Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation
HTML articles powered by AMS MathViewer
- by Atsushi Yoshikawa PDF
- Trans. Amer. Math. Soc. 340 (1993), 103-126 Request permission
Abstract:
It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data ${\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf {R}}^n}, \eta \ne 0$. Here ${r_1}(\eta )$ is a characteristic vector, and ${a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $n = 2$ or $3$ and that ${a_0}(\sigma ,x)$ have the vanishing mean with respect to $\sigma$, it is shown that a genuine solution exists in a time interval independent of $\lambda$, and that the formal solution is asymptotic to the genuine solution as $\lambda \to \infty$.References
- Yvonne Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires, J. Math. Pures Appl. (9) 48 (1969), 117–158 (French). MR 255964
- Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
- John K. Hunter and Joseph B. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (1983), no. 5, 547–569. MR 716196, DOI 10.1002/cpa.3160360502
- J. K. Hunter, A. Majda, and R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables, Stud. Appl. Math. 75 (1986), no. 3, 187–226. MR 867874, DOI 10.1002/sapm1986753187
- Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 181–205. MR 390516, DOI 10.1007/BF00280740
- Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
- Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. MR 544044, DOI 10.1002/cpa.3160330104
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
- Andrew Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws, Oscillation theory, computation, and methods of compensated compactness (Minneapolis, Minn., 1985) IMA Vol. Math. Appl., vol. 2, Springer, New York, 1986, pp. 115–165. MR 869824, DOI 10.1007/978-1-4613-8689-6_{6}
- Atsushi Yoshikawa, Note on the Taylor expansion of smooth functions defined on Sobolev spaces, Tsukuba J. Math. 15 (1991), no. 1, 145–149. MR 1118590, DOI 10.21099/tkbjm/1496161575
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 103-126
- MSC: Primary 35L60; Secondary 35A35, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-1993-1208881-X
- MathSciNet review: 1208881