Well-posedness Issues for Nonlinear Partial Differential Equations Appearing in the Modeling of Long Water Waves

Abstract

Modelling of wave motion in a fluid is usually based on classical systems which are obtained by the hypotheses that the flow is irrotational and the bottom is even. In such a context, the influence of vorticity is entirely disregarded in the formulation of the governing equations. Although this consideration is justified in many circumstances, there are also a fair number of observed cases in near-shore hydrodynamics and open channel flow where this approach is unsuitable. In this thesis, the influence of constant background vorticity on the properties of shock waves in a shallow water system is considered and the governing equations are derived. An analysis of the shock-wave solutions of the system detailed in the body of this paper shows that stationary jumps can be described in terms of two non-dimensional parameters, one being the Froude number and the other incorporating the background vorticity. It is shown that these two parameters completely determine the strength of the jump. Moreover, in many practical situations, the assumption of a flat bottom is too restrictive. If this theory is to describe the physics of an underlying problem adequately, then it is important to introduced uneven bed in the formulation of the governing equations. This is done in this thesis where it is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally lead to singular solutions featuring Dirac delta distributions. These singular solutions feature a Rankine-Hugoniot deficit and the method of complex-valued weak asymptotic is used to provide a firm link between the Rankine-Hugoniot deficit and the singular parts of the weak solutions. Furthermore, it is shown that a shallow water system for interfacial waves in the case of a neutrally buoyant two-layer fluid setup ceases to be strictly hyperbolic and the standard theory of hyperbolic conservation laws cannot be used to solve the Riemann problem. Nevertheless, it is shown that the Riemann problem can still be solved uniquely using singular shocks which contain Dirac delta distributions travelling with the shock. The solution is characterized in terms of the complex-valued weak asymptotic method and it is established that the two solution concepts coincide. The thesis also made a significant contribution to the Brio system which is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set v = 0. In the present contribution, the focus is on such an example, a hyperbolic conservation law appearing in ideal magnetohydrodynamics. For this conservation law, solutions cannot be found using the classical techniques of conservation laws. Consequently, certain Riemann problems have no weak solutions in the traditional Lax admissible sense. It was argued by some authors that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Although solutions of this type were exhibited, uniqueness was not obtained. In this thesis, a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws is introduced. In addition, a criterion which leads to an admissibility condition for singular solutions of the original system is developed and it is shown that such admissible solutions are unique in the framework developed in this thesis.