Thesis (Ph. D.)--West Virginia University, 2010. / Title from document title page. Document formatted into pages; contains vii, 48 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 48).

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Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.

In my Dissertation, I consider the system of thermoelasticity endowed with poly- convex energy. I will present the equations in their mathematical and physical con- text, and I will explain the relevant research in the area and the contributions of my work. First, I embed the equations of polyconvex thermoviscoelasticity into an aug- mented, symmetrizable, hyperbolic system which possesses a convex entropy. Using the relative entropy method in the extended variables, I show convergence from ther- moviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions. Then, I prove a measure-valued versus strong uniqueness result for adiabatic poly- convex thermoelasticity in a suitable class of measure-valued solutions, de ned by means of generalized Young measures that describe both oscillatory and concentra- tion e ects. Instead of working directly with the extended variables, I will look at the parent system in the original variables utilizing the weak stability properties of certain transport-stretching identities, which allow to carry out the calculations by placing minimal regularity assumptions in the energy framework. Next, I construct a variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity. I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy. Also, I prove that the scheme converges when the limit- ing solution is smooth. Finally, for completeness and for the reader's convenience, I present the well-established theory for local existence of classical solutions and how it applies to the equations at hand.

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This dissertation presents an analytical study of a class of problems involving discontinuities, also referred to as shocks, propagating through one dimensional flexible objects such as strings and rods. The study entails interrogation of the classical balance laws of momentum, angular momentum, and energy across propagating discontinuities. A major part of this dissertation also concerns itself with a non-classical entity called the ``material momentum''. The balance of material momentum is studied in a variational context, where both the local and singular forms of it are derived from an action principle. A distinguishing aspect of discontinuities propagating in continua is that, unlike in the bulk, the balance of momentum and angular momentum are not sufficient to describe their mechanics, even when the discontinuities are energy conserving. In this work, it is shown that the additional information required to close the system of equations at propagating discontinuities can be obtained from the singular form of energy balance across them. This entails splitting of the energy balance by its invariance properties, and identifying the non-invariant and invariant part of the source term with the power input and energy dissipation respectively at the shock. This approach is in contrast with other treatments of such problems in the literature, where additional non-classical concepts such as ``material momentum'' and ``configurational force'' have been invoked. To further our understanding of the connections between the classical and non-classical approaches to problems involving discontinuities, a detailed exposition of the concept of material momentum is presented. The balance and conservation laws associated with material momentum are derived from an action principle. It is shown that the conservation of material momentum is associated with the material symmetry of the continuum, and that the conditions for the conservation of physical and material momentum are independent of each other. A new classification of the deformed configurations of the planar Euler elastica based on conserved quantities associated with the spatial and material symmetry of the rod is proposed. The manifestation of the balance of material momentum in seemingly unrelated fields of research, such as fracture mechanics, ideal fluids, and the mechanics of rods with discontinuities, is also discussed. / Ph. D. / One dimensional flexible bodies such as strings and rods can exhibit fascinating and counterintuitive behavior when they interact with rigid obstacles. For instance, a chain falling on a rigid surface falls faster than it would have if it were falling freely. When one end of a long chain piled up in a container placed at an elevation is pulled across the rim and let go, the chain flows out of the container like a water fountain. Discontinuities in the cross-sectional properties of an elastic rod contained in a curved frictionless channel can result in the generation of forces that propel the rod along the channel. Such counterintuitive phenomena are a consequence of the physics taking place at the point of partial contact where the flexible body comes in contact with a rigid surface. The purpose of this dissertation is to study the mechanics of such points of discontinuity. Several such phenomena where effectively one dimensional bodies interact with rigid surfaces are all around us. A familiar example is the peeling of an adhesive tape, where the peeling front qualifies as a point of discontinuity propagating through the tape as the peeling progresses. A good understanding of the mechanics of the peeling front is crucial in estimating the strength of the adhesive. Another such example of practical importance is a mooring line being placed on the seabed. In such situations, the existence of a reaction force acting at the touchdown point depends on whether or not the cable develops a kink at that point. Similar questions of importance can be asked in the context of deployment and unspooling of space tethers. In this dissertation, an analytical study of the general physics of the phenomena described above is presented. Standard theoretical tools of classical physics are employed to understand the mechanics of points of partial contact between flexible and rigid bodies. The conditions under which a flexible body could experience sharp changes in its geometry (e.g. a kink) at such points are investigated. In addition to that, we explore the implications of a nonclassical law of physics called the balance of “material momentum” in the context of such problems.

We present a mathematical study of two conservative systems in fluid mechanics. First, we study a fragmentation model for sprays. The model takes into account the break-up of spray droplets due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system. Next, with an interest in understanding hydrodynamical limits in discontinuous regimes, we study a classical model for shock waves. Specifically, we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L² norm of a perturbed solution relative to the shock wave is bounded above by the L² norm of the initial perturbation. Finally, we include some preliminary relative entropy estimates which are suitable for a study of shock wave solutions to n x n systems of conservation laws having a convex entropy. / text

In a preamble, a quick summary of the line of thought from Noether's theorems to modern views on conserved charges in gauge theories is attempted. Most of the background material needed for the thesis is set out through a small survey of the literature. Emphasis is put on the concepts more than on the formalism, which is relegated to the appendices. The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by cohomological results, covariant phase space methods and by the Hamiltonian formalism. The thermodynamical properties of black holes, especially the first law, are studied in a general geometrical setting and are worked out for several black objects: black holes, strings and rings. Also, the geometrical and thermodynamical properties of a new family of black holes with closed timelike curves in three dimensions are described. The second part of the thesis is the natural generalization of the first part to asymptotic analyses. We start with a general construction of covariant phase spaces admitting asymptotically conserved charges. The representation of the asymptotic symmetry algebra by a covariant Poisson bracket among the conserved charges is then defined and is shown to admit generically central extensions. The asymptotic structures of three three-dimensional spacetimes are then studied in detail and the consequences for quantum gravity in three dimensions are discussed.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.

ABSTRACT In jet problems the conserved quantity plays a central role in the solution process. The conserved quantities for laminar jets have been established either from physical arguments or by integrating Prandtl's momentum boundary layer equation across the jet and using the boundary conditions and the continuity equation. This method of deriving conserved quantities is not entirely systematic and in problems such as the wall jet requires considerable mathematical and physical insight. A systematic way to derive the conserved quantities for jet °ows using conservation laws is presented in this dissertation. Two-dimensional, ra- dial and axisymmetric °ows are considered and conserved quantities for liquid, free and wall jets for each type of °ow are derived. The jet °ows are described by Prandtl's momentum boundary layer equation and the continuity equation. The stream function transforms Prandtl's momentum boundary layer equation and the continuity equation into a single third- order partial di®erential equation for the stream function. The multiplier approach is used to derive conserved vectors for the system as well as for the third-order partial di®erential equation for the stream function for each jet °ow. The liquid jet, the free jet and the wall jet satisfy the same partial di®erential equations but the boundary conditions for each jet are di®erent. The conserved vectors depend only on the partial di®erential equations. The derivation of the conserved quantity depends on the boundary conditions as well as on the di®erential equations. The boundary condi- tions therefore determine which conserved vector is associated with which jet. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived. This approach gives a uni¯ed treatment to the derivation of conserved quantities for jet °ows and may lead to a new classi¯cation of jets through conserved vectors. The conservation laws for second order scalar partial di®erential equations and systems of partial di®erential equations which occur in °uid mechanics are constructed using di®erent approaches. The direct method, Noether's theorem, the characteristic method, the variational derivative method (mul- tiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example. The conservation laws for the non-linear di®usion equa- tion for the spreading of an axisymmetric thin liquid drop, the system of two partial di®erential equations governing °ow in the laminar two-dimensional jet and the system of two partial di®erential equations governing °ow in the laminar radial jet are discussed via these approaches. The group invariant solutions for the system of equations governing °ow in two-dimensional and radial free jets are derived. It is shown that the group invariant solution and similarity solution are the same. The similarity solution to Prandtl's boundary layer equations for two- dimensional and radial °ows with vanishing or constant mainstream velocity gives rise to a third-order ordinary di®erential equation which depends on a parameter. For speci¯c values of the parameter the symmetry solutions for the third-order ordinary di®erential equation are constructed. The invariant solutions of the third-order ordinary di®erential equation are also derived.