The development and applications of the asymptotic method of Lyusternik and Vishik

The asymptotic method of Lyusternik and Vishik has now been greatly expanded and has found important applications in many fields of mechanics. The principal virtues of the method lie in its conceptual simplicity and in the fact that it can be applied to broad classes of partial differential equations with a small parameter occurring in the highest derivatives. The present paper has two chapters. In the first the method is illustrated by the example of the degeneration into elliptical equations of elliptical equations of higher orders, and the basic papers in which the method is developed and extended are surveyed. The cases when the boundary layer functions are defined by partial differential equations (parabolic, elliptic and hyperbolic boundary layers), and also the phenomenon of the interior boundary layer, are illustrated in greater detail. Recommendations are made for the application of the Newton diagram method in boundary layer problems when there is a small parameter entering arbitrarily. Among the most interesting applications of the method we note the problem of the passage past a muffled body of the ultrasonic flow of a viscous gas (Markov and Chudov); number of problems in the non-linear theory of thin flexible plates and shells (Srubshchik and Yudovich); and problems in the motion of a rigid body with cavities containing a viscous fluid (Moiseev, Chernous'ko, and others). The second chapter consists mainly of the author's results on differential equations in a Banach space containing a small parameter in the leading derivatives. Here the Lyusternik–Vishik method has been very little applied so far despite its undoubted advantages in this type of problem. Of special interest are the cases when the boundary problem is situated on the spectrum. The results of the chapter can be applied, in particular, to systems of ordinary differential equations, to integral-differential equations, and also to parabolic equations.