Propagation of singularities, Hamilton-Jacobi equations and numerical applications

Author:
Eduard Harabetian

Journal:
Trans. Amer. Math. Soc. **337** (1993), 59-71

MSC:
Primary 35A20; Secondary 35F20

DOI:
https://doi.org/10.1090/S0002-9947-1993-1179395-0

MathSciNet review:
1179395

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Abstract: We consider applications of Hamilton-Jacobi 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 Hamilton-Jacobi equation which belongs to a special class. For such Hamilton-Jacobi 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 Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.

**[1]**S. Alhinac,*Existence d'ondes de rarefaction pour les systems quasi-lineaires hyperboliques multidimensionnels*, Comm. Partial Differential Equations**12**(1989), 173-230.**[2]**R. Courant and D. Hilbert,*Methods in mathematical physics*. II, Interscience, 1962.**[3]**Michael G. Crandall and Pierre-Louis Lions,*Viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 1–42. MR**690039**, https://doi.org/10.1090/S0002-9947-1983-0690039-8**[4]**M. G. Crandall, L. C. Evans, and P.-L. Lions,*Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 487–502. MR**732102**, https://doi.org/10.1090/S0002-9947-1984-0732102-X**[5]**M. G. Crandall and P.-L. Lions,*Two approximations of solutions of Hamilton-Jacobi equations*, Math. Comp.**43**(1984), no. 167, 1–19. MR**744921**, https://doi.org/10.1090/S0025-5718-1984-0744921-8**[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**, https://doi.org/10.1090/S0002-9947-1986-0825712-4**[8]**S. N. Kružkov,*Generalized solutions of the Hamilton-Jacobi equations of eikonal type*. I, Math. USSR-Sb.**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****[11]**Andrew Majda,*The stability of multidimensional shock fronts*, Mem. Amer. Math. Soc.**41**(1983), no. 275, iv+95. MR**683422**, https://doi.org/10.1090/memo/0275**[12]**Andrew Majda,*The existence of multidimensional shock fronts*, Mem. Amer. Math. Soc.**43**(1983), no. 281, v+93. MR**699241**, https://doi.org/10.1090/memo/0281**[13]**S. Osher and L. Rudin,*Feature-oriented image enhancement using shock filters*, SIAM J. Numer. Anal.**27**(1990), 919-940.**[14]**Stanley Osher and James A. Sethian,*Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations*, J. Comput. Phys.**79**(1988), no. 1, 12–49. MR**965860**, https://doi.org/10.1016/0021-9991(88)90002-2**[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 reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1179395-0

Keywords:
Hamilton-Jacobi,
singularities,
Euler equations

Article copyright:
© Copyright 1993
American Mathematical Society