Celebratio Mathematica

C. S. Morawetz: “On the non-existence of continuous transonic flows past profiles, I,” Comm. Pure Appl. Math. 9 (1956), pp. 45–68. MR 78130 Zbl 0070.20206 article

C. S. Morawetz: “On the non-existence of continuous transonic flows past profiles, II,” Comm. Pure Appl. Math. 10 (1957), pp. 107–131. MR 88253 Zbl 0077.18901 article

C. S. Morawetz: “On the non-existence of continuous transonic flows past profiles, III,” Comm. Pure Appl. Math. 11 : 1 (1958), pp. 129–144. MR 96478 article

C. S. Morawetz: “A weak solution for a system of equations of elliptic-hyperbolic type,” Comm. Pure Appl. Math. 11 : 3 (August 1958), pp. 315–331. MR 96893 Zbl 0081.31201 article

C. S. Morawetz: “Non-existence of transonic flow past a profile,” Comm. Pure Appl. Math. 17 : 3 (1964), pp. 357–367. MR 184522 Zbl 0125.43101 article

In the study of transonic flow, one of the most illuminating theorems to prove would be:

Given an airfoil profile and a continuous two-dimensional irrotational transonic compressible inviscid flow past it with some given speed at infinity, there does not exist a corresponding flow with a slightly different speed at infinity.

Although this theorem was first formulated in 1954, on the basis of conjectures of Frankl and Guderley, see [1], it has not yet been established. Strong evidence that the theorem is true and proof that smooth transonic flows do not exist generally are given by non-existence theorems in which the profile is varied in the supersonic region and the speed at infinity kept fixed.

In [2], Part 1, there is such a “non-existence” theorem for continuous transonic flows which are considered as disturbances about a given smooth flow. Except for considering only a linear perturbation this theorem is quite general and complete but the proof is tediously long. In [2], Part II, allowance is made for the non-linearity at the expense of still further complication. It seems worthwhile to present here a “non-existence” theorem which covers the physically interesting situation and which is fairly simple to prove. The proof will be made even more elementary by the addition of a few assumptions on the pressure-density relation.

In Section I we describe the unperturbed flow and the assumptions, in Section 2 the perturbation flow, in Section 3 the non-existence theorem, in Section 4 the underlying uniqueness theorem. We begin by discussing the flow which is to be varied by varying the airfoil profile.

C. S. Morawetz: “The Dirichlet problem for the Tricomi equation,” Comm. Pure Appl. Math. 23 (1970), pp. 587–601. MR 280062 Zbl 0192.44605 article

C. S. Morawetz: “The mathematical approach to the sonic barrier,” Bull. Am. Math. Soc. (N.S.) 6 : 2 (1982), pp. 127–145. Josiah Willard Gibbs lecture presented at AMS meeting, San Francisco, 7 January 1981. MR 640941 Zbl 0506.76064 article

C. S. Morawetz: “On a weak solution for a transonic flow problem,” Comm. Pure Appl. Math. 38 : 6 (1985), pp. 797–817. MR 812348 Zbl 0615.76070 article

The objective of this paper is to establish the existence of weak solutions for steady state two-dimensional inviscid irrotational compressible flow in an infinitely long channel or in the exterior of an airfoil given the speed upstream at \( \infty \). If the speed upstream is sufficiently small there exists a strong smooth flow because the governing equations are elliptic. This means that the speed of sound always exceeds the fluid speed. But beyond a certain speed at infinity the flow ceases to be purely elliptic and shocks may and do arise. This paper shows how to establish a weak existence theorem for such a flow by the addition of artificial viscosity \( \nu \). The existence of a certain viscous flow with free boundary is presumed. The speed of flow is then assumed to be bounded away from cavitation speed and stagnation and the angle of flow is assumed bounded independent of the viscosity. Then weak existence of the flow solution in the limit \( \nu\to 0 \) is proved.

C. S. Morawetz: “Transonic flow and compensated compactness,” pp. 248–258 in Wave motion: Theory, modelling, and computation (Berkeley, CA, 9–12 June 1986). Edited by A. J. Chorin and A. J. Majda. Mathematical Sciences Research Institute Publications 7. Springer (New York), 1987. Proceedings of a conference in honor of the 60th birthday of Peter D. Lax. MR 920838 Zbl 0850.76290 incollection

The problem of finding steady flow past an airfoil is an old problem going back to the time of Lord Rayleigh. The understanding that there was a difficulty connected to the transition from subsonic flow to supersonic flow must surely, however, be attributed to Chaplygin [1944], whose famous thesis describing solutions of the equations with such transitions was written in 1904. The first mathematical study of such transitions which force a change of type for the differential equations from elliptic to hyperbolic began with the work of Tricomi [1923] in 1923. In 1930 at the International Mechanics Congress, Busemann [1930] with wind tunnel data, and G. I. Taylor [1930] with some computations, presented opposing views of the airfoil problem, the former suggesting that perhaps no steady flow existed and the latter than a series expansion in Mach number gave no evidence of a breakdown when the type changed.

Andrew Joseph Majda	Related
Alexandre J. Chorin	Related
Peter D. Lax	Related

C. S. Morawetz: “An alternative proof of DiPerna’s theorem,” Comm. Pure Appl. Math. 44 : 8–9 (1991), pp. 1081–1090. To Natascha in love and affection. MR 1127051 Zbl 0763.35056 article

The method of compensated compactness is the most elegant and general way to prove the existence of a solution of the initial boundary value problem for a genuinely nonlinear, strictly hyperbolic system of equations for two unknowns and two independent variables. The technique is as follows: Add viscous terms and prove the existence of the solution to the new system. Establish estimates independent of the coefficient of viscosity that are sufficient to yield a weak limit to the viscous solutions as the viscosity goes to zero. Represent the weak limit and the weak limit of functions of the solutions in terms of the Young measure. Construct a large family of entropy pairs which satisfy certain conservation inequalities. Apply the Tartar–Murat relation (see [2], [3]) for such entropy pairs and conclude that the Young measure is a Dirac measure. From this it is easy to see that the weak limit does satisfy the differential equations and initial data weakly. Finding the right family and concluding that the Young measure is Dirac was the work of Di Perna (see [1]) and it is this theorem that is represented in the title. Although no new theorems about nonlinear hyperbolic pairs of equations resulted, the method provided much insight and a new challenge to find the necessary estimates to apply it to other problems. It was extended with some necessary assumptions to the mixed type equations of transonic flow by Morawetz; see [5]. A more elegant way of proving the theorem using entropy pairs of compact support has been provided by D. Serre; see [4]. Here we simplify Serre’s approach and extend it to equations such as those for flow that is not subsonic.

C. S. Morawetz: “On steady transonic flow by compensated compactness,” Methods Appl. Anal. 2 : 3 (1995), pp. 257–268. MR 1362016 Zbl 0868.76042 article

This paper contains a theorem for the mixed equations of potential flow in two space variables that is analogous to DiPerna’s theorem [1983] on the existence of weak solutions for two hyperbolic conservation laws and is based on the Tartar–Murat Lemma for compensated compactness, see [Tartar 1979]. The application is plane flow for which a suitable “viscous” model exists, and this will be discussed in another paper. Some hypothesis about speed must be made. There are other conceivable applications such as axially symmetric flow, plane fluid models for semiconductors, etc. The equations of the viscous model must admit a potential and a stream function, or something like it. This is crucial in proving that the limit at zero viscosity is a genuine weak solution. But also one has to establish some underlying bounds.

I. M. Gamba and C. S. Morawetz: “A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: Existence theorem for potential flow,” Comm. Pure Appl. Math. 49 : 10 (1996), pp. 999–1049. MR 1404324 Zbl 0863.76029 article

In this paper we solve a boundary value problem in a two-dimensional domain \( \Omega \) for a system of equations of Fluid-Poisson type, that is, a viscous approximation to a potential equation for the velocity coupled with an ordinary differential equation along the streamlines for the density and a Poisson equation for the electric field. A particular case of this system is a viscous approximation of transonic flow models. The general case is a model for semiconductors.

We show existence of a density \( \rho \), velocity potential \( \phi \), and electric potential \( \Phi \) in the bounded domain \( \Omega \) that are \( C^{1,\alpha}(\overline{\Omega}) \), \( C^{2,\alpha}(\overline{\Omega}) \), and \( W^{2,\alpha}(\overline{\Omega}) \) functions, respectively, such that \( \rho \), \( \phi \), \( \Phi \), the speed \( |\nabla\phi| \), and the electric field \( E = \nabla\phi \) are uniformly bounded in the viscous parameter. This is a necessary step in the existing programs in order to show existence of a solution for the transonic flow problem.

I. M. Gamba and C. S. Morawetz: “Viscous approximation to transonic gas dynamics: Flow past profiles and charged-particle systems,” pp. 81–102 in Modelling and computation for applications in mathematics, science, and engineering (Evanston, IL, 3–4 May 1996). Edited by J. W. Jerome. Numerical Mathematics and Scientific Computation. Oxford University Press (New York), 1998. MR 1677377 Zbl 0938.76047 incollection

A boundary value problem in a domain \( \Omega \) is considered for a system of equations of Fluid-Poisson type, i.e. a viscous approximation to a potential equation for the velocity coupled with an ordinary differential equation along the streamlines for the density and a Poisson equation for the electric field.

A particular case of this system is a viscous approximation of transonic flow models. The general case is a model for semiconductors.

We present an overview of the problem and, in addition, we show an improvement of the lower bound for the density that controls the rate of approach to cavitation density by a quantity of the order of the viscosity parameter to the power that corresponds to the inverse of the enthalpy function.

This is a necessary step in the existing programs in order to show existence of a solution for the transonic flow problem.

Irene M. Gamba	Related
Joseph Walter Jerome	Related

C. S. Morawetz: “Mixed equations and transonic flow,” J. Hyperbolic Differ. Equ. 1 : 1 (March 2004), pp. 1–26. MR 2052469 Zbl 1055.35093 article

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.

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Cathleen Morawetz

Three decades after
Cathleen Synge Morawetz’s paper
“The mathematical approach to the sonic barrier”

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Three decades afterCathleen Synge Morawetz’s paper“The mathematical approach to the sonic barrier”

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Three decades after
Cathleen Synge Morawetz’s paper
“The mathematical approach to the sonic barrier”