Schochet, Steven Fast singular limits of hyperbolic PDEs. (English) Zbl 0838.35071 J. Differ. Equations 114, No. 2, 476-512 (1994). The symmetric-hyperbolic system containing the small parameter \(\varepsilon\), \[ \partial_t U+ \sum K^i \partial_{\eta_i} U= \varepsilon[\sum A^i(U) \partial_{x_i} U+ \sum D^i(U) \partial_{\eta_i} U+ F(U)],\;\eta\in \mathbb{T}^m,\;x\in \mathbb{R}^n \] is considered. The initial data lies in \(H(\mathbb{T}^m\times \mathbb{R}^n)\). It is well known that, if the additional independent slow variable \(x\) is absent, then the leading term of the asymptotics depends on the fast variables \(\eta\), \(t\) and slow the one \(\tau= \varepsilon t\).In the case under consideration the leading term \(U_0(\eta, t, x, \tau)\) is constructed in the form depending on the additional slow variable \(x\). The existence theorem and an estimate of the remainder is obtained as follows: \(U(\eta, t, x, \varepsilon)= U_0(\eta, t, x, \tau)+ o(1)\), as \(\varepsilon\to 0\), uniformly for long time intervals \(0\leq t\leq O\) \((\varepsilon^{- 1})\). Reviewer: L.Kalyakin (Ufa) Cited in 3 ReviewsCited in 165 Documents MSC: 35L60 First-order nonlinear hyperbolic equations 35B25 Singular perturbations in context of PDEs 35C20 Asymptotic expansions of solutions to PDEs Keywords:fast singular limits PDFBibTeX XMLCite \textit{S. Schochet}, J. Differ. Equations 114, No. 2, 476--512 (1994; Zbl 0838.35071) Full Text: DOI