Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities

Abstract

This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators are obtained.This section gives some background on Colombeau techniques used in the sequel of this paper. As main sources we refer to [8,11,12,15].Nets of complex numbers. A net (u ε ) ε in C (0,1] is said to be strictly nonzero if there exist r > 0 and η ∈ (0, 1] such that |u ε | ≥ ε r for all ε ∈ (0, η]. For several regularity issues we will make use of the concept of slow scale net (s.s.n). A slow scale net is a net (r ε ) ε ∈ C (0,1] such thatA net (u ε ) ε in C (0,1] is said to be slow scale-strictly nonzero if there exist a slow scale net (s ε ) ε and η ∈ (0, 1] such that |u ε | ≥ 1/s ε for all ε ∈ (0, η].C-modules of generalised functions based on a locally convex topological vector space. The most common algebras of generalised functions of Colombeau type as well as the spaces of generalised symbols we deal with are introduced by referring to the following general models.Let E be a locally convex topological vector space topologised through the family of seminorms {p i } i∈I .

Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

“…The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results in [26,23,16] and provide aspects accompanying related recent work in [28,10,9]. MSC 2010: 46F30; 35L45, 35D30 Keywords: generalized functions, generalized solutions to hyperbolic systems * The author acknowledges the support of FWF-project grants Y237 and P20525.…”

“…Yet this concept of regularity is restricted to situations where distributional limits exist and therefore not applicable if initial data or right-hand side are not associated to any distribution. Intrinsic regularity theory in Colombeau algebras is based on the subalgebra G ∞ (Ω) of regular generalized functions in G(Ω) and has been investigated in the context of hyperbolic partial differential equations in [21,19,28,10,9]. In the study of intrinsic regularity of generalized solutions to partial differential equations, the notion of slow scale nets was introduced in [20] and has proven to be essential in many circumstances (cf.…”

“…In the study of intrinsic regularity of generalized solutions to partial differential equations, the notion of slow scale nets was introduced in [20] and has proven to be essential in many circumstances (cf. [21,10,9]). A net (r ε ) ε of complex numbers is said to be of slow scale if |r ε | p = O(1/ε) as ε → 0 for all p ≥ 0.…”

“…Problems of the type (1)-(2) play a prominent role in models of wave propagation in highly heterogeneous media with non-smooth variation of physical properties such as density, sound speed etc. For more details on motivations from the natural sciences and for further mathematical aspects in the context of the theory of generalized functions we may refer to the papers mentioned above as well as to the following series of papers on closely related research [5,22,18,16,28,10,9]. Second-order wave equations in a similar mathematical context have been discussed in [11,32,12].…”

A Regularization Approach to Non-smooth Symplectic Geometry

“…[6,24]) as well as on an analysis of microlocal mapping properties of generalized Fourier integral solution operators in terms of the wave front sets of their kernels, which are to be generalized Lagrangian submanifolds (cf. [7,8]).…”