Nonuniqueness for the Yang–Mills heat flow

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.

The theorems in Chapter 1 are results for small data. The necessity for dealing with small perturbations of the linearized equations is underlined in the sequel by examples which show that, in general, one has to expect the development of singularities in finite time. In particular neither the smallness of the initial data nor the smoothness of data including the coefficients can prevent a solution from blowing up. We shall not go into the details here but we just present an illustration of the typical hyperbolic phenomenon that the solution and/or derivatives of the solution become singular after some time. This will mean in general that norms like the L∞-norm of the local regular solution or of its derivatives become infinite. The only way to avoid a blow-up are smallness of the data in connection with a sufficiently strong vanishing of the nonlinearity near zero and a sufficiently high space dimension. This is the message of the Theorems 1.1 and 1.2. Moreover we have learned from Theorem 1.2 that a solution of the nonlinear wave equation with a quadratic nonlinearity in ℝ3 lives at least exponentially long, although the examples below show that in general a blow-up occurs. Nevertheless this result justifies the notion of “almost global existence” in this case (cf. the paper of John & Klainerman [75]). We mention that for large data a blow-up may occur also in the cases where one has global existence for small data, see Example 1 in Chapter 1 and the remarks below.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction-diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction-diffusion system.

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.

In this thesis I present three separate studies on three different topics in General Relativity. The first study investigates the accuracy with which the mass and angular momentum of a black hole can be determined by measurements of gravitational waves from the hole, using a laser-interferometer gravitational-wave detector. The black hole is assumed to have been strongly perturbed, perhaps by coalescence with a binary companion, and the detector measures the waves produced by its resulting vibration and ring-down. The uncertainties in the measured mass and angular momentum arise from the unavoidable presence of noise in the detector. It is found that the faster the hole rotates, the more accurate the measurements will be, with the uncertainty in the angular momentum decreasing rapidly with increasing rotation speed. It is also found that the errors in the mass and angular momentum are highly correlated. The second study is an analysis of the gravitational collapse of an infinitely long, cylindrical dust shell. This analysis is expected to be helpful in understanding the behavior during collapse of more realistic, finite-length bodies. It is found that the collapse evolves into a naked singularity in finite time, as measured by a distant observer or by one riding on the shell. Analytical expressions for the variables describing the collapse are found at late times, near the singularity. The picture is completed with a numerical simulation that follows the collapse from the start until very close to the singularity. The singularity is found to be strong, in the sense that an observer riding on the shell will be infinitely stretched in the direction parallel to the symmetry axis, and infinitely compressed in the azimuthal direction. The gravitational waves emitted from the collapse are also analyzed. The last study focuses on a different kind of phenomenon, namely, the consequences of the existence of closed timelike curves in a spacetime that contains a wormhole. One might expect that the closed timelike curves would cause difficulty for the initial value problem for systems that evolve in such a spacetime: a system with apparently well-posed initial conditions might have no self-consistent solutions to its evolution equations. We study the simple case of a macroscopic, classical particle with a hard-sphere potential (a billiard ball), and we focus attention on initial conditions for which the evolution, if followed naively, is self- inconsistent: the ball enters one mouth of the wormhole and then comes out of the other mouth at an earlier time, then collides with its younger self, preventing itself from ever entering the first mouth. We find, surprisingly, that for all such dangerous initial conditions, there are an infinite number of self-consistent evolutionary solutions, involving a glancing collision and any number of wormhole traversals. We also find that for many non-dangerous initial conditions, there also exist an infinity of possible evolutions.

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.

Whether smooth solutions of partially dissipative 2D magnetohydrodynamic (MHD) equations develop singularities in finite time has been a long-standing open problem. Since the pioneering work by Cao and Wu [Adv. Math. 226, 1803–1822 (2011)], such problem has garnered much more attention. When the magnetic field is close to an equilibrium state, the study of stability and large-time behavior provides a significant example of the stabilizing effects of the magnetic field on electrically conducting fluids. In this paper, we consider the global well-posedness for 2D partially dissipation MHD equations around a uniform magnetic field in R+2 with no-slip boundary condition. An efficient Stokes estimate is developed, which can be used to obtain the global low order (H2) energy for the partially dissipative MHD system with physical boundary. In addition, we obtain an explicit decay rate for the linear system using spectral analysis methods. We also observe that the half-space shares some similarities with the Cauchy problem, but exhibits some differences, especially when considering spatial derivatives.

Stationary or self similar solutions with suitable homogeneity often play a crucial role in the regularity theory of nonlinear problems, which are physically or geometrically interesting. This has been manifested in the regularity theory of harmonic maps and minimal surfaces. The local partial regularity theorem in [CKN] implies that there are no self-similar solutions with small local energy (also see [TX] for generalizations). Making use of some arguments in [NRS], Tsai has ruled out the existence of any self-similar solutions with a finite local energy. Yet it is unclear whether or not solutions of the incompressible Navier– Stokes equation in three space dimensions would develop singularities in finite time. Therefore, it may be still interesting to construct special solutions of the 3-dimensional Navier–Stokes equation. In this note, we construct a one-parameter family of explicit smooth solutions of the 3-dimensional impressible Navier–Stokes equation on R\p, where p is any given point. These solutions are axisymmetric, homogeneous of degree −1. They are steady solutions to the Navier–Stokes equations and also solve the self-similar

We study the null curves and their motion in 3‐dimensional flat space‐time . We show that when the motion of null curves forms two surfaces in , the integrability conditions lead to the well‐known Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. In this case, we obtain all the geometrical quantities of the surfaces arising from the whole hierarchy but we particularly focus on the surfaces of the modified Korteweg‐de Vries (MKdV) and KdV equations. We obtain one‐ and two‐soliton surfaces associated with the MKdV equation and show that the Gauss and mean curvatures of these surfaces develop singularities in finite time. We also show that the triad vectors on the curves satisfy the spin vector equation in the ferromagnetism model of Heisenberg.

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.

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

Let M be a compact Riemannian manifold and the metrics g = g(t) evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for (M, g(t)) for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results.

This paper is concerned with the following three types of geometric evolution equations: the volume preserving mean curvature flow, the intermediate surface diffusion flow, and the surface diffusion flow. Important common properties of these flows are the preservation of volume and the decrease of perimeter. It is shown in this paper that the intermediate surface diffusion flow can lose convexity. Hence the volume preserving mean curvature flow is the only flow among the evolution equations under consideration which preserves convexity, cf. [11, 16, 14, 17]. Moreover, several sufficient conditions are presented, which illustrate that each of the above mentioned flows can move smooth initial configurations into singularities in finite time.

We explore the low energy dynamics of charge two instantons and dyonic instantons in SU(2) 5-dimensional Yang-Mills. We make use of the moduli space approximation and first calculate the moduli space metric for two instantons. For dyonic instantons the effective action of the moduli space approximation also includes a potential term which we calculate. Using the ADHM construction we are able to understand some aspects of the topology and structure of the moduli space. We find that instantons undergo right angled scattering after a head on collision and we are able to give an analytic description of this in terms of a quotient of the moduli space by symmetries of the ADHM data. We also explore the scattering of instantons and dyonic instantons numerically in a constrained region of the moduli space. Finally we exhibit some examples of closed geodesics on the moduli space, and geodesics which hit the moduli space singularities in finite time.