## 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)

### 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
Full Text: