Celebratio Mathematica

It is im­possible to re­view all of Cath­leen Mor­awetz’s para­mount con­tri­bu­tions to pure and ap­plied math­em­at­ics and to fully as­sess their im­pact on twen­ti­eth cen­tury math­em­at­ics and the math­em­at­ic­al com­munity in gen­er­al. In this art­icle, we fo­cus on Mor­awetz’s deep and in­flu­en­tial work on the ana­lys­is of par­tial dif­fer­en­tial equa­tions (PDEs) of mixed el­lipt­ic-hy­per­bol­ic type, most not­ably in the math­em­at­ic­al the­ory of tran­son­ic flows and shock waves. We also dis­cuss the pro­found im­pact of Mor­awetz’s work on some re­cent de­vel­op­ments and break­throughs in these re­search dir­ec­tions and re­lated areas in pure and ap­plied math­em­at­ics.

Mor­awetz’s early work on tran­son­ic flows has not only provided a new un­der­stand­ing of mixed-type PDEs, but has also led to new meth­ods of ef­fi­cient air­craft design. Mor­awetz’s pro­gram for con­struct­ing glob­al steady weak tran­son­ic flow solu­tions past pro­files has been a source of mo­tiv­a­tion for nu­mer­ous re­cent de­vel­op­ments in the ana­lys­is of non­lin­ear PDEs of mixed type and re­lated mixed-type prob­lems through weak con­ver­gence meth­ods. Fur­ther­more, her work on the po­ten­tial the­ory for reg­u­lar and Mach re­flec­tion of a shock at a wedge (now known as the von Neu­mann prob­lem) has been an in­spir­a­tion for the re­cent com­plete solu­tion of the von Neu­mann con­jec­tures re­gard­ing glob­al shock reg­u­lar re­flec­tion-dif­frac­tion con­fig­ur­a­tions, all the way up to the de­tach­ment angle of the wedge.

As a gradu­ate stu­dent, I learned a great deal from Cath­leen’s pa­pers [1], [2], [3], [4], [5], [6], which were a true in­spir­a­tion to me. My aca­dem­ic jour­ney took a sig­ni­fic­ant turn when I joined the Cour­ant In­sti­tute of Math­em­at­ic­al Sci­ences (New York Uni­versity) as a postdoc­tor­al fel­low un­der the dir­ec­tion of Peter Lax. Dur­ing this time, I had the ex­traordin­ary op­por­tun­ity to learn dir­ectly from Cath­leen about the chal­len­ging and fun­da­ment­al re­search field that had, un­til that time, re­mained largely un­ex­plored. I was im­mensely grate­ful to Cath­leen for ded­ic­at­ing count­less hours to dis­cuss and ana­lyze with me a long list of open prob­lems in this field. Her in­sights were both il­lu­min­at­ing and pro­lif­ic, and I learned im­mensely from her dur­ing my these years at Cour­ant. Mak­ing sub­stan­tial pro­gress on some of these long­stand­ing open prob­lems, however, was a jour­ney that spanned over 10 years, on and off. In­deed, this field has proven to be truly chal­len­ging. As a res­ult, I ex­per­i­enced great joy when I had the hon­or of present­ing our first solu­tion of the von Neu­mann prob­lem in [e1] to Cath­leen dur­ing my lec­ture at the Con­fer­ence on Non­lin­ear Phe­nom­ena in Math­em­at­ic­al Phys­ics, ded­ic­ated to her on the oc­ca­sion of her 85th birth­day, held at the Fields In­sti­tute in Toronto, Canada, from the 18th to 20th of Septem­ber 2008.

[Ed­it­or’s note: The text above is from the In­tro­duc­tion of “Mor­awetz’s con­tri­bu­tions to the math­em­at­ic­al the­ory of tran­son­ic flows, shock waves, and par­tial dif­fer­en­tial equa­tions of mixed type” by Gui Qi­ang G. Chen, pub­lished in the Bul­let­in 61:1 (2024), 1151–171. For the full art­icle, click on the PDF link at the up­per right of this page.]