On some improperly posed problems for quasilinear equations of mixed type

Abstract

The Chaplygin equation, which arises in the study of transonic flow, is one of the simplest equations of mixed type. It was shown that with appropriate hypotheses on the function h a problem of this type could be stabilized (i.e., the solution made to depend continuously on the data) by restricting the solution to lie in the class of functions whose L2 integrals are bounded by some prescribed constant M. Similar restrictions have previously been shown (see e.g., [2], [7]) to guarantee continuous dependence in certain improperly posed problems for elliptic and parabolic equations. However, little study has up to now been devoted to improperly posed problems for equations of mixed type. For that matter one is rarely able to determine just what is a well set problem for such an equation. In this paper we demonstrate that the convexity arguments which led to stability inequalities for the Chaplygin equation [6] actually can be carried over with some modification to yield stability inequalities and error bounds in improperly posed problems for a wide class of quasilinear partial differential equations of mixed type. We shall be concerned here only with the question of obtaining such inequalities. We leave aside the difficult question of existence. The particular equation which we consider is by no means the most general one amenable to the convexity method. We restrict our attention to a relatively simple equation but one which exhibits most of the troublesome features which would appear in the more general operators. A survey of the literature on improperly posed problems will appear in a forthcoming paper by Payne [5]. In this paper we consider the following initial-boundary value problem for a cylindrical domain Q = D x (0, Y), where D is an arbitrary n-dimensional domain with smooth boundary D: