Schochet, Steven Asymptotics for symmetric hyperbolic systems with a large parameter. (English) Zbl 0685.35014 J. Differ. Equations 75, No. 1, 1-27 (1988). From author’s abstract: The asymptotic behavior as \(\lambda\) \(\to \infty\) is analysed for solutions u(\(\lambda)\) of quasilinear symmetric hyperbolic systems of the form \[ A^ 0u_ t+A^ ju_{x_ j}+\lambda C^ ju_{x_ j}=F;\quad u(0,x,\lambda)=u_ 0(x,\lambda), \] where the \(A^ k\) and F depend on t,x,u, and u/\(\lambda\), but the \(C^ j\) are constant. The results are applied to the equations of slightly compressible fluid dynamics to obtain a generalized form of the acoustics equations that describe the first-order correction in incompressible flow. The special case of a barotropic fluid was analyzed previously by Klainerman and Majda. Reviewer: T.C.T.Ting Cited in 25 Documents MSC: 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35L45 Initial value problems for first-order hyperbolic systems Keywords:parameter; acoustics PDFBibTeX XMLCite \textit{S. Schochet}, J. Differ. Equations 75, No. 1, 1--27 (1988; Zbl 0685.35014) Full Text: DOI References: [2] Browning, G.; Kreiss, H.-O, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math., 42, 704-718 (1982) · Zbl 0506.35006 [3] Caflisch, R.; Papanicolaou, G., The Fluid-dynamical limit of a nonlinear model Boltzmann equation, Comm. Pure Appl. Math., 32, 589-616 (1979) · Zbl 0438.76059 [4] Constantin, P., Note on loss of regularity for solutions of the 3-D in compressible Euler and related equations, Comm. Math. Phys., 104, 311-326 (1986) · Zbl 0655.76041 [5] Gustafsson, B., Asymptotic expansions for hyperbolic problems with different time scales, SIAM J. Numer. Anal., 17, 623-634 (1980) · Zbl 0454.35017 [6] Van Harten, A.; Van Hassel, R., A quasi-linear singular perturbation problem of hyperbolic type, SIAM J. Math. Anal., 16, 1258-1267 (1985) · Zbl 0612.35007 [7] Klainerman, S.; Majda, A., Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34, 481-524 (1981) · Zbl 0476.76068 [8] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Comm. Pure Appl. Math., 35, 629-651 (1982) · Zbl 0478.76091 [9] Kreiss, H.-O, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math., 33, 399-439 (1980) · Zbl 0439.35043 [10] Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973), SIAM: SIAM Philadelphia · Zbl 0268.35062 [11] Schochet, S., The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104, 49-75 (1986) · Zbl 0612.76082 [12] Schochet, S., Symmetric hyperbolic systems with a large parameter, Comm. Partial Differential Equations, 11, 1627-1651 (1986) · Zbl 0651.35047 [13] Schochet, S., Hyperbolic-hyperbolic singular limits, Comm. Partial Differential Equations, 12, 589-632 (1987) · Zbl 0629.35079 [14] Tadmor, E., Hyperbolic systems with different time scales, Comm. Pure Appl. Math., 35, 839-866 (1982) · Zbl 0479.35059 [15] Ukai, S., The incompressible limit and initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26, 323-331 (1986) · Zbl 0618.76074 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.