Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-16T07:06:40.727Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities

Published online by Cambridge University Press:  19 September 2013

Claudia Garetto  and
Michael Oberguggenberger
Claudia Garetto
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Michael Oberguggenberger
Affiliation:
Institut für Grundlagen der Technischen Wissenschaften, Leopold-Franzens-Universität, Technikerstrasse 13, 6020 Innsbruck, Austria, (michael.oberguggenberger@uibk.ac.at)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalized functions. We employ the recently developed theory of generalized Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalized solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalized Fourier integral operators are obtained.

Keywords

hyperbolic equationsColombeau generalized functionsmicrolocal analysisPrimary 35S3046F30Secondary 35B65
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 426 - 463
Copyright
Copyright © Edinburgh Mathematical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227–260.Google Scholar
2.Ambrosio, L., Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of variations and nonlinear partial differential equations (ed. Ambrosio, L.et al.), Lecture Notes in Mathematics, Volume 1927, pp. 1–41 (Springer, 2008).Google Scholar
3.Biagioni, H. A., Generalized solutions to nonlinear first-order systems, Monatsh. Math. 118 (1994), 7–20.Google Scholar
4.Biagioni, H. A. and Oberguggenberger, M., Generalized solutions to the Korteweg–de Vries and the regularized long-wave equations, SIAM J. Math. Analysis 23 (1992), 923–940.Google Scholar
5.Bouchut, F. and James, F., One-dimensional transport equations with discontinuous coefficients, Nonlin. Analysis 32 (1998), 891–933.CrossRefGoogle Scholar
6.Cicognani, M., Esistenza, unicità e propagazione della regolarità della soluzione del problema di Cauchy per certi operatori strettamente iperbolici con coefficienti lipschitziani rispetto al tempo, Ann. Univ. Ferrara Sci. Mat. 33 (1987), 259–292.Google Scholar
7.Colombini, F. and Lerner, N., Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657–698.Google Scholar
8.Colombini, F. and Lerner, N., Uniqueness of continuous solutions for BV vector fields, Duke Math. J. 111 (2002), 357–384.CrossRefGoogle Scholar
9.Colombini, F. and Métivier, G., The Cauchy problem for wave equations with non-Lipschitz coefficients: application to continuation of solutions of some nonlinear wave equations, Annales Scient. Éc. Norm. Sup. 41 (2008), 177–220.Google Scholar
10.Colombini, F., De Giorgi, E. and Spagnolo, S., Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Annali Scuola Norm. Sup. Pisa IV 6 (1979), 511–559.Google Scholar
11.Conway, E. D., Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Analysis Applic. 18 (1967), 238–251.CrossRefGoogle Scholar
12.De Simon, L. and Torelli, G., Linear second-order differential equations with discontinuous coefficients in Hilbert spaces, Annali Scuola Norm. Sup. Pisa IV 1 (1974), 131–154.Google Scholar
13.Diperna, R. J. and Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511–547.Google Scholar
14.Duistermaat, J. J., Fourier integral operators, Progress in Mathematics, Volume 130 (Birkhäuser, 1996).Google Scholar
15.Duistermaat, J. J. and Hörmander, L., Fourier integral operators, II, Acta Math. 128 (1972), 183–269.Google Scholar
16.Erlacher, E. and Grosser, M., Ordinary differential equations in algebras of generalized functions, in Pseudo-differential operators, generalized functions and asymptotics, Operator Theory: Advances and Applications, Volume 231, pp. 253–270 (Birkhäuser, 2012).Google Scholar
17.Fernandez, R., On the Hamilton–Jacobi equation in the framework of generalized functions, J. Math. Analysis Applic. 382 (2011), 487–502.CrossRefGoogle Scholar
18.Garetto, C., Topological structures in Colombeau algebras: topological -modules and duality theory, Acta Appl. Math. 88 (2005), 81–123.CrossRefGoogle Scholar
19.Garetto, C., Topological structures in Colombeau algebras: investigation of the duals of and , Monatsh. Math. 146 (2005), 203–226.Google Scholar
20.Garetto, C., Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity, New York J. Math. 12 (2006), 275–318.Google Scholar
21.Garetto, C., Generalized Fourier integral operators on spaces of Colombeau type, in New developments in pseudo-differential operators (ed. Rodino, L. and Wong, M. W.), Operator Theory: Advances and Applications, Volume 189, pp. 137–184 (Birkhäuser, 2008).Google Scholar
22.Garetto, C., Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory, Acta Appl. Math. 102 (2008), 281–318.Google Scholar
23.Garetto, C. and Hörmann, G., Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities, Proc. Edinb. Math. Soc. 48 (2005), 603–629.Google Scholar
24.Garetto, C. and Oberguggenberger, M., Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients, eprint (arXiv:1104.2281 [math.AP], 2011).Google Scholar
25.Garetto, C., Gramchev, T. and Oberguggenberger, M., Pseudodifferential operators with generalized symbols and regularity theory, Electron. J. Diff. Eqns 2005 (2005), 1–43.Google Scholar
26.Garetto, C., Hörmann, G. and Oberguggenberger, M., Generalized oscillatory integrals and Fourier integral operators, Proc. Edinb. Math. Soc. 52 (2009), 351–386.Google Scholar
27.Gel'fand, I. M., Some questions of analysis and differential equations, Usp. Mat. Nauk 14 (1959), 3–19.Google Scholar
28.Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R., Geometric theory ofgeneralized functions with applications to general relativity, Mathematics and Its Applications, Volume 537 (Kluwer Academic, Dordrecht, 2001).Google Scholar
29.Haller, S. and Hörmann, G., Comparison of some solution concepts for linear firstorder hyperbolic differential equations with non-smooth coefficients, Publ. Inst. Math. 84 (2008), 123–157Google Scholar
30.Hörmander, L., Fourier integral operators, I, Acta Math. 127 (1971), 79–183.Google Scholar
31.Hörmander, L., Lectures on nonlinear hyperbolic differential equations (Springer, 1997).Google Scholar
32.Hörmann, G., First-order hyperbolic pseudodifferential equations with generalized symbols, J. Math. Analysis Applic. 293 (2004), 40–56.Google Scholar
33.Hörmann, G. and De Hoop, M., Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67 (2001), 173–224.Google Scholar
34.Hörmann, G. and Oberguggenberger, M., Elliptic regularity and solvability for partial differential equations with Colombeau coefficients, Electron. J. Diff. Eqns 2004 (2004), 1–30.Google Scholar
35.Hörmann, G. G. and Spreitzer, C., Symmetric hyperbolic systems in algebras of generalized functions and distributional limits, J. Math. Analysis Applic. 388 (2012), 1166–1179.CrossRefGoogle Scholar
36.Hörmann, G., Kunzinger, M. and Steinbauer, R., Wave equations on non-smooth space-times, in Asymptotic properties ofsolutions to hyperbolic equations (ed. Ruzhansky, M. and Wirth, J.), Progress in Mathematics, Volume 301 (Birkhäuser, 2012).Google Scholar
37.Hörmann, G., Oberguggenberger, M. and Pilipović, S., Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients, Trans. Am. Math. Soc. 358 (2006), 3363–3383.Google Scholar
38.Hurd, A. E. and Sattinger, D. H., Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Am. Math. Soc. 132 (1968), 159–174.Google Scholar
39.Kuznecov, N. N., On hyperbolic systems of linear equations with discontinuous coefficients, Computat. Math. Math. Phys. 3 (1963), 394–412.Google Scholar
40.Lafon, F. and Oberguggenberger, M., Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case, J. Math. Analysis Applic. 160 (1991), 93–106.Google Scholar
41.Mascarello, M. and Rodino, L., Partial differential equations with multiple characteristics, Mathematical Topics, Volume 13 (Akademie, Berlin, 1997).Google Scholar
42.Nedeljkov, M., Pilipović, S. and Scarpalézos, D., The linear theory of Colombeau gsneralized functions (Longman Scientific and Technical, Harlow, 1998).Google Scholar
43.Oberguggenberger, M., Hyperbolic systems with discontinuous coefficients: generalized solutions and a transmission problem in acoustics, J. Math. Analysis Applic. 142 (1989), 452–467.Google Scholar
44.Oberguggenberger, M., Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics, Volume 259 (Longman, Harlow, 1992).Google Scholar
45.Oberguggenberger, M., Case study of a nonlinear, nonconservative, non-strictly hyperbolic system, Nonlin. Analysis 19 (1992), 53–79.Google Scholar
46.Oberguggenberger, M., Hyperbolic systems with discontinuous coefficients: generalized wavefront sets, in New developments in pseudo-differential operators (ed. Rodino, L. and Wong, M. W.), Operator Theory: Advances and Applications, Volume 189, pp. 117–136 (Birkhäuser, 2008).Google Scholar
47.Poupaud, F. and Rascle, M., Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Commun. PDEs 22 (1997), 337–358.Google Scholar
48.Ruzhansky, M., On local and global regularity of Fourier integral operators, in New developments in pseudodifferential operators (ed. Rodino, L. and Wong, M. W.), Operator Theory: Advances and Applications, Volume 189, pp. 185–200 (Birkhäuser, 2009).Google Scholar
49.Taylor, M. E., Pseudodifferential operators and nonlinear PDEs (Birkhäuser, 1991).CrossRefGoogle Scholar