Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

Abstract

This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions). Basic notionsThis section collects some preliminary notions concerning the different kinds of nets and of quotient spaces used in the paper. For the sake of brevity we mainly report definitions and basic properties, referring to [13], [14], [18], [34] for further details. Colombeau theory Nets of 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].

The Wave Equation with a Discontinuous Coefficient Depending on Time Only: Generalized Solutions and Propagation of Singularities

“…In case of systems or higher order equations, an incoming singularity will split at the interface into a refracted and a reflected wave. While in many cases it has been shown that the Colombeau solution is associated with the distributional solution of the transmission problem [13,20], the task of identifying the refracted and reflected singularity by means of the Colombeau wave front set -directly in the generalized solution without considering the associated distribution -has remained open.…”

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.…”

“…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].…”

Wave Equations and Symmetric First-order Systems in Case of Low Regularity

“…The theory of generalized solutions to linear hyperbolic first-order systems has been developed over 20 years and has achieved a spectrum of results on existence and uniqueness of solutions to the Cauchy problem, distributional limits and regularity of solutions, and symmetrizability (cf. [Obe89,Obe92,LaOb91,Obe09,HoSp12,GaOb11b]). On the other hand, generalized solutions of wave equations arising via the Laplace-Beltrami operator of a Lorentzian metric of low regularity have been studied in [ViWi00, May06, GMS09, Han11, HKS11].…”