By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.

https://iopscience.iop.org/article/10.1088/0031-9112/10/5/018/pdf

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

This paper reviews the present situation with existence and uniqueness theorems for mixed equations and their application to the problems of transonic flow. Some new problems are introduced and discussed. After a very brief discussion of time-dependent flows (Sec. 1) the steady state and its history is described in Sec. 2. In Secs. 3 and 4, early work on mixed equations and their connection to 2D flow are described and Sec. 5 brings up the problem of shocks, the construction of good airfoils and the relevant boundary value problems. In Sec. 6 we look at what two linear perturbation problems could tell us about the flow. In Sec. 7 we describe other examples of fluid problems giving rise to similar problems. Section 8 is devoted to the uniqueness by a conservation law and Secs. 9–11 to the existence proofs by Friedrichs' multipliers. In Sec. 12 a proof is given of the existence of a steady flow corresponding to some of the previous examples but the equations have been modified to a higher order system with a small parameter which when set to zero yields the equations of transonic flow. It remains to show that this formal limit really holds. Much has been left out especially modern computational results and the text reflects the particular interests of the author.

Mixed equations and transonic flow

A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ℝ n+1, n > 1. We give some sufficient conditions for existence of the global weak solutions. These conditions tie together nonlinearity with the speed of propagation and with the dimension n. We also prove necessity of these (or close) conditions. In fact, we extend these necessity results to the nonlocal semilinear equations.

Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations

For partial differential equations of mixed elliptic-hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet-conormal types. Such problems are of interest for applications to transonicflow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a − b − c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. c

/pdf/on-closed-boundary-value-problems-for-equations-of-mixed-1kupjinbfk.pdf

On Closed Boundary Value Problems for Equations of Mixed Elliptic-Hyperbolic Type

In this paper we explicitly calculate fundamental solutions for the Tricomi operator, relative to an arbitrary point in the plane, and show that all such fundamental solutions originate from the hypergeometric function F(1/6,1/6;1;ζ)$ that is obtained when we look for homogeneous solutions to the reduced hyperbolic Tricomi equation.

I. M. Gelfand

Papers

Fundamental solutions for the tricomi operator