Non-symmetric hyperbolic problems with different time scales
The authors study well posedness and the limiting behavior when \(\varepsilon \to 0^+\) of solutions of the Cauchy problem for the singularly perturbed hyperbolic system \[ \varepsilon \partial_t u(x,t) = \bigl[ P_0 (x,t, \partial_x) + \varepsilon P_1 (x,t, \partial_x) \bigr] u(x,t) + \varepsilon F(x,t); \quad u (x,0) = f(x) \tag{*} \] where \(0 < \varepsilon \ll 1\) is a small parameter, \(u \in R^s\), \((x,t) \in R^n \times (0,T]\), \(P_i\), \(i = 1,2\) are in general non-symmetric matrices, \(P_i, F,f\) are \(2 \pi\) periodic in \(x\). The main purpose is to show that: 1) system \((*)\) is well-posed in a sense that a chosen norm of any solution can be estimated in terms of norms of \(f\) and \(F\) independently of \(\varepsilon\); 2) there are smooth initial data \(f\) such that any but fixed number of time derivatives at \(t = 0\) are bounded uniformly for \(\varepsilon\). The method of the proofs is based on the theory of pseudodifferential operators. The results obtained are applied to the linearized system of equations describing a motion of a slightly compressible viscoelastic fluid.
scientific article; zbMATH DE number 775653