First-order hyperbolic partial differential equations

To deal with the multiplication problem for distributions, Colombeau (1981) introduced a space of generalized functions. This space is a differential algebra containing the space of distributions as a linear subspace and having the space of smooth functions as a subalgebra. In addition, nonlinear operations more general than the multiplication make sense in Colombeau`s algebra. Thus, Colombeau`s theory provides an appropriate setting for finding and studying solutions of linear and nonlinear differential equations with singular data and coefficients. In this project, I will deal with the initial value problem for first-order hyperbolic equations with discontinuous coefficients in the framework of Colombeau`s algebra. Owing to the discontinuous coefficients, there is no general way of giving a meaning to this initial value problem in the sense of distributions. However, the notion of a solution makes sense in the framework of Colombeau`s algebra. The goal of this project is to settle the following three questions for this initial value problem: (a) existence and uniqueness of generalized solutions; (b) behavior of generalized solutions in the framework of distribution theory; (c) regularity of generalized solutions. We point out that an alternative to the subalgebra of regular generalized functions introduced by Oberguggenberger (1992) is required to study question (c) in the nonlinear case, since this subalgebra is not invariant under nonlinear maps. An alternative is offered by the subalgebra of elements of total slow scale type, which was introduced by Oberguggenberger (2004). As a further candidate, I introduce the subalgebra of elements of totally bounded type. By using these two subalgebras, I will address question (c) in the nonlinear case. Questions (a), (b) and (c) will be discussed in the linear case for the first half year and in the nonlinear case for the second half year.