Generalized functions and multiple characteristics

The past decade has seen the emergence of a differential-algebraic theory of generalized functions that answered a wealth of questions on solutions to linear and nonlinear partial differential equations involving non-smooth coefficients and strongly singular data. In such cases, the theory of distributions does not provide a general framework in which solutions exist due to inherent constraints in dealing with nonlinear operations. An alternative framework is provided by the theory of Colombeau algebras of generalized functions. Interpreting the non-smooth coefficients and data as elements of the Colombeau algebra, existence and uniqueness has been established for many classes of equations by now. In particular, in order to study the regularity of solutions, microlocal techniques have been introduced into this setting and a calculus of generalized pseudodifferential operators, defined through Colombeau-regularization of non-smooth symbols, has been developed. A theory of generalized Fourier integral operators acting on Colombeau algebras, where both the phase function and amplitude are objects of Colombeau type, has been recently initiated. Starting from the preliminary work on generalized FIOs, this project will focus on solving hyperbolic equations and systems, generated by highly singular coefficients and data, by means of generalized FIO techniques and will provide a careful microlocal investigation of the solution by studying the microlocal mapping properties of these operators. The first part of the project will be devoted to a notion of generalized strict hyperbolicity for systems of pseudodifferential equations which assures the well-posedness of the corresponding Cauchy problem in the Colombeau context via construction of a generalized FIO parametrix. In other words we will express the Colombeau solution of a generalized hyperbolic system as action of a generalized FIO on the Cauchy data (modulo some smoothing error). This method has to be preferred to the proof via energy estimates, used so far for hyperbolic systems of PDEs in the Colombeau setting, because it delivers better insight into qualitative properties. More precisely, we will proceed to a microlocal investigation of the solution, by making use of generalized Hamiltonian flows and suitable notions of generalized wave front sets. The second part of the project will elaborate a Colombeau approach to systems of PDEs with multiple characteristics of variable multiplicity and singular coefficients. Via an approximation procedure, we will perform a transformation into a generalized strictly hyperbolic system for which the Cauchy problem is well-posed in the Colombeau context. Our method will not only allow a wide generality in the choice of coefficients and Cauchy data but will also provide generalized FIO techniques for systems with multiplicities. The Colombeau approach seems a promising way of overcoming the notorious difficulties of working with standard Fourier integral operators in case of systems with multiplicities.