## 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