Generalized Solutions in Hydrodynamic Stability

Abstract

This chapter surveys the results obtained in the generalized frame of hydrodynamic stability theory. After an introductory paragraph (2.1), which deals with the mathematical facts required in the sequel, the generalized solutions which appear in the study of hydrodynamic stability are discussed in § 2.2. Then two problems related to linear stability, which justify from the mathematical point of view the investigations of classical linear stability theory, are discussed: the completeness of the normal modes (§ 2.3) and the linearization principle (§ 2.4). In § 2.5 the principle of exchange of stabilities connecting hydrodynamic stability with the branching of solutions of the Navier-Stokes equations is analysed, and § 2.6 is devoted to nonlinear stability theory, namely, to universal criteria of hydrodynamic stability.KeywordsBanach SpaceWeak SolutionStrong SolutionCouette FlowGeneralize DerivativeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.