Stable singularity formation for the inviscid primitive equations

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.

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

Nonuniqueness for the Yang–Mills heat flow

Well-posedness and blow-up phenomena for a higher order shallow water equation

Global solutions and blow-up phenomena to a shallow water equation

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.

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 here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.

Singularities of a Variational Wave Equation

This paper provides a new general method for establishing a finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. This methodology is applied to two different problems. The first problem considered is the two-phase vortex sheet problem with surface tension, for which, under suitable assumptions of smallness of the initial height of the heaviest phase and velocity fields, is proved the finite-time singularity of the natural norm of the problem. This is in striking contrast with the case of finite-time splash and splat singularity formation for the one-phase Euler equations of [4] and [8], for which the natural norm (in the one-phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the inviscid fluid. For a very general set of geometries (essentially the contact zone being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. Compared to the previous paper [20] for the rigid body case, the present paper allows for small square integrable vorticity and provides a characterization of acceleration at contact. A surface energy is shown to blow up and acceleration at contact is shown to oppose the motion: it is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.

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.

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.