» Rosinger E.E. - Generalized Solutions of Nonlinear Partial Differential Equations
Rosinger E.E. - Generalized Solutions of Nonlinear Partial Differential Equations
||Generalized Solutions of Nonlinear Partial Differential Equations
||The role played by generalized. in particular distribution solutions of linear partial differential equations is by now well established and known. A recent detailed survey of this
subject is presented for instance in Hormander . As is known, the foundation of this field started with the linear theory of distributions of Schwartz , and during the last three decades a vast number of studies concerning distribution solutions of linear, especially constant coefficient, partial differential equations has emerged, a somewhat earlier survey
being presented in Treves .
Unfortunately, an early so called impossibility result of Schwartz , seemed to block the use of distributions in a convenient systematic nonlinear theory of generalized solutions for nonlinear partial differential equations, as it proved the inexistence of a multiplication for arbitrary pairs of distributions in a way which would preserve certain usual properties of the multiplication of continuous and other functions (see details in Part 1, Chapter 2).
This result has usually been misinterpreted by saying that a convenient multiplication of arbitrary distributions was not possible, and that certainly delayed the emergence of sufficiently general and systematic nonlinear theories of generalized solutions needed in a rigorous study of general nonlinear partial differential equations.
The situation was not necessarily made easier by a subsequent insufficiency result of Lewy, which showed that the linear theory of distributions was not sufficient even for the solution of rather simple linear partial differential equations (see details in Part 1, Chapter 3, Section 1).
Nevertheless, there has been an ever increasing interest in nonlinear partial differential equations and some of the most basic and simple ones - for instance, the shock wave equations - proved to have nonclassical solutions among their most important and typical ones. In view of that, various ad-hoc weak solution methods for certain classes of nonlinear partial differential equations have been developed, as surveyed for instance in Lions . Unfortunately, some of these ad-hoc nonlinear methods can lead to so called stability paradoxes, such as for instance the existence of simultaneous weak and
strong solutions for the system u = 0, u2 = 1, as shown in Rosinger [2,3] and Part 1, Chapter 3, Section 2 in the sequel.