Yoshikawa, Atsushi Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation. (English) Zbl 0794.35099 Trans. Am. Math. Soc. 340, No. 1, 103-126 (1993). Summary: It is well known that a quasilinear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data \(\lambda^{-1}a_ 0(\lambda x\cdot\eta,x)r_ 1(\eta)\), \(\lambda>0\), \(x,\eta\in\mathbb{R}^ n\), \(\eta\neq 0\). 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 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 \(\lambda\to\infty\). Cited in 2 Documents MSC: 35L60 First-order nonlinear hyperbolic equations 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 41A63 Multidimensional problems Keywords:nonlinear geometric optics approximation; quasilinear first order strictly hyperbolic system; genuine solution × Cite Format Result Cite Review PDF Full Text: DOI References: [1] 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). · Zbl 0177.36404 [2] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 · doi:10.1002/cpa.3160410705 [3] John K. Hunter and Joseph B. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (1983), no. 5, 547 – 569. · Zbl 0547.35070 · doi:10.1002/cpa.3160360502 [4] 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. · Zbl 0657.35084 · doi:10.1002/sapm1986753187 [5] Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 181 – 205. · Zbl 0343.35056 · doi:10.1007/BF00280740 [6] 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), Springer, Berlin, 1975, pp. 25 – 70. Lecture Notes in Math., Vol. 448. [7] Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43 – 101. · Zbl 0405.35056 · doi:10.1002/cpa.3160330104 [8] 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. · Zbl 0268.35062 [9] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. · Zbl 0537.76001 [10] 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. · doi:10.1007/978-1-4613-8689-6_6 [11] Atsushi Yoshikawa, Note on the Taylor expansion of smooth functions defined on Sobolev spaces, Tsukuba J. Math. 15 (1991), no. 1, 145 – 149. · Zbl 0785.46036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.