Preliminaries.- Regular and Singular Perturbations.- Evolution Equations in Hilbert Spaces.- Singularly Perturbed Hyperbolic Problems.- Presentation of the Problems.- Hyperbolic Systems with Algebraic Boundary Conditions.- Hyperbolic Systems with Dynamic Boundary Conditions.- Singularly Perturbed Coupled Boundary Value Problems.- Presentation of the Problems.- The Stationary Case.- The Evolutionary Case.- Elliptic and Hyperbolic Regularizations of Parabolic Problems.- Presentation of the Problems.- The Linear Case.- The Nonlinear Case.
Chapter 5 Hyperbolic Systems with Dynamic Boundary Conditions(p. 61-62)
In this chapter we investigate the first four problems presented in Chapter 3. All these problems include dynamic boundary conditions (which involve the derivatives of v(0, t), v(1, t), (BC.2) also include integrals of these functions). Note that all the four problems are singularly perturbed of the boundary layer type with respect to the sup norm.
The chapter consists of four sections, each of them addressing one of the four problems.
As a first step in our treatment, we construct a formal asymptotic expansion for each of the four problems, by employing the method presented in Chapter 1. For three of the four problems we construct zeroth order asymptotic expansions. In the case of problem (LS), (IC), (BC.1), we construct a first order expansion of the solution in order to offer an example of a higher order asymptotic expansion. It should be pointed out that first or even higher order asymptotic expansions can be constructed for all the problems considered in this chapter but additional assumptions on the data should be required and much more laborious computations are needed.
Once a formal asymptotic expansion is determined, we will continue with its validation. More precisely, as a second step in our analysis, we will formulate and prove results concerning the existence, uniqueness, and higher regularity of the terms which occur in each of the previously determined asymptotic expansions. As in the previous chapter, we need higher regularity to show that our asymptotic expansions are well defined and to derive estimates for the remainder components. Our investigations here are mainly based on classic methods in the theory of evolution equations in Hilbert spaces associated with monotone operators as well as on linear semigroup theory. It should be pointed out that each of the four problems requires a different framework and separate analysis. All the operators associated with the corresponding reduced (unperturbed) problems are subdifferentials, except for the reduced problem in Subsection 5.4.2. When the PDE system under investigation is nonlinear (see Subsections 5.2.2 and 5.3.2), the treatment becomes much more complex.
