for two convolution problems on the unit circle via a variational method. Then we apply the best constants on a nonlinear integrable shallow water equation (the DGH equation) to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Finally, we discuss the blow-up phenomena for the nonperiodic case.

Blow-up of solutions to a nonlinear dispersive rod equation

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions.

New isotropic and anisotropic sudden singularities

We show the existence of an infinite family of finite-time singularities in isotropically expanding universes which obey the weak, strong and dominant energy conditions. We show what new type of energy condition is needed to exclude them ab initio. We also determine the conditions under which finite-time future singularities can arise in a wide class of anisotropic cosmological models. New types of finite-time singularity are possible which are characterized by divergences in the time rate of change of the anisotropic-pressure tensor. We investigate the conditions for the formation of finite-time singularities in a Bianchi-type VII0 universe with anisotropic pressures and construct specific examples of anisotropic sudden singularities in these universes.

Axisymmetric Flows in Hall-MHD: A Tendency Towards Finite-Time Singularity Formation

Spontaneous development of shock-like singularities in axisymmetric solutions of the Hall-MHD equations is discussed. It is shown that the Hall-term in Ohm's law leads to a Burgers-type equation for the magnetic field evolution in weakly compressible regime. Numerical simulations are used to investigate the validity of this approximation for a particular class of initial conditions.

Unsteady three-dimensional marginal separation caused by surface-mounted obstacles and/or local suction

Earlier investigations of steady two-dimensional marginally separated laminar boundary layers have shown that the non-dimensional wall shear (or equivalently the negative non-dimensional perturbation displacement thickness) is governed by a nonlinear integro-differential equation. This equation contains a single controlling parameter . This in turn leads to a significant simplification of the problem allowing, among other things, a systematic study of devices used in boundary-layer control and an analytical investigation of the conditions leading to the formation of finite-time singularities which have been observed in earlier numerical studies of unsteady two-dimensional and three-dimensional flows in the vicinity of a line of symmetry. Also, it is found that it is possible to construct exact solutions which describe waves of constant form travelling in the spanwise direction. These waves may contain singularities which can be interpreted as vortex sheets. The existence of these solutions strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thereby generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation bubbles. This is supported by asymptotic analysis.

Current-sheet formation in incompressible electron magnetohydrodynamics.

The nonlinear dynamics of axisymmetric, as well as helical, frozen-in vortex structures is investigated by the Hamiltonian method in the framework of ideal incompressible electron magnetohydrodynamics. For description of current-sheet formation from a smooth initial magnetic field, local and nonlocal nonlinear approximations are introduced and partially analyzed that are generalizations of the previously known exactly solvable local model neglecting electron inertia.

Collapse of a class of three-dimensional Euler vortices

Substitution of the flow field U ( x,y,z,t )={ u ( x, y, t ), v ( x, y, t ), z γ ( x, y, t )} into the three-dimensional incompressible Euler equations generates a closed system of evolution equations, for the strain rate γ ( x, y, t ) and the two-dimensional vorticity ω ( x, y, t ), which are uniform in the z -direction. The system models a class of dynamical, stretched three-dimensional vortex flows that include Burgers9 vortices. Recent numerical simulations by Ohkitani & Gibbon have revealed that the strain rate γ ( x, y, t ) appears to develop a finite-time singularity, from smooth initial data, in the region where gamma is negative. Here, we prove that, for a large class of initial data, the support of γ - := max{0, - γ } necessarily collapses to zero in a finite time, while at the same time, the L 1 norm of γ - remains non-zero. Hence, γ - must necessarily become singular before or at the time of collapse. Our vortex flow represents one of a subclass of Euler solutions that have infinite energy. The fundamental question of finite-time singularity formation from smooth initial data for finite-energy three-dimensional Euler solutions remains the important open question.