Time Singularity Research Articles

Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with χ 3 nonlinear response

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

Bounded Diameter Under Mean Curvature Flow

We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case the diameter always stays bounded. We also obtain sharp \(L^{n-1}\) bound for the curvature. The key ingredients for our proof are the Łojasiewicz inequalities by Colding–Minicozzi and Chodosh–Schulze, and the solution of the mean-convex neighbourhood conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the prior results by Gianniotis–Haslhofer, where diameter and curvature control has been obtained under the more restrictive assumption that the flow is globally two-convex.

The Chern–Ricci flow on primary Hopf surfaces

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern–Ricci flow. We use a construction of Gauduchon–Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern–Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round\(S^1\). Uniform \(C^{1+\beta }\) estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

AbstractWe present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We also prove self‐similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on ℝ or for Hölder‐continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self‐similar singularity into the problem of establishing the nonlinear stability of an approximate self‐similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self‐similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well‐posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations. © 2021 Wiley Periodicals LLC.

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^{\theta})^2, u_0^r, u_0^z \in C^{1,\alpha}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,\alpha}$, our interpretation that the velocity is $C^{1,\alpha}$ is not correct since the velocity in 3D also depends on $u^{\theta}$, which is not $C^{1,\alpha}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden\u2013Fowler equation generated by derivative with non-singular kernel

In this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s>1, and suggests that a suitable space for the well-posedness is Gevrey class of order s=1, which is exactly the space of analytic functions.

On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family of transport equations given by a parameter \(\alpha \), \(0< \alpha <2\), with the cases \(\alpha =0\) and \(\alpha =1\) corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for \(H^{2}\) patches in the half-space setting for \(0<\alpha < 1/3\), allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of \(\alpha \) for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for \(0<\alpha <2\) as long as the arc-chord condition and the regularity of order \(C^{1+\delta }\) for \(\delta >\alpha /2\) are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in \(H^{2}\) for \(0<\alpha < 1\) and in \(H^3\) for \(1<\alpha <2\).

Curve Flows with a Global Forcing Term

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. We prove an analogue to Huisken’s distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below -pi and show that this condition is sharp. With that, we can exclude singularities in finite time for bounded forcing terms. For immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle. In particular, all of the above holds for the area preserving curve shortening flow.

Local Regularity for the Harmonic Map and Yang\u2013Mills Heat Flows

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

First Return Time to the contact hyperplane for $ N $-degree-of-freedom vibro-impact systems

&lt;p style='text-indent:20px;'&gt;The paper deals with the dynamics of conservative &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ N $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.&lt;/p&gt;

Error Analysis of Nonlinear Time Fractional Mobile/Immobile Advection-Diffusion Equation with Weakly Singular Solutions

Abstract In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. The memory requirement and computational cost are O(Q) and O(QK), respectively, where K is the number of the final time step and Q is the number of quadrature points used in the trapezoidal rule. Some numerical experiments are given to confirm our theoretical analysis and the effectiveness of the presented methods.

On the conditions for the breaking of oscillations in a cold plasma

The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the existence of a global in time solutions is obtained. For the original problem, a sufficient condition for the loss of smoothness is found, as well as a sufficient condition for the solution to remain smooth at least for time $ 2 \pi $. In addition, it is shown that in the general case, arbitrarily small perturbations of the trivial state lead to the formation of singularities in a finite time. It is further proved that there are special initial data such that the respective solution remains smooth for all time, even in the relativistic case. Periodic in space traveling wave gives an example of such a solution. In order for such a wave to be smooth, the velocity of the wave must be greater than a certain constant that depends on the initial data. Nevertheless, arbitrary small perturbation of general form destroys these global in time smooth solutions. The nature of the singularities of the solutions is illustrated by numerical examples.

On Weak Super Ricci Flow through Neckpinch

Abstract In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊 n . We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.

Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $

&lt;abstract&gt;&lt;p&gt;In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.&lt;/p&gt;&lt;/abstract&gt;

How Do Degenerate Mobilities Determine Singularity Formation in Cahn--Hilliard Equations?

Cahn--Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have nonconstant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn--Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory and enables the systematic development of robust numerical methods for this family of model equations.

On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case

&lt;p style='text-indent:20px;'&gt;A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \lambda $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \lambda $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.&lt;/p&gt;

The Fuchsian approach to global existence for hyperbolic equations

We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions toward the singular time under a small initial data assumption. We, then, apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.

Short-time existence of the α-Dirac-harmonic map flow and applications

In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for α-Dirac-harmonic maps. The obstacle to the global existence is the singular time when the kernel of the Dirac operator no longer stays minimal along the flow. In this case, the kernel may not be continuous even if the map is smooth with respect to time. To overcome this issue, we use the analyticity of the target manifold to obtain the density of the maps along which the Dirac operator has minimal kernel in the homotopy class of the given initial map. Then, when we arrive at the singular time, this density allows us to pick another map which has lower energy to restart the flow. Thus, we get a flow which may not be continuous at a set of isolated points. Furthermore, with the help of small energy regularity and blow-up analysis, we finally get the existence of nontrivial α-Dirac-harmonic maps () from closed surfaces. Moreover, if the target manifold does not admit any nontrivial harmonic sphere, then the map part stays in the same homotopy class as the given initial map.

Stability of switched singular time delay systems with switching induced state jumps

This study addresses the problem of exponential stability for switched singular state-delayed systems with switching induced state jumps, which has not been studied up to now. The considered state delay varies in a time-varying interval. On the basis of equivalent dynamics decomposition, a model of state jump at switching instants is firstly established under an assumption that the time length between arbitrary two adjacent switches is larger than the upper bound of the state delay. Then, a sufficient condition on exponential stability of the system under the reranged dwell-time switching constraint is presented. The key idea is the design of a dwell-time-dependent generalised Lyapunov function as well as a dwell-time-dependent function with respect to algebraic variables and application of the Razumikhin approach. The obtained stability condition exploits the lower bound and the upper bound of the dwell time. In addition, it is independent of the size of the state delay and allows the delay to be a fast time-varying function. Finally, two numerical examples are given to show the efficacy of the derived result.