scholarly journals Type II singularities on complete non-compact Yamabe flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Beomjun Choi ◽  
Panagiota Daskalopoulos ◽  
John King
Keyword(s):  
Blow Up ◽  
Type Ii ◽  
Conformally Flat ◽  
Time Singularity ◽  
Yamabe Flow ◽  
Blowing Up ◽  
Rotationally Symmetric ◽  
Blow Up Rate ◽  
Finite Time Singularity ◽  
First Time

AbstractThis work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

scholarly journals Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

2019 ◽  
Vol 2019 (754) ◽  
pp. 225-251 ◽  
Author(s):  
James Isenberg ◽  
Haotian Wu
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type Ii ◽  
Matched Asymptotics ◽  
Curve Shortening Flow ◽  
Initial Hypersurface ◽  
Rate Dependent ◽  
Blowing Up

Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Linrui Li ◽  
Shu Wang
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Smooth Solution ◽  
Blow Up ◽  
Nonlocal Term ◽  
Convection Term ◽  
Large Initial Data ◽  
Singularity Formation ◽  
Time Singularity ◽  
Finite Time Singularity

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.

scholarly journals Type-II singularities of two-convex immersed mean curvature flow

2016 ◽  
Vol 2 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Theodora Bourni ◽  
Mat Langford
Keyword(s):  
Mean Curvature ◽  
General Class ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Analogous Result ◽  
Curvature Flow ◽  
Type Ii ◽  
Rotationally Symmetric ◽  
The Mean ◽  
Mean Convex

AbstractWe show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of twoconvex hypersurfaces by the two-harmonic mean curvature.

scholarly journals Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mauricio Bogoya ◽  
Julio D. Rossi
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Asymptotic Bounds ◽  
Blowing Up ◽  
Blow Up Rate ◽  
Blowing Up Solutions

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

scholarly journals Blow Up of Quintic Power 1-D Nonlinear Schroedinger Equation

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Blow Up ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Heat Equations ◽  
Adaptive Grids ◽  
Time Singularity ◽  
One Dimensional ◽  
Blowing Up ◽  
Similarity Scaling ◽  
The One

This work studies an adaptive finite difference approximation to the one dimensional nonlinear Schroedinger equiation with quintic power, with special emphasis on the case when the solution blows up with finite blowing-up time $T_\infty.$ The adaptivity is utilizing similarity scaling adaptive grids studied by Berger and Kohn to study numerical solution of semilinear heat equations with finite blowing-up time.Furthermore, we reports an asymptotic behavior of the blow-up solution approaching $T_\infty$ time singularity.

scholarly journals Evidence of log-periodic oscillations and increasing icequake activity during the breaking-off of large ice masses

2008 ◽  
Vol 54 (187) ◽  
pp. 725-737 ◽  
Author(s):  
Jérome Faillettaz ◽  
Antoine Pralong ◽  
Martin Funk ◽  
Nicholas Deichmann
Keyword(s):  
Snow Avalanches ◽  
Velocity Measurements ◽  
Time Singularity ◽  
Periodic Oscillations ◽  
Physical Mechanisms ◽  
Naturally Occurring ◽  
Combined Motion ◽  
Power Law Function ◽  
Finite Time Singularity ◽  
First Time

AbstractIn 1973, surface velocities were measured for the first time on an unstable hanging glacier to predict its collapse. The observed velocities have been shown to increase as a power-law function of time up to infinity at the theoretical time of failure (known as ‘finite time singularity’). This is the characteristic signature of critical phenomena and has been observed in the case of various other naturally occurring ruptures such as earthquakes, landslides and snow avalanches. Recent velocity measurements performed on Weisshorn and Mönch hanging glaciers, Switzerland, confirmed this behaviour, while log-periodic oscillations superimposed on this general acceleration were also detected. Despite different rupture mechanisms in both cases, the log frequency of the oscillations is shown to be the same. The seismic activity was recorded near the unstable Weisshorn hanging glacier, simultaneously with the velocity measurements. Results show dramatically increasing icequake activity 3 days before the final collapse. Combined motion–seismic monitoring seems to be a promising way to accurately predict the breaking-off of hanging glaciers. Such a combined analysis is also useful for capturing the physical mechanisms of rupture in natural heterogeneous materials.

scholarly journals The Laplacian Flow of Locally Conformal Calibrated G2-Structures

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 7
Author(s):  
Marisa Fernández ◽  
Victor Manero ◽  
Jonatan Sánchez
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Natural Extension ◽  
Time Dependent ◽  
Time Interval ◽  
Time Singularity ◽  
Ancient Solutions ◽  
Locally Conformal ◽  
Laplacian Flow ◽  
Finite Time Singularity

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( − ∞ , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to − ∞ , and they blow-up at a finite-time singularity.

scholarly journals Type II ancient compact solutions to the Yamabe flow

2018 ◽  
Vol 2018 (738) ◽  
pp. 1-71 ◽  
Author(s):  
Panagiota Daskalopoulos ◽  
Manuel del Pino ◽  
Natasa Sesum
Keyword(s):  
Curvature Operator ◽  
Type Ii ◽  
Toda System ◽  
Yamabe Flow ◽  
Sharp Estimates ◽  
Ancient Solutions ◽  
Rotationally Symmetric ◽  
New Type ◽  
Error Terms ◽  
The Right

AbstractWe construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as{t\to{-}\infty}, to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments, based on sharp estimates on ancient solutions of the approximated linear equation and careful estimation of the error terms which allow us to make the right choice of parameters. Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result generalizes to the gluing ofkspheres for any{k\geq 2}, in such a way the configuration of radii of the spheres glued is driven as{t\to{-}\infty}by aFirst order Toda system.

Ionized AGN outflows are less powerful than assumed: A multi-wavelength census of outflows in type II AGN

2019 ◽  
Vol 15 (S356) ◽  
pp. 225-225
Author(s):  
Dalya Baron
Keyword(s):  
Optical Emission ◽  
Infrared Emission ◽  
Type Ii ◽  
Ionized Gas ◽  
Kinetic Coupling ◽  
Mass Outflow ◽  
Multi Wavelength ◽  
Novel Method ◽  
First Time ◽  
Ionization Parameter

AbstractIn this talk I will show that multi-wavelength observations can provide novel constraints on the properties of ionized gas outflows in AGN. I will present evidence that the infrared emission in active galaxies includes a contribution from dust which is mixed with the outflow and is heated by the AGN. We detect this infrared component in thousands of AGN for the first time, and use it to constrain the outflow location. By combining this with optical emission lines, we constrain the mass outflow rates and energetics in a sample of 234 type II AGN, the largest such sample to date. The key ingredient of our new outflow measurements is a novel method to estimate the electron density using the ionization parameter and location of the flow. The inferred electron densities, ∼104.5 cm−3, are two orders of magnitude larger than found in most other cases of ionized outflows. We argue that the discrepancy is due to the fact that the commonly-used [SII]-based method underestimates the true density by a large factor. As a result, the inferred mass outflow rates and kinetic coupling efficiencies are 1–2 orders of magnitude lower than previous estimates, and 3–4 orders of magnitude lower than the typical requirement in hydrodynamic cosmological simulations. These results have significant implications for the relative importance of ionized outflows feedback in this population.

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