Time Singularity Research Articles

Uniqueness of asymptotically conical tangent flows

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to "flow through" such a singularity.

Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1

We prove non-uniqueness for a class of weak solutions to the Navier–Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1.

Singularities in nearly uniform one-dimensional condensates due to quantum diffusion

Dissipative systems can often exhibit wavelength-dependent loss rates. One prominent example is Rydberg polaritons formed by electromagnetically-induced transparency, which have long been a leading candidate for studying the physics of interacting photons and also hold promise as a platform for quantum information. In this system, dissipation is in the form of quantum diffusion, i.e., proportional to $k^2$ ($k$ being the wavevector) and vanishing at long wavelengths as $k\to 0$. Here, we show that one-dimensional condensates subject to this type of loss are unstable to long-wavelength density fluctuations in an unusual manner: after a prolonged period in which the condensate appears to relax to a uniform state, local depleted regions quickly form and spread ballistically throughout the system. We connect this behavior to the leading-order equation for the nearly-uniform condensate -- a dispersive analogue to the Kardar-Parisi-Zhang (KPZ) equation -- which develops singularities in finite time. Furthermore, we show that the wavefronts of the depleted regions are described by purely dissipative solitons within a pair of hydrodynamic equations, with no counterpart in lossless condensates. We close by discussing conditions under which such singularities and the resulting solitons can be physically realized.

The Cauchy-Lagrange method for 3D-axisymmetric wall-bounded and potentially singular incompressible Euler flows

We treat the incompressible, and axisymmetric Euler equations for a three-dimensional cylindrical domain with boundaries. The equations are solved by the novel Cauchy-Lagrange algorithm (CLA), which uses the time-analyticity of the Lagrangian trajectories of an incompressible Euler flow and computes the time-Taylor coefficients of the Lagrangian map via recursion relations. This semi-Lagrangian algorithm uses a pseudo-spectral type approach in space by approximating the flow fields by Chebyshev-Fourier polynomials. New methods are presented to solve the resulting Poisson problems directly for their second-order space derivatives. The flow fields, known on the Lagrangian trajectories after one time-step, are interpolated back onto the Eulerian grid to start a new recursion cycle. The time-step is only limited by the radius of convergence and, thus, independent of any Courant-Friedrichs-Lewy (CFL) condition. This allows to advance the flow with larger time-steps independently of the mesh. Stationary and swirl-free flows are used to thoroughly test our implementation for the given geometry. In this work, our ultimate goal is to apply the CLA to a flow, which might develop finite time singularities, resulting in a loss of smoothness. This demands either 3D, or 2D with swirl, and may well require the presence of solid boundaries, as indicated in recent numerical work (Luo and Hou (2014) [45]).

Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems

In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., & Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019. Furthermore, we also prove that the solution to the \(abcd-\)Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for the \(abcd-\)Boussinesq system. These latter mathematical results are related to the numerics in Bona, and Chen, Singular solutions of a Boussinesq system for water waves. J. Math. Study, 49(3), 205-220, 2016.

Gevrey regularity and finite time singularities for the Kakutani–Matsuuchi model

In this paper, we deal with the Kakutani–Matsuuchi model which describes the surface elevation η of the water-waves under the effect of viscosity. We first derive the decay rate of weak solutions. This can be used to obtain the decay rate of ‖η(t)‖Ḣ1 when initial data is sufficiently small in Ḣ1. We next show the existence, uniqueness, Gevrey regularity and decay rates of η with sufficiently small initial data in B2,11. To do so, we derive a commutator estimate involving Gevrey operator. We then apply our method to the supercritical quasi-geostrophic equations. We finally show the formation of singularities of smooth solutions in finite time for a certain class of initial data.

On nonlinear singular BVPs with nonsmooth data. Part 2: Convergence of collocation methods

We discuss numerical solution of boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,x′(t)=M(t)tx(t)+f(t,x(t))t,t∈(0,1],b(x(0),x(1))=0, where M:[0,1]→Rn×n and f:[0,1]×Rn→Rn are continuous matrix-valued and vector-valued functions, respectively. Moreover, b:Rn×Rn→Rn is a continuous nonlinear mapping which is specified according to a spectrum of the matrix M(0) to guarantee the BVP to be well-posed. For the case where M(0) has eigenvalues with nonzero real parts, we prove new convergence results for the collocation method and analytical results about the necessary smoothness of the solution to the problem required in the numerical analysis.We illustrate the theory by means of numerical examples.

Psychosocial Effect of Newborn Genomic Sequencing on Families in the BabySeq Project

Newborn genomic sequencing (nGS) may provide health benefits throughout the life span, but there are concerns that it could also have an unfavorable (ie, negative) psychosocial effect on families. To assess the psychosocial effect of nGS on families from the BabySeq Project, a randomized clinical trial evaluating the effect of nGS on the clinical care of newborns from well-baby nurseries and intensive care units. In this randomized clinical trial conducted from May 14, 2015, to May 21, 2019, at well-baby nurseries and intensive care units at 3 Boston, Massachusetts, area hospitals, 519 parents of 325 infants completed surveys at enrollment, immediately after disclosure of nGS results, and 3 and 10 months after results disclosure. Statistical analysis was performed on a per-protocol basis from January 16, 2019, to December 1, 2019. Newborns were randomized to receive either standard newborn screening and a family history report (control group) or the same plus an nGS report of childhood-onset conditions and highly actionable adult-onset conditions (nGS group). Mean responses were compared between groups and, within the nGS group, between parents of children who received a monogenic disease risk finding and those who did not in 3 domains of psychosocial impact: parent-child relationship (Mother-to-Infant Bonding Scale), parents' relationship (Kansas Marital Satisfaction Scale), and parents' psychological distress (Edinburgh Postnatal Depression Scale anxiety subscale). A total of 519 parents (275 women [53.0%]; mean [SD] age, 35.1 [4.5] years) were included in this study. Although mean scores differed for some outcomes at singular time points, generalized estimating equations models did not show meaningful differences in parent-child relationship (between-group difference in adjusted mean [SE] Mother-to-Infant Bonding Scale scores: postdisclosure, 0.04 [0.15]; 3 months, -0.18 [0.18]; 10 months, -0.07 [0.20]; joint P = .57) or parents' psychological distress (between-group ratio of adjusted mean [SE] Edinburgh Postnatal Depression Scale anxiety subscale scores: postdisclosure, 1.04 [0.08]; 3 months, 1.07 [0.11]; joint P = .80) response patterns between study groups over time for any measures analyzed in these 2 domains. Response patterns on one parents' relationship measure differed between groups over time (between-group difference in adjusted mean [SE] Kansas Marital Satisfaction Scale scores: postdisclosure, -0.19 [0.07]; 3 months, -0.04 [0.07]; and 10 months, -0.01 [0.08]; joint P = .02), but the effect decreased over time and no difference was observed on the conflict measure responses over time. We found no evidence of persistent negative psychosocial effect in any domain. In this randomized clinical trial of nGS, there was no persistent negative psychosocial effect on families among those who received nGS nor among those who received a monogenic disease risk finding for their infant. ClinicalTrials.gov Identifier: NCT02422511.

Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457\ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at $a=1/2$ as well as prove the existence of a leading order complex singularity for general values of $a$ in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For $a_c<a\leq1$, we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For $a \gtrsim 1.3$, we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for $a<a_c$ which are similar to the real line case. For $a_c<a\le0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $ a\ge 1,$ we identify a global existence of solutions.

Disturbance estimator based dynamic compensator design for fractional order fuzzy control systems

The robust stabilization problem for singular fractional order time delay T-S fuzzy systems with nonlinearities and unknown external disturbances is addressed in this paper. An equivalent-input-disturbance (EID) estimator is used to estimate the impact of external disturbances and nonlinearities on the system output. Based on this EID approach, a dynamic compensator is designed to solve the stabilization problem of the considered system. Moreover, by considering a relevant Lyapunov-Krasovskii functional candidate and by using Lyapunov technique, the stability conditions in terms of LMIs are acquired for the considered closed-loop system. At last, to validate the effectiveness of the proposed result, two numerical examples are provided.

Infinite-time singularities models and possible avoidance

In this work, we explore two models of future Infinite time singularities in f(T) theory, where T is the torsion scalar. We consider the algebraic function g(T) as the teleparallel term T plus arbitrary function f(T). A suitable Hubble parameter expression is assumed to characterize both of the two singularities models and an expanding universe is provided from imposed constraints. We solve differential equations of g(T) and obtain the algebraic g(T) models of each type of future infinite singularity. In order to study the possible avoidance of singularities, we take into account the viscosity in the fluid and explore three interesting cases: constant viscosity δo, non-constant viscosity proportional to energy density δoρα with α a real parameter and the general case where the viscosity is proportional to (−T)n2, where n is a natural number. We notice that in the first case, the Little Rip singularity may be avoided but the Pseudo Rip is robust against the viscous fluid while for the non-constant viscosity and general case, both singularities cannot be avoided.

The Euler disk and its dynamic finite time singularity: Investigating a fascinating acoustical and mechanical phenomenon with simple means

Many have spun a coin or other disk on a table, observing — and listening! — that — while its motion comes to rest, the angular velocity of the contact point and the frequency of the emitted sound raise in a striking way. This dynamical system is known as the "Euler disk," and its physics is theoretically understood as a "finite time singularity" in form of an inverse power law. Various dissipation mechanisms lead to different exponents in the power law. The present investigation is about different configurations of the Euler disk using acoustical and mechanical data. We present results for the well-known Euler disk toy, well in accord with a model of viscous air dissipation. This is the first measurement using the acoustical signal for the dynamics of the Euler disk. Moreover, we present new cases where other dissipation mechanism must be dominant. Experiments are carried out with simple means and using a smartphone for recording and analysing data. This and interesting physics of the Euler disc make it a good candidate e.g. for undergraduate lab-work courses or research projects, which can spark students interest in acoustics, and more generally in applying physics to phenomena of everyday life.

The harmonic heat flow of almost complex structures

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure ( M , g ) (M, g) . This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J J has small energy (depending on the norm | ∇ J | |\nabla J| ), then the flow exists for all time and converges to a Kähler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kähler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.

Big bounce and future time singularity resolution in Bianchi I cosmologies: The projective invariant Nieh-Yan case

We extend the notion of the Nieh-Yan invariant to generic metric-affine geometries, where both torsion and nonmetricity are taken into account. Notably, we show that the properties of projective invariance and topologicity can be independently accommodated by a suitable choice of the parameters featuring this new Nieh-Yan term. We then consider a special class of modified theories of gravity able to promote the Immirzi parameter to a dynamical scalar field coupled to the Nieh-Yan form, and we discuss in more detail the dynamics of the effective scalar tensor theory stemming from such a revised theoretical framework. We focus, in particular, on cosmological Bianchi I models and we derive classical solutions where the initial singularity is safely removed in favor of a big bounce, which is ultimately driven by the nonminimal coupling with the Immirzi field. These solutions, moreover, turn out to be characterized by finite time singularities, but we show that such critical points do not spoil the geodesic completeness and wave regularity of these spacetimes.

On the Slightly Perturbed De Gregorio Model on $$S^1$$

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $\omega_t + a u\omega_x = u_x \omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $|| \omega(t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type-$I$ mean curvature

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in ${\\mathbb R}^3$ is of type-$I$, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-$I$. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in ${\\mathbb R}^3$ is at least of type-$I$.

De Sitter vacua in the string landscape

The late-time behavior of our universe is one of accelerated expansion, or that of a de Sitter space, and therefore motivates us to look for time-dependent backgrounds. Finding such backgrounds in string theory has always been a challenging problem. An even harder problem is to find time-dependent backgrounds that allow positive dark energies. As a first step to handle such scenarios, we study a time-dependent background in type IIB theory, with four-dimensional de Sitter isometries, by uplifting it to M-theory and then realizing it as a coherent, or squeezed-coherent, state over an appropriate solitonic configuration. While classically such a background does not solve the equations of motion, the corresponding Schwinger-Dyson equations reveal that there are deeper issues that may even prohibit a solution to exist at the quantum level, as long as the internal space remains time-independent. A more generic analysis is then called for, where both the effective four-dimensional space-time, the internal space, and the background fluxes are all time-dependent. We study in details such a background by including perturbative and non-perturbative as well as local and non-local quantum terms. Our analysis reveals a distinct possibility of the emergence of a four-dimensional positive curvature space-time with de Sitter isometries and time-independent Newton's constant in the landscape of type IIB string theory. We argue how the no-go and the swampland criteria are avoided in generating such a background, and compare it with other possibilities involving backgrounds with time-dependent Newton constants. These time-varying Newton constant backgrounds typically lead to unavoidable late time singularities, amongst other issues.

Smoothing Transforms for Dynamical Systems with Non-smooth Input

Many control systems incorporate input from actuators. The actuators can be represented as force or displacement input. When an actuator is modeled as a displacement input, there are many practical applications for which the displacement is not smooth and its time derivative, i.e. the velocity, is not continuous everywhere. Numerical study for such displacement input becomes tedious since the state variables experience a finite jump at the time of acceleration singularity. Although the acceleration singularity can be avoided if the actuator dynamics is modeled by a force input, such an approach increases the order of the overall system. To preserve the simplicity of the displacement input of the actuators, we propose transforms of state variables so that only the velocity of the displacement input appears explicitly in the equations of the dynamical system. In short, we introduce transforms that reduce the order of time derivatives of the input. This approach simplifies the numerical solutions for non-smooth displacement input. Moreover, when the transform is applied to a parametrically excited pendulum, the stabilizing effect of the parametric excitation is shown explicitly.

Energy conditions for inhomogeneous EOS and its thermodynamics analysis with the resolution on finite time future singularity problems

In this paper, we study two different cases of inhomogeneous EOS of the form [Formula: see text]. We derive the energy density of dark fluid and dark matter component for both the cases. Further, we calculate the evolution of energy density, gravitational constant and cosmological constant. We also explore the finite time singularity and thermodynamic stability conditions for the two cases of EOS. Finally, we discuss the thermodynamics of inhomogeneous EOS with the derivation of internal energy, Temperature and entropy and also show that all the stability conditions are satisfied for the two cases of EOS.