New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System

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.

Numerical investigation of tsunami-like wave hydrodynamic characteristics and its comparison with solitary wave

Nonuniqueness for the Yang–Mills heat flow

With the consideration of the superthermal electron distribution, we present a theoretical investigation about the nonlinear propagation of electron-acoustic solitary and shock waves in a dissipative, nonplanar non-Maxwellian plasma comprised of cold electrons, superthermal hot electrons, and stationary ions. The reductive perturbation technique is used to obtain a modified Korteweg-de Vries Burgers equation for nonlinear waves in this plasma. We discuss the effects of various plasma parameters on the time evolution of nonplanar solitary waves, the profile of shock waves, and the nonlinear structure induced by the collision between planar solitary waves. It is found that these parameters have significant effects on the properties of nonlinear waves and collision-induced nonlinear structure.

We derive an explicit formula that permits the recovery of the profile of an irrotational solitary water wave from pressure data measured at the flat bed of the fluid domain. The formula is valid for the governing equations and applies to waves of small and large amplitude.

Solitary wave interactions of the GRLW equation

The propagation of dust acoustic waves, both solitary and shock waves of arbitrary amplitude, is revisited for the first time in a dusty plasma composed of Kaniadakis‐distributed electrons, positive ions, and dust grains with variable charge. The ‐generalized electron charging current is derived using the orbit‐motion‐limited (OML) theory. It is observed that the Kaniadakis electron distribution plays a significant role in shaping the behavior of variable‐charge solitary waves. Specifically, in the adiabatic regime, this distribution results in solitary structures that are less sharply peaked and promotes enhanced electron collection by the dust grains, or equivalently, suppresses the collection of positive ions. In the non‐adiabatic regime, the system supports dissipative shock waves, and the presence of ‐distributed electrons facilitates a transition in the wave profile from oscillatory to monotonic behavior.

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.

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.

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.