Singularly perturbed linear and semilinear hyperbolic systems: Kinetic theory approach to some folk’s theorems.

*(English)*Zbl 0893.35009An asymptotic procedure is presented to give a systematic approach to singularly perturbed hyperbolic problems. In fact, in numerous branches of science the approximation of a wave equation with strong damping by the solution of a suitable diffusion equation has been developed by using various ad hoc methods.

The method used by the author is a modified version of the well known Chapman-Enskog procedure and has been first applied in kinetic theory. In the paper the procedure is used for three basic types of scaling, which models for example strongly damped wave equation, hyperbolic heat conduction (Maxwell-Cattaneo model) and random walk theory. Results of semigroup theory of operators are used to perform the approximation. Also Lipschitz perturbation of a singularly perturbed system are studied.

In both linear and nonlinear cases estimates of the order to the error of the diffusion approximation are given.

The method used by the author is a modified version of the well known Chapman-Enskog procedure and has been first applied in kinetic theory. In the paper the procedure is used for three basic types of scaling, which models for example strongly damped wave equation, hyperbolic heat conduction (Maxwell-Cattaneo model) and random walk theory. Results of semigroup theory of operators are used to perform the approximation. Also Lipschitz perturbation of a singularly perturbed system are studied.

In both linear and nonlinear cases estimates of the order to the error of the diffusion approximation are given.

Reviewer: S.Totaro (Firenze)

##### MSC:

35B25 | Singular perturbations in context of PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

35L60 | First-order nonlinear hyperbolic equations |

47D06 | One-parameter semigroups and linear evolution equations |