The energy method, stability, and nonlinear convection
A brief outline of the contents is given. In chapter 2 we develop the ideas of the energy method on some simple one-space dimensional equations. Chapter 3 then extends the energy method to the equations for a linear viscous, incompressible fluid, and to the standard Bénard problem for a linear viscous, incompressible, heat-conducting fluid. In chapters 4 to 10 we systematically develop the energy method in a variety of contexts: half-space problems, geophysical problems, convection driven by surface tension, convection in other classes of fluid, time-dependent convection problems, and we study the connection with the Lyapunov method in partial differential equations. Chapters 11 to 13 adopt a different line of attack. They concentrate on technologically relevant and relatively new theories: dielectric fluids, ferrohydrodynamics, and chemically reacting mixtures. Where appropriate, a thermodynamic derivation of the necessary equations is given. Very little on energy stability theory has been developed on these theories and could profitably be done so in future. The main body of the book closes with a brief concluding chapter, number 14. Two appendices are included. The first collects some inequalities frequently used in energy stability theory. The second is a brief account of numerical solutions to the type of eigenvalue problem one encounters in linear and in nonlinear energy stability theory. Particular attention is paid to the compound matrix method.
scientific article; zbMATH DE number 52179