#### by Irene M. Gamba

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.