Propagation of singularities, HamiltonJacobi equations and numerical applications
Author:
Eduard Harabetian
Journal:
Trans. Amer. Math. Soc. 337 (1993), 5971
MSC:
Primary 35A20; Secondary 35F20
MathSciNet review:
1179395
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider applications of HamiltonJacobi equations for which the initial data is only assumed to be in . Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a HamiltonJacobi equation which belongs to a special class. For such HamiltonJacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using HamiltonJacobi equations for capturing fronts has been used in [14] for fronts propagating with curvaturedependent speed.
 [1]
S. Alhinac, Existence d'ondes de rarefaction pour les systems quasilineaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 12 (1989), 173230.
 [2]
R. Courant and D. Hilbert, Methods in mathematical physics. II, Interscience, 1962.
 [3]
Michael
G. Crandall and PierreLouis
Lions, Viscosity solutions of HamiltonJacobi
equations, Trans. Amer. Math. Soc.
277 (1983), no. 1,
1–42. MR
690039 (85g:35029), http://dx.doi.org/10.1090/S00029947198306900398
 [4]
M.
G. Crandall, L.
C. Evans, and P.L.
Lions, Some properties of viscosity solutions
of HamiltonJacobi equations, Trans. Amer.
Math. Soc. 282 (1984), no. 2, 487–502. MR 732102
(86a:35031), http://dx.doi.org/10.1090/S0002994719840732102X
 [5]
M.
G. Crandall and P.L.
Lions, Two approximations of solutions of
HamiltonJacobi equations, Math. Comp.
43 (1984), no. 167, 1–19. MR 744921
(86j:65121), http://dx.doi.org/10.1090/S00255718198407449218
 [6]
M. G. Crandall, Private communication.
 [7]
Eduard
Harabetian, A convergent series expansion for
hyperbolic systems of conservation laws, Trans.
Amer. Math. Soc. 294 (1986), no. 2, 383–424. MR 825712
(87k:35160), http://dx.doi.org/10.1090/S00029947198608257124
 [8]
S. N. Kružkov, Generalized solutions of the HamiltonJacobi equations of eikonal type. I, Math. USSRSb. 27 (1975).
 [9]
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, R. I., 1968.
 [10]
Peter
D. Lax, Hyperbolic systems of conservation laws and the
mathematical theory of shock waves, Society for Industrial and Applied
Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical
Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
(50 #2709)
 [11]
Andrew
Majda, The stability of multidimensional shock fronts, Mem.
Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR 683422
(84e:35100)
 [12]
Andrew
Majda, The existence of multidimensional shock fronts, Mem.
Amer. Math. Soc. 43 (1983), no. 281, v+93. MR 699241
(85f:35139)
 [13]
S. Osher and L. Rudin, Featureoriented image enhancement using shock filters, SIAM J. Numer. Anal. 27 (1990), 919940.
 [14]
Stanley
Osher and James
A. Sethian, Fronts propagating with curvaturedependent speed:
algorithms based on HamiltonJacobi formulations, J. Comput. Phys.
79 (1988), no. 1, 12–49. MR 965860
(89h:80012), http://dx.doi.org/10.1016/00219991(88)900022
 [15]
L. Rudin, Images, numerical analysis of singularities, and shock filters, Ph.D. Thesis, Computer Science Dept., CalTech, Pasadena, Calif., 1987.
 [16]
Joel
Smoller, Shock waves and reactiondiffusion equations,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Science], vol. 258, SpringerVerlag, New York, 1983. MR 688146
(84d:35002)
 [1]
 S. Alhinac, Existence d'ondes de rarefaction pour les systems quasilineaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 12 (1989), 173230.
 [2]
 R. Courant and D. Hilbert, Methods in mathematical physics. II, Interscience, 1962.
 [3]
 M. G. Crandall and P. L. Lions, Viscosity solutions of HamiltonJacobi equations. Trans. Amer. Math. Soc. 277 (1983). MR 690039 (85g:35029)
 [4]
 M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of HamiltonJacobi equations, Trans. Amer. Math. Soc. 282 (1984). MR 732102 (86a:35031)
 [5]
 M. G. Crandall and P. L. Lions, Two approximations of solutions to HamiltonJacobi equations, Math. Comp. 43 (1984). MR 744921 (86j:65121)
 [6]
 M. G. Crandall, Private communication.
 [7]
 E. Harabetian, A convergent series expansion for hyperbolic systems of conservation laws, Trans. Amer. Math. Soc. 294 (1986). MR 825712 (87k:35160)
 [8]
 S. N. Kružkov, Generalized solutions of the HamiltonJacobi equations of eikonal type. I, Math. USSRSb. 27 (1975).
 [9]
 O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, R. I., 1968.
 [10]
 P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM Regional Conf. Ser. in Appl. Math., no. 11, 1973. MR 0350216 (50:2709)
 [11]
 A. Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc., No. 275 (1983). MR 683422 (84e:35100)
 [12]
 , The existence of multidimensional shock fronts, Mem. Amer. Math. Soc., No. 281 (1983). MR 699241 (85f:35139)
 [13]
 S. Osher and L. Rudin, Featureoriented image enhancement using shock filters, SIAM J. Numer. Anal. 27 (1990), 919940.
 [14]
 S. Osher and J. Sethian, Fronts propagating with curvaturedependent speed: Algorithms based on HamiltonJacobi formulations, J. Comput. Phys. 79 (1988), 1249. MR 965860 (89h:80012)
 [15]
 L. Rudin, Images, numerical analysis of singularities, and shock filters, Ph.D. Thesis, Computer Science Dept., CalTech, Pasadena, Calif., 1987.
 [16]
 J. Smoller, Shock waves and reaction diffusion equations, SpringerVerlag, 1983. MR 688146 (84d:35002)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
35A20,
35F20
Retrieve articles in all journals
with MSC:
35A20,
35F20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199311793950
PII:
S 00029947(1993)11793950
Keywords:
HamiltonJacobi,
singularities,
Euler equations
Article copyright:
© Copyright 1993 American Mathematical Society
