Celebratio Mathematica

C. S. Morawetz: Contracting spherical shocks treated by a perturbation method. Ph.D. thesis, New York University, 1951. Advised by K. O. Friedrichs. MR 2594107 phdthesis

C. S. Morawetz: “The eigenvalues of some stability problems involving viscosity,” J. Rational Mech. Anal. 1 (1952), pp. 579–603. MR 51648 Zbl 0048.19205 article

Small disturbances of the form \( \phi(y)e^{i\alpha(x-ct)} \) are assumed to occur in a viscous flow (1) between moving walls, (2) with a symmetrical velocity profile between two fixed walls and (3) in a boundary layer along a flat plate. Corresponding to each problem for the viscous fluid, there is a simpler problem with viscosity zero. It is natural to conjecture that, for large Reynolds numbers, the eigenvalues in the complete viscous problem are related to those in the corresponding inviscid problem. The present investigation justifies the relationship between inviscid and viscous eigenvalues in most cases and in fact expresses these viscous eigenvalues asymptotically with respect to the Reynolds number in terms of the inviscid eigenvalues.

C. S. Morawetz and I. I. Kolodner: “On the non-existence of limiting lines in transonic flows,” Comm. Pure Appl. Math. 6 : 1 (February 1953), pp. 97–102. MR 55132 Zbl 0050.19803 article

C. S. Morawetz: “Asymptotic solutions of the stability equations of a compressible fluid,” J. Math. Physics, Mass. Inst. Techn. 33 : 1–4 (April 1954), pp. 1–26. MR 60664 Zbl 0059.20101 article

In the study of the stability of a compressible fluid carried out by Lees and Lin, use is made of formal asymptotic series which are formal solutions of a sixth order system of differential equations involving a large parameter. The principal feature of this system is that if we let the parameter go to infinity the system is reduced to one of second order with a singular point. In this paper we shall prove that these formal series actually represent solutions of the given system, and investigate some of the properties of these solutions. In many problems of this type, it is sufficient to know that there exists some domain of validity for the formal series solutions. Here, however, our principal object will be to find a sufficiently large domain of validity so that the boundary value problem arising from the stability problem can be treated asymptotically. For this purpose, we need a large domain of validity for a complete set of asymptotic solutions.

C. S. Morawetz: “A uniqueness theorem for Frankl’s problem,” Comm. Pure Appl. Math. 7 : 4 (November 1954), pp. 697–703. MR 65791 Zbl 0056.31904 article

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: “Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation,” Proc. Roy. Soc. London. Ser. A. 236 (1956), pp. 141–144. MR 79712 Zbl 0070.31802 article

C. S. Morawetz: “Uniqueness for the analogue of the Neumann problem for mixed equations,” Mich. Math. J. 4 : 1 (1957), pp. 5–14. MR 85441 Zbl 0077.09602 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: “The decay of solutions of the exterior initial-boundary value problem for the wave equation,” Comm. Pure Appl. Math. 14 : 3 (August 1961), pp. 561–568. MR 132908 Zbl 0101.07701 article

C. S. Morawetz: “Magnetohydrodynamic shock structure without collisions,” Phys. Fluids 4 : 8 (1961), pp. 988–1006. Zbl 0111.39603 article

The internal structure of a magnetohydrodynamic shock is examined under the condition that there are no collisions among the plasma particles. The equations to be solved are the collisionless, steady Boltzmann equations for ions and electrons coupled with Maxwell’s equation for the fields (a self-consistent system). There is one space variable \( x \) and all quantities are prescribed constant as \( x = -\infty \). Under appropriate conditions at \( -\infty \), e.g., no transverse magnetic field, low ion pressure, and Alfvén–Mach number roughly less than 2, the state at \( +\infty \) has oscillating fields, density, etc. The length scale is a mean phase length. Thus a change of state is possible without collisions.

P. D. Lax, C. S. Morawetz, and R. S. Phillips: “The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle,” Bull. Am. Math. Soc. 68 : 6 (1962), pp. 593–595. Shorter, early version of an article eventually published in Comm. Pure Appl. Math. 16:4 (1963). MR 142890 Zbl 0108.28301 article

Ralph Saul Phillips	Related
Peter D. Lax	Related

C. S. Morawetz: “The limiting amplitude principle,” Comm. Pure Appl. Math. 15 : 3 (1962), pp. 349–361. MR 151712 Zbl 0196.41202 article

In this paper a rigorous mathematical proof is given of the so-called limiting amplitude principle for reflecting bodies. This principle states that every solution of the wave equation with a harmonic forcing term, \[ \Box U = \Delta U - U_{tt} = e^{i\omega t}g(x), \] in the exterior of a reflecting body tends to the steady state solution \[ e^{i\omega t}V(x), \] uniformly on bounded sets as \( t \) tends to infinity. Here \( V \) satisfies the reduced wave equation \[ \Delta V + \omega^2 V = g, \] vanishes on the body and satisfies the Sommerfeld radiation condition at infinity. In the precise formulation of the result some conditions are imposed of which the most important is that the body is star-shaped.

C. S. Morawetz: “Modification for magnetohydrodynamic shock structure without collisions,” Phys. Fluids 5 : 11 (1962), pp. 1447–1450. Zbl 0117.21204 article

P. D. Lax, C. S. Morawetz, and R. S. Phillips: “Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle,” Comm. Pure Appl. Math. 16 : 4 (November 1963), pp. 477–486. A shorter, early version of this article was published in Bull. Am. Math. Soc. 68:6 (1962). MR 155091 Zbl 0161.08001 article

Ralph Saul Phillips	Related
Peter D. Lax	Related

C. S. Morawetz: “A uniqueness theorem for the relativistic wave equation,” Comm. Pure Appl. Math. 16 : 3 (August 1963), pp. 353–362. MR 162057 Zbl 0117.06402 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 limiting amplitude principle for arbitrary finite bodies,” Comm. Pure Appl. Math. 18 : 1–2 (1965), pp. 183–189. MR 190516 Zbl 0161.08201 article

Consider a dynamical system driven by an externally imposed harmonic force. If there is some kind of dissipation, then for large times the system is expected to tend to a harmonic motion, with the period of the external force, regardless of its initial configuration. A classical example of this behavior is furnished by the motion of a damped spring subject to a harmonically varying external force. The purpose of this paper is to prove such a result for a system governed by the wave equation in the exterior of a finite reflecting body. The dissipative mechanism in this case is radiation to infinity. Accordingly, we show that the spatial part of the harmonic motion which occurs in the limit is outgoing in the sense of Sommerfeld.

C. S. Morawetz: “Exponential decay of solutions of the wave equation,” Comm. Pure Appl. Math. 19 : 4 (November 1966), pp. 439–444. MR 204828 Zbl 0161.08002 article

Suppose a signal over a finite region propagates in free space according to the wave equation or some corresponding system of equations for which Huyghen’s principle holds. At a fixed point in space the signal will pass and leave the medium undisturbed. If Huyghen’s principle does not hold, some residual signal is left and slowly decays to zero. If there is a body which does not absorb any of the energy of the signal, we may expect that the signal is altered by the body but that, if Huyghen’s principle holds and if the body cannot hold energy, the residual signal decays much faster than if the principle does not hold.

This in fact is the case and it was proved in [1] for star-shaped reflecting bodies that the decay is exponential. This proof contains, although not explicitly, Theorem I which says that, if there is any rate of energy decay for a given body, then there is an exponential rate of decay. Here we give a simple proof of the same theorem.

C. S. Morawetz: “Mixed equations and transonic flow,” Rend. Mat. e Appl. (5) 25 (1966), pp. 482–509. MR 225020 Zbl 0317.35063 article

C. S. Morawetz and D. Ludwig: “An inequality for the reduced wave operator and the justification of geometrical optics,” Comm. Pure Appl. Math. 21 : 2 (March 1968), pp. 187–203. MR 223136 Zbl 0157.18701 article

We present an inequality for the reduced wave operator in the exterior of a star-shaped surface in \( n \)-space, with a Dirichlet boundary condition on the surface and a radiation condition at infinity. This inequality is used to demonstrate the continuous dependence (in a suitable norm) of the solution of a scattering problem upon the boundary data and inhomogeneous term in the differential equation. This basic result is then used together with the results of D. Ludwig [7] to prove that the formal solution of the scattering problem for a convex body, which is given by geometrical optics, is asymptotic to the exact solution. Similar results have been given in two dimensions by V. S. Buslaev [1] and R. Grimshaw [2], using different methods, who also consider the Neumann problem. Unfortunately the methods used here are inapplicable in that case.

C. S. Morawetz: “Time decay for the nonlinear Klein–Gordon equations,” Proc. Roy. Soc. Ser. A 306 (1968), pp. 291–296. MR 234136 Zbl 0157.41502 article

C. S. Morawetz: “Two \( L^p \) inequalities,” Bull. Am. Math. Soc. 75 : 6 (1969), pp. 1299–1302. MR 256149 Zbl 0186.18901 article

C. S. Morawetz and D. Ludwig: “The generalized Huyghens’ principle for reflecting bodies,” Comm. Pure Appl. Math. 22 : 2 (March 1969), pp. 189–205. MR 605677 Zbl 0167.10102 article

C. S. Morawetz: “Energy flow: Wave motion and geometrical optics,” Bull. Am. Math. Soc. 76 : 4 (1970), pp. 661–674. MR 267283 Zbl 0212.44102 article

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: “Profile problems for transonic flows with shocks,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 49 : 6 (1970), pp. 347–356. MR 295673 Zbl 0235.76033 article

C. S. Morawetz and W. A. Strauss: “Asymptotics of a nonlinear relativistic wave equation,” Bull. Am. Math. Soc. 77 : 5 (September 1971), pp. 797–798. A later version of this was published in Partial differential equations (1973). MR 294894 Zbl 0216.37404 article

C. S. Morawetz: “Well-posed problems and transonic flow,” pp. 325–333 in Proceedings of the X-th symposium on advanced problems and methods in fluid mechanics (Rynia, Poland, 6–11 September 1971), part 1. Edited by W. Fiszdon, Z. Płochocki, and M. Bratos. Fluid dynamics transactions 6. Państwowe Wydawnictwo Naukowe (Warsaw), 1971. MR 502811 incollection

Z. Płochocki	Related
M. Bratos-Anderson	Related
Władysław Fiszdon	Related

C. S. Morawetz: “On the modes of decay for the wave equation in the exterior of a reflecting body,” Proc. R. Ir. Acad., Sect. A 72 (1972), pp. 113–120. MR 303095 Zbl 0239.35057 article

C. S. Morawetz and W. A. Strauss: “Decay and scattering of solutions of a nonlinear relativistic wave equation,” Comm. Pure Appl. Math. 25 (1972), pp. 1–31. MR 303097 Zbl 0228.35055 article

C. S. Morawetz and W. A. Strauss: “Asymptotics of a nonlinear relativistic wave equation,” pp. 365–368 in Partial differential equations (Berkeley, CA, 9–27 August 1971). Edited by D. C. Spencer. Proceedings of Symposia in Pure Mathematics 23. American Mathematical Society (Providence, RI), 1973. An earlier version of this was published in Bull. Am. Math. Soc. 77:5 (1971). MR 333492 Zbl 0269.35060 incollection

C. S. Morawetz: “Estimates for a slowly-varying wave equation with a periodic potential,” Comm. Pure Appl. Math. 26 (1973), pp. 803–817. Collection of articles dedicated to Wilhelm Magnus. MR 336087 Zbl 0271.35047 article

C. S. Morawetz and W. A. Strauss: “On a nonlinear scattering operator,” Comm. Pure Appl. Math. 26 (1973), pp. 47–54. MR 348299 Zbl 0265.35057 article

K. S. Moravec: “A decay theorem for Maxwell’s equations,” pp. 233–240 in Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I, published as Uspehi Mat. Nauk 29 : 2 (176) (1974). The English version was published in Russ. Math. Surv. 29:2 (1974). MR 404833 incollection

C. S. Morawetz: “A decay theorem for Maxwell’s equations,” Russ. Math. Surv. 29 : 2 (1974), pp. 242–250. A Russian version was published in Uspehi Mat. Nauk 29:2(176) (1974). Zbl 0293.35010 article

C. S. Morawetz: “Decay for solutions of the exterior problem for the wave equation,” Comm. Pure Appl. Math. 28 (1975), pp. 229–264. MR 372432 Zbl 0304.35064 article

C. S. Morawetz: “Nouveaux problèmes sur les équations mixtes” [New problems on mixed equations], pp. 1–10 in Séminaire Goulaouic–Lions–Schwartz 1974–1975: Équations aux derivées partielles linéaires et non linéaires [Goulaouic–Lions–Schwartz Seminar 1974–1975: Linear and nonlinear partial derivative equations]. Centre de Mathématiques, École Polytechnique (Paris), March 1975. Exposé no. 15. MR 397189 Zbl 0307.35066 incollection

C. S. Morawetz: Notes on time decay and scattering for some hyperbolic problems. Regional Conference Series in Applied Mathematics 19. Society for Industrial and Applied Mathematics (Philadelphia), 1975. MR 492919 Zbl 0303.35002 book

C. S. Morawetz: “Time decay and relaxation schemes,” Advances in Math. 24 : 1 (April 1977), pp. 63–73. To Norman Levinson who introduced me to the theory of partial differential equations. MR 436618 Zbl 0443.35013 article

C. S. Morawetz, J. V. Ralston, and W. A. Strauss: “Decay of solutions of the wave equation outside nontrapping obstacles,” Comm. Pure Appl. Math. 30 : 4 (1977), pp. 447–508. A correction to this article was published in Comm. Pure Appl. Math. 31:6 (1978). MR 509770 Zbl 0372.35008 article

James Vickroy Ralston, Jr.	Related
Walter Alexander Strauss	Related

G. Kriegsmann and C. S. Morawetz: “Numerical solutions of exterior problems with the reduced wave equation,” J. Comput. Phys. 28 : 2 (August 1978), pp. 181–197. MR 502981 Zbl 0393.65042 article

C. S. Morawetz, J. V. Ralston, and W. A. Strauss: “Correction to: ‘Decay of solutions of the wave equation outside nontrapping obstacles’,” Comm. Pure Appl. Math. 31 : 6 (1978), pp. 795. Correction to an article published in Comm. Pure Appl. Math. 30:4 (1977). MR 509771 Zbl 0404.35015 article

James Vickroy Ralston, Jr.	Related
Walter Alexander Strauss	Related

C. S. Morawetz: “Geometrical optics and the singing of whales,” Am. Math. Monthly 85 : 7 (August–September 1978), pp. 548–554. MR 521793 Zbl 0429.76047 article

G. B. Kolata: “Cathleen Morawetz: The mathematics of waves,” Science 206 : 4415 (12 October 1979), pp. 206–207. MR 546415 Zbl 1225.01075 article

C. S. Morawetz: “Nonlinear conservation equations,” Am. Math. Mon. 86 : 4 (April 1979), pp. 284–287. MR 1539008 Zbl 0404.35072 article

C. S. Morawetz: “A regularization for a simple model of transonic flow,” Comm. Partial Differential Equations 4 : 1 (1979), pp. 79–111. MR 514720 Zbl 0448.35067 article

G. A. Kriegsmann and C. S. Morawetz: “Numerical methods for solving the wave equation with variable index of refraction,” pp. 118–123 in Boundary and interior layers — computational and asymptotic methods (Dublin, 3–6 June 1980). Edited by J. J. H. Miller. Boole Press Conference Series 2. Boole Press (Dún Laoghaire, Ireland), 1980. MR 589356 Zbl 0448.65065 incollection

John James Henry Miller	Related
Gregory Anthony Kriegsmann	Related

G. A. Kriegsmann and C. S. Morawetz: “Solving the Helmholtz equation for exterior problems with variable index of refraction, I,” SIAM J. Sci. Statist. Comput. 1 : 3 (1980), pp. 371–385. MR 596031 Zbl 0469.65084 article

A new technique for numerically solving the reduced wave equation on exterior domains is presented. The method is basically a relaxation scheme which exploits the limiting amplitude principle. A modified boundary condition at “infinity” is also given. The technique is tested on several model problems: the scattering of a plane wave off a metal cylinder, a metal strip, a Helmholtz resonator, an inhomogeneous cylinder (lens), and a nonlinear plasma column. The results are in good qualitative agreement with previously calculated values. In particular, the numerical solutions exhibit the correct refractive and diffractive effects at moderate frequencies.

C. S. Morawetz: “A formulation for higher-dimensional inverse problems for the wave equation,” Comput. Math. Appl. 7 : 4 (1981), pp. 319–331. MR 615305 Zbl 0465.35083 article

Two particular inverse scattering problems are of special interest. The first concerns the discovery of a perturbation in the speed of sound by analyzing the return signal from a blast wave set off above it. The second concerns the determination of a potential from the back scattering of plane waves. In one dimensional problems the two cases are very closely related. In higher dimensions the situation becomes much more complicated.

We present here a new approach to four such classical higher dimensional inverse problems for determining the coefficients of a partial differential equation, including the two mentioned. The idea stems from the work of Deift and Trubowitz [1979], in one space dimension. No conclusive theorem is found but the approach provides an algorithm for iteration which might lead to an existence theorem and which will be explored numerically elsewhere.

G. A. Kriegsmann and C. S. Morawetz: “Computations with the nonlinear Helmholtz equation,” J. Opt. Soc. Am. 71 : 8 (1981), pp. 1015–1019. MR 623362 article

C. S. Morawetz: Lectures on nonlinear waves and shocks (Bangalore, India). Tata Institute of Fundamental Research Lectures on Mathematics and Physics 67. Springer (Berlin), 1981. Notes by P. S. Datti. MR 660637 Zbl 0498.76001 book

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

A. Bayliss, G. A. Kriegsmann, and C. S. Morawetz: “Strange boundary layer effects on the edge of a nonlinear plasma,” pp. 3–12 in Computational and asymptotic methods for boundary and interior layers (Dublin, 16–18 June 1982). Edited by J. J. H. Miller. Boole Press Conference Series 4. Boole Press (Dún Laoghaire, Ireland), 1982. MR 737566 Zbl 0513.76127 incollection

John James Henry Miller	Related
Alvin Bayliss	Related
Gregory Anthony Kriegsmann	Related

A. Bayliss, G. A. Kriegsmann, and C. S. Morawetz: “The nonlinear interaction of a laser beam with a plasma pellet,” Comm. Pure Appl. Math. 36 : 4 (1983), pp. 399–414. MR 709642 Zbl 0534.65083 article

Alvin Bayliss	Related
Gregory Anthony Kriegsmann	Related

C. S. Morawetz and G. A. Kriegsmann: “The calculations of an inverse potential problem,” SIAM J. Appl. Math. 43 : 4 (1983), pp. 844–854. MR 709741 Zbl 0542.65079 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.

K. O. Friedrichs: Selecta, vol. 2. Edited by C. S. Morawetz. Contemporary Mathematicians. Birkhäuser (Boston), 1986. With commentaries by Tosio Kato, Fritz John, Louis Nirenberg and David Isaacson. book

David Isaacson	Related
Tosio Kato	Related
Kurt Otto Friedrichs	Related
Louis Nirenberg	Related
Fritz John	Related

K. O. Friedrichs: Selecta, vol. 1. Edited by C. S. Morawetz. Contemporary Mathematicians. Birkhäuser (Boston), 1986. With a foreword by Cathleen S. Morawetz, a biography by Constance Reid, and commentaries by Peter D. Lax, Wolfgang Wasow, Harold Weitzner. MR 897749 Zbl 0613.01020 book

Wolfgang R. Wasow	Related
Harold Weitzner	Related
Kurt Otto Friedrichs	Related
Peter D. Lax	Related
Constance Reid	Related

C. S. Morawetz: “Mathematical problems in transonic flow,” Canad. Math. Bull. 29 : 2 (1986), pp. 129–139. MR 844890 Zbl 0572.76055 article

C. S. Morawetz: “On the transonic flow past an airfoil,” pp. 109 in Nonstrictly hyperbolic conservation laws (Anaheim, CA, 9–10 January 1985). Edited by B. L. Keyfitz and H. C. Kranzer. Contemporary Mathematics 60. American Mathematical Society (Providence, RI), 1987. MR 873535 incollection

Barbara Lee Keyfitz	Related
Herbert C. Kranzer	Related

C. S. Morawetz: “Weak solutions of transonic flow by compensated compactness,” pp. 253–256 in Dynamical problems in continuum physics (Minneapolis, 1985). Edited by J. L. Bona, C. Dafermos, J. L. Ericksen, and D. Kinderlehrer. IMA Volumes in Mathematics and its Applications 4. Springer (New York), 1987. MR 893400 Zbl 0617.76063 incollection

Constantine Michael Dafermos	Related
Jerald L. Ericksen	Related
David Samuel Kinderlehrer	Related
Jerry Lloyd Bona	Related

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

J. D. Patterson: “Cathleen Synge Morawetz,” pp. 152–155 in Women of mathematics: A bibliographic sourcebook. Edited by L. S. Grinstein and P. J. Campbell. Greenwood Press (Westport, CT), 1987. incollection

Paul Jude Campbell	Related
Louise S. Grinstein	Related
James D. Patterson	Related

C. S. Morawetz: “The Courant Institute of Mathematical Sciences,” pp. 303–307 in A century of mathematics in America, part II. Edited by P. Duren and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003135 Zbl 0667.01021 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related

C. S. Morawetz: Transonic flow and mixed equations, 1989. 60 minute VHS videocassette, AMS-MAA Joint Lecture Series. lecture recorded 12 January 1989, Phoenix, AZ. MR 1056292 Zbl 0923.76072 misc

A. Bayliss, Y. Y. Li, and C. S. Morawetz: “Scattering by a potential using hyperbolic methods,” Math. Comp. 52 : 186 (1989), pp. 321–338. MR 958869 Zbl 0692.65068 article

Yan Yan Li	Related
Alvin Bayliss	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, D. C. Stevens, and H. Weitzner: “A numerical experiment on a second-order partial differential equation of mixed type,” Comm. Pure Appl. Math. 44 : 8–9 (1991), pp. 1091–1106. In appreciation of Natascha Brunswick’s many contributions to the Courant Institute and particularly to this journal. MR 1127052 Zbl 0836.35102 article

In an earlier article in this journal one of us examined a problem of linear electromagnetic wave propagation in a plasma based on a particular plasma model (see [1]) and commented on the mathematically rather non-standard system of linear partial differential equations for the problem. Since then our attempts to solve the system numerically have not led to entirely satisfactory results. In this note we report on numerical experiments on a much simpler model partial differential equation which can be studied in much greater detail and which clarifies the numerical and mathematical difficulties inherent in the original problem.

Natascha Artin Brunswick	Related
Harold Weitzner	Related
Deborah C. Stevens	Related

C. S. Morawetz: The last 75 years: Giants of applied mathematics, 1991. 45 minute VHS videocassette, AMS-MAA Joint Lecture Series. A lecture presented at the seventy-fifth anniversary celebration of the founding of the MAA, 8 August 1990, Columbus, OH. An article based on this lecture was published in Am. Math. Monthly 99:9 (1992). MR 1153664 misc

Geoffrey Ingram Taylor	Related
Theodore von Kármán	Related
John von Neumann	Related
Norbert Wiener	Related

C. S. Morawetz: “Giants,” Am. Math. Monthly 99 : 9 (November 1992), pp. 819–828. Based on a lecture presented at the seventy-fifth anniversary celebration of the founding of the MAA. A video of the lecture this article was based on was published by the MAA in 1991. MR 1191701 Zbl 0778.01011 article

A. J. Majda: “Nomination for Cathleen S. Morawetz for President of the AMS,” Notices Am. Math. Soc. 40 : 7 (1993), pp. 816–817. article

“Cathleen S. Morawetz elected AMS President,” AWM Newsletter 24 : 1 (January–February 1994), pp. 6. article

C. S. Morawetz: “Potential theory for regular and Mach reflection of a shock at a wedge,” Comm. Pure Appl. Math. 47 : 5 (1994), pp. 593–624. MR 1278346 Zbl 0807.76033 article

If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength \( b \)) and small wedge angles \( 2\theta_w \) through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter \[\beta = c_1\theta^2_w(\gamma + 1)b \] ranges from 0 to \( \infty \). Here \( \gamma \) is the polytropic constant and \( c_1 \) is the sound speed behind the incident shock.

For \( \beta > 2 \) regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For \( \beta < \frac{1}{2} \) Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case \( \beta = 0 \) can be solved. For \( \frac{1}{2} < \beta < 2 \) or even larger \( \beta \) the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens.

The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for \( \beta \) sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point.

C. S. Morawetz, W. Abikoff, C. Corillon, I. Kra, T. Weinstein, and J. Gilman: “Remembering Lipman Bers,” Notices Am. Math. Soc. 42 : 1 (1995), pp. 8–25. This was later reprinted in Lipman Bers: A life in mathematics (2015). MR 1306867 Zbl 1042.01527 article

Irwin Kra	Related
William Abikoff	Related
Carol Corillon	Related
Lipman Bers	Related
Tilla Weinstein	Related
Jane Piore Gilman	Related

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.

C. S. Morawetz: “Memories of a wartime student of math and physics at the University of Toronto,” pp. 307–311 in Canadian Mathematical Society, 1945–1995: Mathematics in Canada, vol. 1. Edited by P. Fillmore. Canadian Mathematical Society (Ottawa, ON), 1995. MR 1661627 Zbl 0934.01009 incollection

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

T. Perl: “Cathleen Synge Morawetz,” pp. 147–152 in Notable women in mathematics: A biographical dictionary. Edited by C. Morrow and T. Perl. Greenwood Press (Westport, CT), 1998. incollection

Charlene Morrow	Related
Teri Hoch Perl	Related

A. Jackson: “Cathleen Morawetz receives National Medal of Science,” Notices Am. Math. Soc. 46 : 3 (March 1999), pp. 352. article

C. S. Morawetz: “Mathematics to the rescue (retiring Presidential address),” Notices Am. Math. Soc. 46 : 1 (January 1999), pp. 9–16. MR 1658886 Zbl 1194.01037 article

“Morawetz receives National Science Medal,” SIAM News (22 January 1999). article

C. S. Morawetz: “Variations on conservation laws for the wave equation,” Bull. Am. Math. Soc. (N.S.) 37 : 2 (2000), pp. 141–154. MR 1751947 Zbl 0957.35100 article

The first part of this paper, presented as an Emmy Noether lecture in connection with the ICM in Berlin in August 1998, gives some examples of using Noether’s theorem for conservation laws for Tricomi-like equations and for the wave equation. It is also shown that equations which are semilinear variations of the wave equation can very often be handled similarly. The type of estimate obtained can even be used to get otherwise unobtainable local estimates for regularity.

The fourth part is an introduction to the relation of black holes to the wave equation mainly showing the results of D. Christodoulou. His results use much more difficult estimates not corresponding at all to those in the first part of the paper.

Cathleen Morawetz: A great mathematician, published as Methods Appl. Anal. 7 : 2. International Press (Somerville, MA), June 2000. Issues 2 and 3 are dedicated to Cathleen Morawetz on her seventy-seventh birthday. An identical preface (concerning Morawetz) to the one here appeared in issue 3. MR 1869283 Zbl 0995.00024 book

“Curriculum vitae: Cathleen Synge Morawetz,” pp. ii–vii in Cathleen Morawetz: A great mathematician, published as Methods Appl. Anal. 7 : 2. International Press (Somerville, MA), June 2000. MR 1869284 Zbl 1159.01342 incollection

Cathleen Morawetz: A great mathematician, published as Methods Appl. Anal. 7 : 3. International Press (Somerville, MA), June 2000. Issues 2 and 3 are dedicated to Cathleen Morawetz on her seventy-seventh birthday. An identical preface (concerning Morawetz) to the one here appeared in issue 2. MR 1869294 Zbl 0997.00566 book

E. Hopf: Selected works of Eberhard Hopf: With commentaries. Edited by C. S. Morawetz, J. B. Serrin, and Y. G. Sinai. AMS Collected Works Series 17. American Mathematical Society (Providence, RI), 2002. MR 1985954 Zbl 1011.01017 book

Ya. G. Sinai	Related
Eberhard Friedrich Ferdinand Hopf	Related
James Burton Serrin, Jr.	Related

W. Jäger, P. Lax, and C. S. Morawetz: “Olga Arsen’evna Oleĭnik (1925–2001),” Notices Am. Math. Soc. 50 : 2 (2003), pp. 220–223. English version of a Russian language article in Trudy Seminara imeni I. G. Petrovskogo 2003:23 (2003). Also published in J. Math. Sci. 120:3 (2004). MR 1951108 Zbl 1159.01335 article

Willi Jäger	Related
Peter D. Lax	Related
Olga Arsen’evna Oleĭnik	Related

V. Eger, P. Laks, and K. Moravets: “Olga Arsen’evna Oleĭnik,” Tr. Semin. im. I. G. Petrovskogo 2003 : 23 (2003), pp. 7–15. An English version of this article appeared in J. Math. Sci. 120:3 (2004) and in Notices Am. Math. Soc. 50:2 (2003). MR 2085178 article

V. Eger	Related
Peter D. Lax	Related
Olga Arsen’evna Oleĭnik	Related

“2004 Steele Prizes,” Notices Am. Math. Soc. 51 : 4 (2004), pp. 421–425. Cathleen Morawetz received a Lifetime Achievement award. MR 2039815 Zbl 1168.01303 article

Lawrence Craig Evans	Related
Nicolai Vladimirovich Krylov	Related
John W. Milnor	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.

S. Friedlander, P. Lax, C. Morawetz, L. Nirenberg, G. Seregin, N. Ural’tseva, and M. Vishik: “Olga Alexandrovna Ladyzhenskaya (1922–2004),” Notices Am. Math. Soc. 51 : 11 (December 2004), pp. 1320–1331. MR 2105237 Zbl 1168.01327 article

Gregory A. Seregin	Related
Susan Jean Friedlander	Related
Nina Nikolaevna Ural’tseva	Related
Mark Iosifovich Vishik	Related
Peter D. Lax	Related
Olga Alexandrovna Ladyzhenskaya	Related
Louis Nirenberg	Related

W. Jäger, P. Lax, and C. S. Morawetz: “Olga Arsen’evna Oleinik,” J. Math. Sci. 120 : 3 (2004), pp. 1242–1246. English version of a Russian language article in Tr. Semin. im. I. G. Petrovskogo 2003:23 (2003). Also published in Notices Am. Math. Soc. 50:2 (2003). Zbl 1077.01014 article

Willi Jäger	Related
Peter D. Lax	Related
Olga Arsen’evna Oleĭnik	Related

C. S. Morawetz: “Problems, including mathematical problems, from my early years,” pp. 267–272 in Complexities: Women in mathematics. Edited by B. A. Case and A. Leggett. Princeton University Press, 2005. from the Olga Taussky Todd Celebration of Careers for Women In Mathematics. incollection

Anne M. Leggett	Related
Olga Taussky-Todd	Related
Bettye Anne Case	Related

C. S. Morawetz and B. Thomases: “A decay theorem for some symmetric hyperbolic systems,” J. Hyperbolic Differ. Equ. 3 : 3 (2006), pp. 475–480. MR 2238738 Zbl 1100.35065 article

C. S. Morawetz: “A memory of Gaetano Fichera,” Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 30 : 1 (2006), pp. 3–5. MR 2489589 article

D. Lupo, C. S. Morawetz, and K. R. Payne: “On closed boundary value problems for equations of mixed elliptic-hyperbolic type,” Comm. Pure Appl. Math. 60 : 9 (2007), pp. 1319–1348. An erratum to this article was published in Comm. Pure Appl. Math. 61:4 (2008). MR 2337506 Zbl 1125.35066 article

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 transonic flow 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.

Daniela Lupo	Related
Kevin Ray Payne	Related

D. Lupo, C. S. Morawetz, and K. R. Payne: “Erratum: ‘On closed boundary value problems for equations of mixed elliptic-hyperbolic type’,” Comm. Pure Appl. Math. 61 : 4 (2008), pp. 594. Erratum to an article published in Comm. Pure Appl. Math. 60:9 (2007). MR 2383934 Zbl 1144.35451 article

Daniela Lupo	Related
Kevin Ray Payne	Related

T. Cheng: Cathleen Morawetz (interviewed by Tiffany K. Cheng), 8 July 2009. Interview conducted as part of the Margaret MacVicar Memorial AMITA (Association of MIT Alumnae) Oral History Project. misc

C. Morawetz: “Introduction: Dinner speech,” Q. Appl. Math. 68 : 1 (2010), pp. 3. Zbl 1183.01031 article

Cathleen Morawetz, December 2012. Simons Foundation “Science Lives” online video. misc

A. Jackson: “Happy 91st, Cathleen Synge Morawetz,” Notices Am. Math. Soc. 61 : 5 (2014), pp. 510–511. Zbl 1338.01042 article

C. S. Morawetz: “Remembering Lipman Bers,” pp. 299–309 in Lipman Bers: A life in mathematics. Edited by L. Keen, I. Kra, and R. E. Rodríguez. American Mathematical Society (Providence, RI), 2015. This originally appeared in Notices Am. Math. Soc. 42:1 (1995). Zbl 06686715 incollection

Irwin Kra	Related
Rubí E. Rodríguez	Related
Lipman Bers	Related
Linda Jo Goldway Keen	Related

K. Chang: “Cathleen Morawetz, mathematician with real-world impact, dies at 94,” NYT (11 August 2017). article

E. Langer: “Her problem-solving theorems helped pave the way for women in mathematics,” Washington Post (14 August 2017). This obituary was later published under a different title by the Chicago Tribune (16 August 2017). article

E. Langer: “Cathleen Morawetz, first female mathemetician to win National Medal of Science, dies at 94,” Chicago Tribune (16 August 2017). This obituary was originally published with a different title by the Washington Post (14 August 2017). article

Cathleen Morawetz

Complete Bibliography

Filter the Bibliography List

year	title	people

			clear