On the system of Hamilton-Jacobi and transport equations arising in geometrical optics

Communications in Partial Differential Equations

Volumn 28, Issue 5-6, 2003, Pages 1085-1111

  Ben Moussa, B ^a   Kossioris, G T ^a,b  

^a INSTITUTE OF COMPUTER SCIENCE   (Greece)

^b UNIVERSITY OF CRETE   (Greece)

Author keywords

Blow up; Caustics; Eikonal equation; High frequency asymptotics; Measure solutions; Paraxial approximation; Singularities: Multivalued solutions; Transport equation; Viscosity solutions; WKB approximation

Indexed keywords

EID: 0037704917     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-120021187     Document Type: Article

References (42)

1. Albano, P., Cannarsa, P. (2002). Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch. Rat. Mech. Analysis 162:1-23.
2. Aubin, J. P., Cellina, A. (1984). Differential inclusions. A Series of Comprehensive Studies in Mathematics. Springer-Verlag, 264.
3. Barles, G. (1994). Solutions des viscosité des équations de Hamilton-Jacobi. In: Mathématiques & Applications. Springer-Verlarg, 17.
4. Ben Moussa, B., Vila J. P. (2000). Convergence of SPH method for scalar nonlinear conservation laws. SIAM J. Numer. Anal. 37(3):863-887.
5. Ben Moussa, B. (1999). Meshless particle methods: recent developments for nonlinear conservation laws in bounded domain. In: Proceedings Int. Conference on Godunov methods: theory and applications. October 18-22. Oxford: UK, 97-108.
6. Ben Moussa, B. (1998). Analyse numérique de méthodes particulaires de type SPH pour les lois de conservation. Ph.D. Thesis. INSA-Toulouse.
7. Benamou, J. D. (1996). Big ray tracing: multivalued travel time field computation using viscosity solutions of the eikonal equation. J. Comp. Phys. 128:463-474.
8. Benamou, J. D. (1999). Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52: 1443-1475.
9. Bensoussan, A., Lions, J. L., Papanicolaou, G. (1978). Asymptotic Analysis of Periodic Structures. North-Holland.
10. Bouchut, F., James, F. (1998). One-dimensional transport equations with discontinuous coefficients. Nonlinear Analysis TMA 32(7):891-933.
11. Cannarsa, P., Mennucci, A., Sinestrari, C. (1997). Regularity results for solutions of a class of Hamilton-Jacobi Equations. Arch. Rat. Mech. Analysis 140:197-223.
12. Cannarsa, P., Soner, H. M. (1989). Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications. Nonlinear Anal. TMA 13:305-323.
13. Cěrvenỳ, V., Molotkov, I. A., Pšenčik, I. (1977). Ray Method in Seismology. Praha:Univerzita Karlova.
14. Clarke, F. H., Ledyaev, Yu, S., Stern, R. J., Wolenski, P. R. (1998). Nonsmooth Analysis and Control Theory. Springer.
15. Crandall, M. G., Ishii, H., Lions, P. -L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. AMS 27:1-67.
16. Crandall, M. G., Lions, P. L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277:1-42.
17. Crandall, M. G., Lions, P. L. (1984). Two approximations of solutions of Hamilton-Jacobi Equations. Math. Comp. 43(167):1-19.
18. Dafermos, C. M. (2000). Hyperbolic Conservations Laws in Continuum Physics. A Series of Comprehensive Studies in Mathematics. Springer, 325.
19. DiPerna, R. J., Lions, P. L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98:511-547.
20. Fatemi, E., Engquist, B., Osher, S. (1995). Numerical solution of the high frequency asymptotic expansion for the scalar wave equation. J. Comput. Phys. 120(1):145-155.
21. Filippov, A. F. (1989). Differential equations with discontinuous right-hand side. Amer. Math. Soc. Transl. 42:199-231.
22. Filippov, A. F. (1988). Differential equations with discontinuous right-hand sides. Mathematics and its Applications. Kluwer Academic Publishers.
23. Flattè, S. M. (1986). The Schrödinger equation in classical physics. Am. J. Phys. 54(12):1088-1092.
24. Fleming, W. H. (1971). The stochastic control for small noise intensities. SIAM J. Control 9(3):473-517.
25. Fleming, W. H., Souganidis, P. E. (1986). Asymptotic series and the method of vanishing viscosity. Indiana U. Math. J. 35(2):425-447.
26. Fock, V. (1950). Theory of radia-wave propagation in an inhomogeneous atmosphere for a raised source. Bull. Acd. Sci. USSR, Ser. Phys. 14.
27. Gosse, L., James, F. (2002). Convergence results for an inhomogeneous system arising in various high frequency approximations. Num. Math. 90:721-753.
28. Izumiya, S., Kossioris, G. T. (1997). Realization theorems for geometric singularities for Hamilton-Jacobi equations. Comm. Anal. Geom. 5(3):255-290.
29. Izumiya, S., Kossioris, G. T. (1997). Bifurcations of shock waves for viscosity solutions of Hamilton-Jacobi equations of one space variable. Bull. Sci. Math. 121:619-667.
30. Izumiya, S., Kossioris, G. T., Makrakis, G. N. (2001). Multivalued solutions to the eikonal equation in stratified media. Quart. Appl. Math. 59(2):365-390.
31. Katsaounis, T., Kossioris, G. T., Makrakis, G. N. (2001). Computation of high frequency fields near caustics. Math. Models Methods Appl. Sci. 11:199-228.
32. Kossioris, G. T. (1993). Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions. Comm. P.D.E. 18:1085-1108.
33. Kruzkov, S. N. (1967). Generalized solutions of nonlinear first order equations with several variables II. Math. USSR Sbornik 1:93-116.
34. Leontovich, M., Fock, V. (1946). Solution of the problem of propagation of electromagnetic waves along the Earth's surface. J. Phys. 10:13-24.
35. Lin, C. T., Tadmor, E. (2001). L1-Stability and error estimates for approximate Hamilton-Jacobi solutions. Numer. Math. 87:701-735.
36. Lions, P. L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics. Boston: Pitman, pp. 69.
37. Maslov, V. P., Fedoryuk, V. M. (1981). Semi-classical approximations in quantum mechanics. Reidel D, Doordrecht.
38. Messiah, A. (1961, 1962). Quantum Mechanics. (2 vols). New York: John Wiley.
39. Oleinik, O. A. (1963). Discontinuous solutions of nonlinear differential equations. Transi. AMS 26:95-172.
40. Poupaud, F., Rascle, M. (1997). Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm. in P.D.E. 22:337-358.
41. Souganidis, P. E. (1985). Existence of viscosity solutions of Hamilton-Jacobi equations. J. Diff. Equations. 56:345-390.
42. Tappert, F. D. (1977). The parabolic approximation method. In: Keller, J.B., Papadakis, J.S., eds. Wave Propagation and Underwater Acoustics. Lecture Notes in Physics. Berlin: Springer, 70.