Celebratio Mathematica

Cath­leen Synge Mor­awetz wrote this art­icle in con­nec­tion with The Jo­si­ah Wil­lard Gibbs lec­ture she presen­ted at the Amer­ic­an Math­em­at­ic­al So­ci­ety meet­ing in San Fran­cisco, Cali­for­nia, Janu­ary 7, 1981. This is a beau­ti­ful piece on a sub­ject at the core of ap­plied math­em­at­ic­al ana­lys­is and nu­mer­ic­al meth­ods mo­tiv­ated by the press­ing en­gin­eer­ing tech­no­logy of the mid-twen­ti­eth cen­tury and the hu­man urge to travel fast at ef­fi­cient cost. From the math­em­at­ic­al view­point this prob­lem com­prises the un­der­stand­ing of mod­els of non­lin­ear par­tial dif­fer­en­tial equa­tions arising in com­press­ible flu­id mech­an­ics, as much as un­der­stand­ing how to ob­tain nu­mer­ic­al ap­prox­im­a­tions to a mod­el dis­cret­iz­a­tion that res­ult both in find­ing nu­mer­ic­ally com­puted sur­faces close to the mod­el’s solu­tions (if such ex­ists) but also in match­ing these com­puted mod­el out­puts to ex­per­i­ments from en­gin­eer­ing or ex­per­i­ment­al ob­ser­va­tion view­points.

This com­ment­ary starts with a de­scrip­tion of the state-of-the-art up to 1982, from a very com­pre­hens­ive ex­plan­a­tion for any sci­ent­ist of what it takes to fly an ob­ject with wings and the is­sues of in­stabil­it­ies that arise as we try to fly too fast, to the de­scrip­tion of the ad­equate mod­el giv­en by the sys­tem of Hamilton–Jac­obi frame­work of con­ser­va­tion of mass and mo­mentum for a com­press­ible po­ten­tial is­en­trop­ic in­vis­cid flu­id, for­mu­lated by the coupled non­lin­ear sys­tem of con­ser­va­tion of mass to the Bernoulli law as­so­ci­ated to such a flu­id mod­el.

More spe­cific­ally, de­fin­ing the state vari­ables by dens­ity \( \rho \), ve­lo­city \( \vec{q} \) and pres­sure \( p = p(\rho) \), con­sider the re­l­at­ive mo­tion of hav­ing an obstacle (such as an air­foil) at rest, so that the ve­lo­city at in­fin­ity, \( \vec{q}_\infty \), is ac­tu­ally the speed as­so­ci­ated to that obstacle. If the flow is ir­rota­tion­al, there ex­ists a po­ten­tial func­tion \( \rho \) that gov­erns the ve­lo­city by the re­la­tion \( \vec{q}= \nabla\Phi \), and so the flow speed is \( \mathbf{q} = |\nabla\Phi| \). The cor­res­pond­ing con­ser­va­tion of mo­mentum re­la­tion, renor­mal­ized by the dens­ity and in­teg­rated along non­cross­ing stream­lines (i.e., or­tho­gon­al level sur­faces to the po­ten­tial func­tion level sur­faces), yields the Bernoulli’s law that ex­presses a bal­ance law for \( \rho \), \( p \), \( |\nabla\Phi| \). Hence, one can ob­tain a closed sys­tem of first-or­der equa­tions by coup­ling con­ser­va­tion of mass with Bernoulli’s law to ob­tain a quasi­lin­ear Hamilton–Jac­obi sys­tem for steady, ir­rota­tion­al, in­vis­cid is­en­trop­ic gas flow \begin{equation}\label{eqon} \mathrm{div}(\rho\nabla\Phi)=0 \quad\text{and}\quad \frac{|\nabla\Phi|^2}{2}+p(\rho)=K, \end{equation} where \( K \) is the isoen­er­get­ic con­stant. Bernoulli’s law al­lows for the re­la­tion \( \rho = \rho(\nabla\Phi) \), so that the Hamilton–Jac­obi sys­tem is re­duced to a quasi­lin­ear scal­ar equa­tion that, when writ­ten in two di­men­sions, takes the form \begin{equation}\label{eqtw} (c^2-u^2)\Phi_{xx}-2uv\Phi_{xy}+(c^2-v^2)\Phi_{yy}=0, \end{equation} for \[ \vec{q}=\nabla\Phi= (u,v) \] the ve­lo­city field, and \[ c = \frac{dp}{d\rho}c(|\nabla\Phi|) \] the speed of sound, giv­en through Bernoulli’s law. Then the quo­tient \( M = \mathbf{q}/c \), re­ferred to as the Mach num­ber, de­term­ines the loc­al speed of sound. Then the fol­low­ing oc­curs (de­scribed us­ing a slight change of word­ing from the Mor­awetz art­icle [7]):

…if \( \mathbf{q} \) is small, so both \( u \), \( v \) are small, then \( \Delta\Phi = 0 \) mean­ing flow is es­sen­tially in­com­press­ible. Choos­ing loc­al co­ordin­ates with \( v = 0 \) then, for \( \mathbf{q} < c \) the equa­tion is el­lipt­ic and for \( \mathbf{q} > c \), the equa­tion is hy­per­bol­ic. That means the flow is ana­log­ous to the in­com­press­ible case with loc­ally smooth solu­tions for \( M < 1 \), but when \( M > 1 \), all the dif­fi­cult fea­tures of non­lin­ear hy­per­bol­ic equa­tions oc­cur.

Yet, a solu­tion across the two re­gions with a nonempty con­tact set needs to be un­der­stood as well. This prob­lem is at the core of “passing the son­ic bar­ri­er”.

This re­gime is called tran­son­ic when the emer­gence of strong shocks are ex­pec­ted to be dis­con­tinu­ous solu­tions to this mixed type sys­tem for a sta­tion­ary flow frame­work (see Fig­ure 2 and its de­scrip­tion from the wind tun­nel ex­per­i­ments in the Mor­awetz pa­per [7]). While ex­ist­ence of solu­tions for the tran­son­ic flow prob­lem may be rather simple in one space di­men­sion, their non­trivi­al solu­tions in two or more di­men­sions re­main one of the most haunt­ing prob­lems in flu­id dy­nam­ics, with strong im­plic­a­tions that range from the mod­el­ing of air­flow past wing pro­files in aerospace ap­plic­a­tions to wave propaga­tion and sin­gu­lar­ity form­a­tion in re­lativ­ity the­ory.

Mor­awetz’s pa­per is a mas­terly ex­plan­a­tion of why lin­ear meth­ods fail as shown by means of Friedrichs’ mul­ti­pli­er meth­od [e1] and her own work on [4], but also dis­cussed per­turb­a­tion the­ory that yields the Tricomi equa­tion as an ap­prox­im­a­tion to the tran­son­ic flow mod­el in equa­tion \eqref{eqtw}, re­lated to her own con­tri­bu­tions [1], [2], [3], [5], [6], [14]. She also presen­ted her vis­ion on how in­sight­ful nu­mer­ic­al ap­prox­im­a­tions and ap­plied ana­lys­is led to sig­ni­fic­ant res­ults that im­pacted lin­ear and non­lin­ear wave the­ory for hy­per­bol­ic sys­tems.

Cath­leen con­cludes her 23-page present­a­tion stat­ing,

We are left with the gen­er­al weak ex­ist­ence the­or­em for the full non­lin­ear prob­lem un­solved. There are lots of ap­proaches to try: Show the dif­fer­ence scheme con­verges. Ex­tend the vari­ation­al prin­ciples of el­lipt­ic the­ory. Per­haps something quite new…

Dur­ing the years since 1982, there have been sev­er­al sig­ni­fic­ant is­sues that have been ad­dressed and brought pro­gress to this area. There has been pro­gress in solv­ing the tran­son­ic flow mod­el for the full sys­tem \eqref{eqon} for small per­turb­a­tions on po­ten­tial strength by draw­ing con­nec­tions to the obstacle prob­lem by means of solv­ing a suit­able free bound­ary prob­lem. Chen and Feld­man con­struc­ted full tran­son­ic solu­tions ([e4], [e5]) for weak shock con­di­tions.

Yet their as­sump­tions do not fully cov­er the strong dis­con­tinu­ity re­gime that Mor­awetz en­vi­sioned in her title as “The math­em­at­ic­al ap­proach to the son­ic bar­ri­er” that would match ex­per­i­ment­al data, so many as­pects of the math­em­at­ics for tran­son­ic flow mod­els re­mains un­solved.

Cath­leen and I worked for five years in the mid-1990s, and we pro­posed an ap­proach for solv­ing the prob­lem that would ad­mit large shocks. But cer­tainly we run short of claim­ing the ex­ist­ence of solu­tion to the steady, ir­rota­tion­al, in­vis­cid is­en­trop­ic gas flow mod­el in two di­men­sions for the non­trivi­al obstacle do­main for large shocks. Our tech­niques are based on a couple of manuscripts that Cath­leen had de­veloped in the mid-1980s and the early 1990s on solv­ing a vis­cous ap­prox­im­a­tion to equa­tions \eqref{eqon} in a non­trivi­al do­main [8] with enough good es­tim­ates, uni­form in the vis­cos­ity para­met­er, and study­ing their in­vis­cid lim­it by means of com­pensated com­pact­ness tech­niques by Mur­at, Tar­tar, and Di­Per­na, ad­jus­ted to sys­tem \eqref{eqon} and de­scribed in her work [9], [10], [11], and [14]. Later, we were able to con­struct \( C^\infty(\Omega) \)-solu­tions, \( \Omega \) in \( \mathbb{R}^2 \), for the vec­tor field \( \mathbf{q} \) and dens­ity \( \rho \) solv­ing an up­wind vis­cous ap­prox­im­a­tion to sys­tem \eqref{eqon}, some­how in­spired by the Jameson nu­mer­ic­al ap­prox­im­a­tion, for both the neut­ral gases cases as well as the case of charged gases, by coup­ling the flu­id equa­tions to an elec­tro­stat­ic mean field po­ten­tial. It was im­port­ant to show that such a vis­cous sys­tem was solv­able. We ac­com­plished this task by con­sid­er­ing non-New­to­ni­an vis­cos­it­ies and non­lin­ear bound­ary con­di­tions [e3], [12] that al­lowed sharp con­trol for the speed from above and the dens­ity from be­low [e2], [13]. Much is left to do, such as the pas­sage to the in­vis­cid lim­it, to show that weak en­trop­ic solu­tions do ex­ist in the whole do­main.

In our con­ver­sa­tions through the last twenty years, Cath­leen and I wondered wheth­er the lack of suc­cess­ful pro­gress was due to a lack of more avail­able tech­niques bey­ond the ones already used. These avail­able tech­niques in­clude com­pact­ness by com­par­is­ons the­or­em, reg­u­lar­iz­a­tion with de­gree the­ory for the Leray–Schaud­er fixed point the­or­em, op­tim­al uni­form bounds rates, and the pas­sage to the in­vis­cid lim­it by com­pensated com­pact­ness.

Or per­haps the quasi­lin­ear sys­tem \eqref{eqon} is either too simple or in­com­plete to de­scribe the phe­nom­ena ob­served by the wind tun­nel ex­per­i­ments from Fig­ure 2 of [7]. In such a case, mod­el cor­rec­tions and new ex­per­i­ments may be needed.

To­ward the end of the art­icle, Cath­leen wrote

Let me close now…to say that I have left a lot un­said and a lot un­quoted. But I would like to thank for their help my tran­son­ic col­leagues, Kurt O. Friedrichs, Lip­man Bers, Paul R. Ga­rabedi­an and Ant­ony Jameson.

I hope many of my col­leagues would have been able to in­ter­act as I have done with Mor­awetz and these cham­pi­ons whose minds were filled day after day with the math­em­at­ics and nu­mer­ics of tran­son­ic flow mod­els. And I hope for more in­di­vidu­als whose curi­os­ity will arise to com­plete Mor­awetz’s en­vi­sioned math­em­at­ic­al path.