zbMATH — the first resource for mathematics

Fast singular limits of hyperbolic PDEs. (English) Zbl 0838.35071
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)

35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI