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Shock Waves and PDEs

A Conversation with Barbara Lee Keyfitz

Fariba Fahroo
Reza Malek-Madani

Communicated by Notices Associate Editor Reza Malek-Madani

1. Introduction

For the past half century, hyperbolic conservation laws have served as an important subfield of applied mathematics, especially in its relation to physics, chemistry, and biology. The way we model the behavior of elastic materials and fluid flows is inherently through partial differential equations (PDEs). When we explore dynamics (wave propagation), we end up analyzing a special class of PDEs, which are often hyperbolic and nonlinear, and are capable of modeling features such as shocks, dislocations, and fracture, because they are able to support discontinuous functions as their solutions. One of the grand challenges in mathematics in the past fifty years has been to develop a framework to make sense of what it means to have a discontinuous solution of a differential equation.

Barbara Keyfitz has spent her entire career in this field. One of the goals of this article is to highlight her contributions. We have had extensive conversations with Barbara and will describe some of her substantial contributions in her own voice. While we will present some of the details of her technical contributions, our emphasis will be on allowing Barbara to share her mathematical journey over the past several decades, first by describing her triumphs, and some of the challenges she has encountered, but also by sharing some of her unique achievements in providing service to our community, at the national and international level.

Barbara’s education started at the University of Toronto in 1962. After earning her bachelors degree in mathematics, she enrolled as a graduate student in the Courant Institute of Mathematical Sciences in 1966 where she studied conservation laws under the direction of Peter D. Lax. Our conversation with Barbara began from this stage of her career, and we proceeded to learn about the path that took her to Columbia University, Princeton, Arizona State University, University of Houston, and Ohio State, where she is now an emeritus professor. In addition, while she has served on numerous national and international committees, we will concentrate on hearing from Barbara about her contributions as the president (2005–2007) of the Association for Women in Mathematics, the president of the International Council for Industrial and Applied Mathematics (2011–2015), and as director of the Fields Institute (2004–2008).

But first a few definitions.

1.1. Hyperbolicity and PDEs

The simplest conservation law is the scalar partial differential equation

which is typically supplemented with the initial condition . In 1, stands for time, for space, and for a physical concentration. Standard generalizations of 1 are to one-dimensional systems of equations

where , and to multidimensional systems

where . Many fundamental equations of mathematical physics fall in the category of 1, 2, or 3.

Equation 2 is called hyperbolic if the matrix has real eigenvalues, which in the case of 1 reduces to simply existing. The corresponding definition for 3 is that have real eigenvalues.

Equation 2 shows why conservation laws are special. A standard way of solving the wave equation 2 is to follow the behavior of solutions along characteristics, curves defined in the plane by the equations

where is an eigenvalue of . In the case of 1, the characteristic curves are defined by the ordinary differential equations

Unlike the linear wave equation, where is a constant and the characteristics are therefore parallel straight lines and never intersect, the characteristics of a nonlinear conservation law depend on the solution , and could intersect at a finite time . This behavior is the key observation why the analysis of conservation laws is so challenging. To elaborate on this further, consider the case of in 1, the inviscid Burgers equation. Keeping fixed, the characteristic curves of 5 now take the form

A simple computation shows the source of the problem: The quantity , that is, the solution confined to a fixed characteristic curve, remains constant in , as can be seen from

and therefore . So in fact the equation in 6 reduces to , a constant in , so the characteristics of Burgers equation are again straight lines, but their slopes now depend on . Consequently, if the initial state is such that for , then the two characteristics that initiate at and will meet in finite time, and the solution will become multivalued. At this point we say that the solution has developed a discontinuity or a shock. Much research in the past several decades has been dedicated to making sense of how a solution of a PDE with smooth initial data can develop a discontinuity in finite time, and once it does, how we are supposed to extend the solution beyond this point. This is the starting point of much of the research in conservation laws, including Barbara’s research.

1.2. Student of Peter Lax

One of the first questions we asked Barbara Keyfitz was how she came to study at Courant and how she chose conservation laws as her topic.

“When I think about it, there was a certain amount of serendipity involved. I was an undergraduate at the University of Toronto, where the mathematics honors program was quite advanced relative to US schools. In my final year, one of the courses I took (I am pretty sure we did not have a course in partial differential equations) was an advanced ODE course with F. V. Atkinson. He introduced the method of characteristics for quasilinear scalar equations (see above!), as a way of showing how to use ODEs to solve PDEs. I was entranced. I determined to go to graduate school to study partial differential equations. In 1966, this was not that easy. (Remember, in 1966, PDE was considered ‘applied math’, and the focus was not on analysis.) At the time, Canadian graduate programs were still quite small, and some were unwelcoming to female graduate students. So I applied to a few programs in the US … and then another very important person in my life comes into the story. That was Chandler Davis. He said, ‘Go to NYU and work with Peter Lax’. I sent a letter to NYU and nothing happened.

“After a while Chandler asked me how the application was going and I said I haven’t heard anything. He asked me what I had done. I said I wrote to NYU, and he said ‘No, no, no, that’s not the way it works, you need to write to the Courant Institute’. So I wrote to the Courant Institute, and they said ‘you’re late, but since you were recommended by Chandler Davis, we’ll consider it’. So I applied and was accepted.

“The Courant institute had just moved into a new building at 251 Mercer Street, and I had a desk in a beautiful office with windows on two sides (there were four desks there) and a telephone on my desk. I picked up the telephone; I phoned Peter Lax and said ‘I’d like to take your functional analysis course’, and he said, politely, ‘well actually this course is meant for people who know something’. I enrolled anyway. (Fortunately, the course was not graded.)

“On a sunny afternoon, beautiful California weather, at an AMS Summer School in Berkeley on Global Analysis in 1968, after rather little further interaction between us, Peter suggested a research topic to me. He described the evolution of a scalar conservation law by the method of characteristics (see above, again!), and how the characteristics collide; and he said, I remember the phrase, ‘but the flow must go on’.

“Peter Lax’s research has spanned much of modern PDE; he was directing the research of other students, many of whom have gone on to brilliant careers in a number of fields. How did he pick conservation laws for me?

“He talked about weak solutions, and then he left me on my own. When I couldn’t think of anything to do, he suggested, ‘look at the difference between two solutions of a scalar equation. As long as the solutions are smooth, it’s a simple calculation that the difference remains constant, but once you get weak solutions, I don’t know, I think it might go bad.’ So that was the challenge.”

As the earlier example of the inviscid Burgers equation showed, one expects that solutions to the initial-value problems of conservation laws exist in a classical sense on a maximal time interval. Once a discontinuity is formed, however, we need to consider the solution in a weak sense. For the scalar conservation law , , the weak formulation of the problem takes the form

where is any function belonging to , the space of smooth functions with compact support. While this formulation presents a natural generalization from the classical formulation of the problem, with a natural connection to how conservation laws appear in continuum physics, it does not lead to a well-posed problem: simple examples of initial-value problems exist that exhibit multiple weak solutions. The challenge then became to come up with additional conditions capable of selecting a unique solution. These conditions, often called entropy conditions, have had enormous success for two important classes of conservation laws: scalar conservation laws in (such as the equation in 3 with ), and systems of conservation laws in (such as the one in 2). In fact, as shown in Chapter 5 of Daf10, weak solutions of these conservation laws, under appropriate conditions on the nonlinearities and endowed with an entropy condition, are well-posed: they have unique solutions and are stable in .

Figure 1.

Peter Lax and some of his students, at the conference for the 80th birthdays of Peter Lax and Louis Nirenberg, Toledo, Spain. In the picture, BLK leaning over to talk with Peter Lax (seated). Standing, from left to right: part of David Levermore, Alex Chorin, Stephanos Venakides, and Sebastian Noelle. Visible at lower right corner, Haim Brezis.

Graphic without alt text

“I made the naive (and very restrictive) assumption that the solution was piecewise smooth, and that the only irregularities were shocks. It then becomes a simple calculus problem to compute the difference: as long as you have a convex conservation law, like Burgers equation, and the shocks are going in the right direction, which is to say, the left states are higher than the right states, the norm strictly decreases when shocks are present. If the conservation law isn’t convex, then sometimes it decreases and sometimes it doesn’t, but I wrote down a condition on discontinuities that would give the desired inequality. I brought this back to Lax and he said, ‘that condition looks familiar; I think Oleinik had something to say about that.’ So we looked it up and we found that indeed the Oleinik condition was necessary and sufficient for the norm to decrease (Ole63). So that became my thesis … The rule of thumb seems to be that when you prove a result that your adviser didn’t believe, you then have a dissertation.

“The result itself turned into a paper, Qui71, published about the same time I got my degree. It attracted some interest: my result showed that the entropy solutions to a scalar conservation law form a contractive semigroup in . That year, 1970, was a prime time for applying semigroup theory to PDEs. I had found a semigroup, but I had no idea of what people were talking about when they spoke about the generator of a semigroup, which I had not looked for or found. (Crandall’s paper, a few years later, notes this, and does find the generator Cra72.) At the same time, the PDE community was also discovering that Soviet mathematicians had been studying conservation laws a little bit earlier. The well-known and much cited paper of Kruzkov Kru70, translated into English in 1970, had been published in Russian, I think in 1968. There was also work of Vol’pert Vol86; he had done genuine analysis. That is, he had looked at weak solutions that didn’t have the simplified structure I considered. I was rescued a little bit by Dave Schaeffer who showed that, generically, solutions of a convex conservation law do have a piecewise continuous structure, Sch73.

“I spent a couple of years trying to figure out more estimates, which turned out not to be very useful.”

2. Research in Nonstrictly Hyperbolic Systems of Conservation Laws

Our conversation at this point turned to the realities of life after graduate school; how does one get a job, how to balance personal life with building a career, and what research projects to pursue.

“The real problem was that I didn’t know what to do next. For personal reasons I needed to stay in New York. I got a job at Columbia University, in the engineering school. They were willing to hire me because there had been an institute for flight studies, and, okay, conservation laws govern compressible flow. Right after they hired me, that institute was disbanded. And once again, I was rescued by a tremendous piece of good luck.

“Another Courant graduate at Columbia, John (C. K.) Chu, was in the mechanical engineering department. We talked a bit. He tried to get me interested in Tokamak reactors and fusion, but I didn’t find anything to work on. But a couple of years later Herb Kranzer, a friend of his and another Courant graduate, had a sabbatical and decided to spend it with John. John introduced me to Herb. One of Herb’s students, Dennis Korchinski, in his PhD dissertation Kor77 had looked at an interesting problem that he had not been able to analyze completely. It involved an extremely simple system of conservation laws, where he had considered the Riemann problem, a simple initial value problem with two constant states. For states that were close to together (‘small data’), you could get regular solutions, for other states (‘large data’) he couldn’t analytically find a solution, and when he tried to do it numerically … because what else can you do … solutions seemed to blow up; they became unbounded. Herb described the problem to me, and then he said the magic words, ‘Would you like to work on this problem with me?’

“This is now 1974. As we started working on it, we realized that we didn’t know very much about systems of conservation laws. It was the early days of the field. There was Glimm’s theorem, which had been around by then for almost ten years, which almost nobody understood, ourselves included, and there was not much else. There was a lot of work by Joel Smoller, but Smoller’s results didn’t help with our problem. Smoller had been a visitor at Courant my last year as a student there; he was extremely kind, and somebody I could talk to about conservation laws, but his parting warning was that life beyond studenthood was brutal.

“Conservation law research on systems, up to that point, had required that characteristics be distinct, and Korchinski’s problem didn’t have that feature, so we started to think about nonstrictly hyperbolic conservation laws.

In KK80 Keyfitz and Kranzer introduced the following canonical example of a nonstrictly hyperbolic conservation law:

where . This problem is an example of the system described in 2 where the matrix may not have distinct eigenvalues for a subset of -space, and therefore may not be diagonalizable. A motivating example of is

which appears in certain stress-strain laws of nonlinear elasticity. The Riemann problem for 7 is the solution to this system subject to the initial condition

where and and are constants.

“When we started looking at nonstrict hyperbolicity, we observed that there were two things that could happen, and you can easily see what they are if you think about a matrix with two equal eigenvalues: is it diagonalizable or is it not? We started out by looking at a planar elastic string, modeled by a pair of second order equations, or equivalently, a system of four first-order equations (see AMM88) … we knew we couldn’t handle the four-dimensional problem, so we created two first order equations, instead of second order equations, and when we did that, we got a matrix that was not diagonalizable. (This was, in fact, the result of a modeling error on our part.)

“Reviewing what was known about large data Riemann problems for strictly hyperbolic systems, we discovered a condition which we called opposite variation, somewhat more general than conditions that Smoller and Johnson SJ69 had advanced. If systems satisfied this condition, we could also solve Riemann problems for any pair of nonstrictly hyperbolic equations where the behavior of eigenvectors near characteristic crossings was of a certain type. The solutions weren’t as well-behaved as for strictly hyperbolic Riemann problems, but we could describe the sense in which they satisfied continuous dependence on the data.

“Eventually, we noticed that in fact with the elastic string problem, for the full four-dimensional system, the matrix is diagonalizable where the eigenvalues become equal in pairs. We solved the Riemann problem for that system, as well as our ersatz nondiagonalizable example. The paper appeared in the Archive for Rational Mechanics and Analysis, KK80.”

The paper KK80 has received over four hundred citations. For several years following its publication in 1980, this paper set the stage for much of the research in nonstrictly hyperbolic conservation laws. The paper SS86 is a notable body of research on these equations where the authors present a classification of the behavior of Riemann problems for systems motivated by simulation of oil recovery problems. This particular application, sometimes referred to as the ‘reservoir problem’, was a focus of research by James Glimm and his associates during the decade of the 80’s, and introduced a new set of theoretical and computational challenges, culminating in formulation of the “front tracking” approach for shock waves, first for numerical simulation (see GGL98 for details) and eventually, triumphantly, to Alberto Bressan’s celebrated results providing the first proofs of the well-posedness of the Cauchy problem for general, strictly hyperbolic systems Bre00.

“The intersection of our work with Glimm’s was the nondiagonalizable problem, which we’d solved by mistake but published anyway along with the result relevant to elastic strings. It was one of Glimm’s simple reservoir models. Eli Isaacson and Blake Temple, at that time two of Glimm’s postdocs, had solved it, and planned to give a talk about it in an upcoming Joint Math Meeting. I saw their abstract and wrote to them, ‘you might want to know that we have a published paper that solves this problem’. They withdrew the paper they had submitted for publication (I still have the preprint), and graciously talked about our result instead at the JMM. We had sent our published paper to Glimm, but of course he hadn’t looked at it, and all he said to them was, ‘you ought to read the literature’.

“I still feel badly about it. When this happened I had a tenure-track job and they were postdocs; they were junior and it wouldn’t have hurt me to be kinder. Why did I have to treat them as though they were stealing my result?

“But then another direction opened up. At the time it seemed totally screwball, but it sustained Herb’s and my research for another 15 years or so. We started thinking about what happens if instead of opposite variation, when these characteristics cross, you have the same variation in the two families, and here I give credit to the mathematician Michelle Schatzman, a friend. I was trying to explain this problem to her, that you could have the characteristics crossing in the shape of an , in opposite orientation, or you could have the characteristics crossing still as an , but a distorted , with the lines going from lower left to the upper right, from both sides. Michelle took a marker to my and simply separated the lines …. So now you have a system that is strictly hyperbolic, as the two characteristics are separated. It didn’t take us long to write down an example, to try to solve the Riemann problem, and to find that indeed we couldn’t solve it, even for the strictly hyperbolic equation. When we resorted to numerical methods, the solution blew up, just as Dennis Korchinski’s example had: solutions became unbounded.”

In 2011, Barbara published the paper Key11 which, as its title indicates, gave a retrospective on her contribution to the concept of singular shocks and the inspiration she received from her conversations with Michelle Schatzman.

Singular shocks were first discovered by Keyfitz and Kranzer in their attempt to finding solutions to Riemann problems in certain strictly hyperbolic problems based on a peculiar reformulation of the governing equations for isothermal, isentropic gas dynamics. Despite the fact the system is genuinely nonlinear, they discovered a large region of state space where the Riemann problem cannot be solved using shocks and rarefactions. They produced approximate unbounded solutions which do not satisfy the equation in the classical weak-solution sense. This discovery has led to a rich, new area of conservation laws with deep connections to the Dafermos-DiPerna regularization theory, and to geometric singular perturbation theory (the manifold theory for singular dynamical systems). The paper by Tsikkou Tsi16 gives an excellent review of this field, in addition to describing how singular shocks appear in chromatography. See also Sev07 for a systematic exposition of what was known about systems with singular shock solutions fifteen years ago.

Barbara continued to share her recollections of her mathematical journey.

“After my (deservedly) failing to get tenure at two fine institutions—Columbia and Princeton—but by then happily married and a mother, I and my mathematician husband Marty Golubitsky found jobs at Arizona State University. A brief foray on my part into applications of singularity theory (Marty’s field) was a springboard to jobs for both of us at the University of Houston, and ultimately at Ohio State.

“In 1992, at Houston, I got a State of Texas grant to hire a postdoc, Sunčica (Sunny) Čanić. A colleague of mine, David Wagner, a former student of Smoller’s, had noted an example of a two-dimensional Riemann problem, described in a paper of Brio and Hunter, BH92. This was another lucky chance where something came along at exactly the right moment. At the time very little was known about systems of conservation laws in two space dimensions. David suggested that we ‘bite off a little bit of multi-dimensional systems’. Sunny and I started looking at these 2D Riemann problems and found that we needed to learn about elliptic equations, something that hyperbolic people usually avoid like the plague. And, serendipitously, Sunny moved to Iowa State University, and met Gary Lieberman, who was the best person in the world to help us, and did, ČanićKL00.”

Barbara’s collaboration with Sunny Čanić was another productive period in her mathematical career. In more than a dozen papers, Barbara and Sunny attacked a number of hard problems in conservation laws: The work on degenerate elliptic PDEs appeared in ČanićKL00. The paper proves existence of a solution to a free boundary problem for the transonic small-disturbance equation—the free boundary is the position of a transonic shock dividing two regions of smooth flow; the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. Joined later by another postdoc, Eun Heui Kim, they analyzed a prototype for Mach stems ČanićKK05. Gui-Qiang Chen and Misha Feldman and their students and associates have made great strides with this approach, documented in their monograph, CF18.

One of the remarkable aspects of Barbara’s career is her service and mentoring record, the strength of which parallels her research record. In the remainder of this article, rather than bringing readers up to date on the field of conservation laws, now so much more extensive than in 1970, we report on our conversation on how she prepared herself for the high-level community service positions she has held.

Barbara attributed much of her success to her ability to listen and to engage, abilities she acquired and mastered early in her career. The statement ‘one thing led to another’ was a common refrain as she described how opportunities presented themselves, paralleling the serendipity present in so many of her research collaborations.

3. Community Service and Mentoring

During her long career, Barbara has been on the leadership of many national and international organizations, including serving as the vice president for programs at SIAM and the treasurer of ICIAM; but serving as the president of the Association of Women in Mathematics (see Key22), and as the president of ICIAM, particularly stand out. Her tenure as the director of the Fields Institute, and the many initiatives she was able to launch in that position, ended up being the main focus of our conversation.

Figure 2.

Akershus Castle, Abel Prize ceremony 2014 (the winner was Yakov Sinai). This was a banner year for women. From left to right, Maria J. Esteban (member of the Abel Prize Committee), me (president of ICIAM), Ragni Piene (chair of the Abel Committee), Ingrid Daubechies (president of IMU), and Marta Sanz-Sole (president of the European Mathematical Society). [Not at the ceremony, but rounding out the theme, Ruth Williams had just completed a term as president of IMS.]

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“In terms of highlights, Fields was the highlight. Being the director of Fields was a job that I think I enjoyed more than anything else in my career. For the first six months I just walked around the campus of the University of Toronto smiling—people would stop me to ask, ‘What are you smiling about?’”

During the four and half years of her directorship at Fields, Barbara was able to introduce several programs, many of them intended to improve the participation of women in the research programs at the Fields Institute, as well as in programs at other partner institutions in Ontario.

“I wanted to make Fields more accommodating to women. For example, there were distinguished lecture series that had never had women lecturers. There was no kind of affirmative action. I asked our Board of Directors for a mandate, and they said, ‘Go ahead’. Yes!

“Another function at Fields was organizing workshops at other universities, as well as thematic programs, which are like the ones in the US institutes. A large fraction of the Fields budget was spent on short-term events, typically one-week workshops that took place at different partner institutions in Canada. We would solicit proposals, and we would have to evaluate them. On one occasion my deputy director wrote to a proposer, ‘I see your list of invited speakers doesn’t include a single woman. Could you please explain your thinking to me?’”

At the end of our conversation, we asked Barbara about her impressions related to dedicating workshops or awards solely to women.

“I have mixed feelings about that. Some women mathematicians take exception to being recognized in part because they’re women. Instead of seeing it as an opportunity, a foot in the door, they don’t want to accept an honor that in their interpretation is sullied by this. And they see attending a workshop for women, from which important male scholars are excluded, as an inefficient use of their time. … On the other hand BIRS, the Banff math institute, now has one workshop a year for women. These appear to have been extremely successful. I think it’s because they bring women together physically. There’s a story here.

“For a while, BIRS had been rather unfriendly about women’s participation; for example, they refused to allow people to bring children—no daycare, no accommodations. If a woman didn’t want to come under those conditions, then she didn’t need to come. Since many places had started making daycare arrangements, I felt BIRS could have at least thought about it, or at least not complained about my asking. So out of the blue the director said, ‘how about this year we give you one week for women, you can do whatever you want with it. It is yours.’ It happened it was the week of Rosh Hashanah, which is possibly why nobody wanted it. I was willing to give up Rosh Hashanah for one year and so we ran a workshop. It wasn’t primarily a scientific workshop, and had the unimaginative name ‘Women in Mathematics’. The idea was to discuss ‘what can women do to get ahead.’ We had participants from Canada, US and Mexico, because BIRS is run by all three countries. Our workshop report was titled ‘Women in the Academic Ranks: A Call to Action’.” (It can be found on the BIRS workshop report page at”

4. Afterword

We started planning this project in June of 2023. Barbara has been enormously generous with her time over the past 4 months: In addition to two sessions on July 24 and Aug 1 where we had the opportunity to meet remotely, we have been in frequent contact with Barbara with our questions. This article is built on extracts from those many, many hours of communications. We hope it gives a glimpse of the impact of Barbara’s career.

The article does not do justice to two aspects of our interactions: We haven’t been able to adequately depict Barbara’s sharp and engaging sense of humor. Our sessions were always enlivened by her infectious laughter, which we felt we couldn’t get enough of. Second, while neither of us has been Barbara’s student, it was clear during the past few months that she is a master teacher. Our conversation sessions were filled with Barbara describing details of her work, using examples and analogies that we could embrace, which is reminiscent of the important role canonical and reduced models have played throughout her research career … Barbara’s lectures are now available on social media, which we highly recommend.

For the past several decades Barbara Keyfitz has participated in, has led and organized many national meetings and PDE conferences. Her approach to research in applied mathematics has led to posing new questions and pursuing new directions. Her lifelong desire to work to improve the environment of research for women has already had an enduring effect. We have been so fortunate that she has always been approachable and ready to share and to engage.


Stuart S. Antman and Reza Malek-Madani, Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media, Quart. Appl. Math. 46 (1988), no. 1, 77–93, DOI 10.1090/qam/934683. MR934683,
Show rawAMSref \bib{AntmanMalek-Madani}{article}{ label={AMM88}, author={Antman, Stuart S.}, author={Malek-Madani, Reza}, title={Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media}, journal={Quart. Appl. Math.}, volume={46}, date={1988}, number={1}, pages={77--93}, issn={0033-569X}, review={\MR {934683}}, doi={10.1090/qam/934683}, }
M. Brio and J. K. Hunter, Mach reflection for the two-dimensional Burgers equation, Phys. D 60 (1992), no. 1-4, 194–207, DOI 10.1016/0167-2789(92)90236-G. Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). MR1195600,
Show rawAMSref \bib{BH}{article}{ label={BH92}, author={Brio, M.}, author={Hunter, J. K.}, title={Mach reflection for the two-dimensional Burgers equation}, note={Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991)}, journal={Phys. D}, volume={60}, date={1992}, number={1-4}, pages={194--207}, issn={0167-2789}, review={\MR {1195600}}, doi={10.1016/0167-2789(92)90236-G}, }
Alberto Bressan, Hyperbolic systems of conservation laws: The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. MR1816648,
Show rawAMSref \bib{bressan}{book}{ label={Bre00}, author={Bressan, Alberto}, title={Hyperbolic systems of conservation laws}, series={Oxford Lecture Series in Mathematics and its Applications}, volume={20}, subtitle={The one-dimensional Cauchy problem}, publisher={Oxford University Press, Oxford}, date={2000}, pages={xii+250}, isbn={0-19-850700-3}, review={\MR {1816648}}, }
Sunčica Čanić, Barbara Lee Keyfitz, and Eun Heui Kim, Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, SIAM J. Math. Anal. 37 (2006), no. 6, 1947–1977, DOI 10.1137/S003614100342989X. MR2213401,
Show rawAMSref \bib{CKK}{article}{ label={{\v {C}ani\'{c}}KK05}, author={\v {C}ani\'{c}, Sun\v {c}ica}, author={Keyfitz, Barbara Lee}, author={Kim, Eun Heui}, title={Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks}, journal={SIAM J. Math. Anal.}, volume={37}, date={2006}, number={6}, pages={1947--1977}, issn={0036-1410}, review={\MR {2213401}}, doi={10.1137/S003614100342989X}, }
Sunčica Čanić, Barbara Lee Keyfitz, and Gary M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), no. 4, 484–511, DOI 10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.3.CO;2-B. MR1733695,
Show rawAMSref \bib{Canickeyfitz1}{article}{ label={{\v {C}ani\'{c}}KL00}, author={\v {C}ani\'{c}, Sun\v {c}ica}, author={Keyfitz, Barbara Lee}, author={Lieberman, Gary M.}, title={A proof of existence of perturbed steady transonic shocks via a free boundary problem}, journal={Comm. Pure Appl. Math.}, volume={53}, date={2000}, number={4}, pages={484--511}, issn={0010-3640}, review={\MR {1733695}}, doi={10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.3.CO;2-B}, }
Gui-Qiang G. Chen and Mikhail Feldman, The mathematics of shock reflection-diffraction and von Neumann’s conjectures, Annals of Mathematics Studies, vol. 197, Princeton University Press, Princeton, NJ, 2018. MR3791458,
Show rawAMSref \bib{chenfeldman}{book}{ label={CF18}, author={Chen, Gui-Qiang G.}, author={Feldman, Mikhail}, title={The mathematics of shock reflection-diffraction and von Neumann's conjectures}, series={Annals of Mathematics Studies}, volume={197}, publisher={Princeton University Press, Princeton, NJ}, date={2018}, pages={xiv+814}, isbn={978-0-691-16055-9}, review={\MR {3791458}}, }
Michael G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132, DOI 10.1007/BF02764657. MR316925,
Show rawAMSref \bib{Cran}{article}{ label={Cra72}, author={Crandall, Michael G.}, title={The semigroup approach to first order quasilinear equations in several space variables}, journal={Israel J. Math.}, volume={12}, date={1972}, pages={108--132}, issn={0021-2172}, review={\MR {316925}}, doi={10.1007/BF02764657}, }
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010, DOI 10.1007/978-3-642-04048-1. MR2574377,
Show rawAMSref \bib{Dafermos1}{book}{ label={Daf10}, author={Dafermos, Constantine M.}, title={Hyperbolic conservation laws in continuum physics}, series={Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, volume={325}, edition={3}, publisher={Springer-Verlag, Berlin}, date={2010}, pages={xxxvi+708}, isbn={978-3-642-04047-4}, review={\MR {2574377}}, doi={10.1007/978-3-642-04048-1}, }
James Glimm, John W. Grove, Xiao Lin Li, Keh-Ming Shyue, Yanni Zeng, and Qiang Zhang, Three-dimensional front tracking, SIAM J. Sci. Comput. 19 (1998), no. 3, 703–727, DOI 10.1137/S1064827595293600. MR1616658,
Show rawAMSref \bib{GlimmFrontTracking}{article}{ label={GGL{\etalchar {+}}98}, author={Glimm, James}, author={Grove, John W.}, author={Li, Xiao Lin}, author={Shyue, Keh-Ming}, author={Zeng, Yanni}, author={Zhang, Qiang}, title={Three-dimensional front tracking}, journal={SIAM J. Sci. Comput.}, volume={19}, date={1998}, number={3}, pages={703--727}, issn={1064-8275}, review={\MR {1616658}}, doi={10.1137/S1064827595293600}, }
Barbara Lee Keyfitz, Singular shocks: retrospective and prospective, Confluentes Math. 3 (2011), no. 3, 445–470, DOI 10.1142/S1793744211000424. MR2847239,
Show rawAMSref \bib{KeytfitzSingularShocks}{article}{ label={Key11}, author={Keyfitz, Barbara Lee}, title={Singular shocks: retrospective and prospective}, journal={Confluentes Math.}, volume={3}, date={2011}, number={3}, pages={445--470}, issn={1793-7442}, review={\MR {2847239}}, doi={10.1142/S1793744211000424}, }
Barbara Lee Keyfitz, Interesting times: Reflections of AWM’s seventeenth president, in Fifty Years of Women in Mathematics: Reminiscences, History, and Visions for the Future of AWM, edited by Janet L. Beery, Sarah J. Greenwald, and Cathy Kessel, vol. 28, Springer, 2022, 721–737.
Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219–241, DOI 10.1007/BF00281590. MR549642,
Show rawAMSref \bib{KeyfitzKranzer}{article}{ label={KK80}, author={Keyfitz, Barbara L.}, author={Kranzer, Herbert C.}, title={A system of nonstrictly hyperbolic conservation laws arising in elasticity theory}, journal={Arch. Rational Mech. Anal.}, volume={72}, date={1979/80}, number={3}, pages={219--241}, issn={0003-9527}, review={\MR {549642}}, doi={10.1007/BF00281590}, }
Dennis James Korchinski, SOLUTION OF A RIEMANN PROBLEM FOR A 2 X 2 SYSTEM OF CONSERVATION LAWS POSSESSING NO CLASSICAL WEAK SOLUTION, ProQuest LLC, Ann Arbor, MI, 1977. Thesis (PhD)–Adelphi University. MR2626928,
Show rawAMSref \bib{Korchinski}{book}{ label={Kor77}, author={Korchinski, Dennis James}, title={SOLUTION OF A RIEMANN PROBLEM FOR A 2 X 2 SYSTEM OF CONSERVATION LAWS POSSESSING NO CLASSICAL WEAK SOLUTION}, note={Thesis (PhD)--Adelphi University}, publisher={ProQuest LLC, Ann Arbor, MI}, date={1977}, pages={83}, review={\MR {2626928}}, }
S. N. Kružkov, First order quasilinear equations with several independent variables (Russian), Mat. Sb. (N.S.) 81(123) (1970), 228–255. MR267257,
Show rawAMSref \bib{Kr}{article}{ label={{Kru}70}, author={Kru\v {z}kov, S. N.}, title={First order quasilinear equations with several independent variables}, language={Russian}, journal={Mat. Sb. (N.S.)}, volume={81(123)}, date={1970}, pages={228--255}, issn={0368-8666}, review={\MR {267257}}, }
O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172, DOI 10.1090/trans2/026/05. MR151737,
Show rawAMSref \bib{Oleinik1963}{article}{ label={Ole63}, author={Ole\u {\i }nik, O. A.}, title={Discontinuous solutions of non-linear differential equations}, journal={Amer. Math. Soc. Transl. (2)}, volume={26}, date={1963}, pages={95--172}, review={\MR {151737}}, doi={10.1090/trans2/026/05}, }
Barbara Keyfitz Quinn, Solutions with shocks: An example of an -contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132, DOI 10.1002/cpa.3160240203. MR271545,
Show rawAMSref \bib{Keyfitz1}{article}{ label={Qui71}, author={Quinn, Barbara Keyfitz}, title={Solutions with shocks: An example of an $L_{1}$-contractive semigroup}, journal={Comm. Pure Appl. Math.}, volume={24}, date={1971}, pages={125--132}, issn={0010-3640}, review={\MR {271545}}, doi={10.1002/cpa.3160240203}, }
David G. Schaeffer, A regularity theorem for conservation laws, Advances in Math. 11 (1973), 368–386, DOI 10.1016/0001-8708(73)90018-2. MR326178,
Show rawAMSref \bib{sch}{article}{ label={Sch73}, author={Schaeffer, David G.}, title={A regularity theorem for conservation laws}, journal={Advances in Math.}, volume={11}, date={1973}, pages={368--386}, issn={0001-8708}, review={\MR {326178}}, doi={10.1016/0001-8708(73)90018-2}, }
Michael Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc. 190 (2007), no. 889, viii+163, DOI 10.1090/memo/0889. MR2355635,
Show rawAMSref \bib{Sever}{article}{ label={Sev07}, author={Sever, Michael}, title={Distribution solutions of nonlinear systems of conservation laws}, journal={Mem. Amer. Math. Soc.}, volume={190}, date={2007}, number={889}, pages={viii+163}, issn={0065-9266}, review={\MR {2355635}}, doi={10.1090/memo/0889}, }
J. A. Smoller and J. L. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 32 (1969), 169–189, DOI 10.1007/BF00247508. MR236527,
Show rawAMSref \bib{SmollerJohnson}{article}{ label={SJ69}, author={Smoller, J. A.}, author={Johnson, J. L.}, title={Global solutions for an extended class of hyperbolic systems of conservation laws}, journal={Arch. Rational Mech. Anal.}, volume={32}, date={1969}, pages={169--189}, issn={0003-9527}, review={\MR {236527}}, doi={10.1007/BF00247508}, }
David G. Schaeffer and Michael Shearer, The classification of systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40 (1987), no. 2, 141–178, DOI 10.1002/cpa.3160400202. MR872382,
Show rawAMSref \bib{SchaefferShearer}{article}{ label={SS86}, author={Schaeffer, David G.}, author={Shearer, Michael}, title={The classification of $2\times 2$ systems of nonstrictly hyperbolic conservation laws, with application to oil recovery}, journal={Comm. Pure Appl. Math.}, volume={40}, date={1987}, number={2}, pages={141--178}, issn={0010-3640}, review={\MR {872382}}, doi={10.1002/cpa.3160400202}, }
Charis Tsikkou, Singular shocks in a chromatography model, J. Math. Anal. Appl. 439 (2016), no. 2, 766–797, DOI 10.1016/j.jmaa.2016.03.001. MR3475951,
Show rawAMSref \bib{Tsikkou}{article}{ label={Tsi16}, author={Tsikkou, Charis}, title={Singular shocks in a chromatography model}, journal={J. Math. Anal. Appl.}, volume={439}, date={2016}, number={2}, pages={766--797}, issn={0022-247X}, review={\MR {3475951}}, doi={10.1016/j.jmaa.2016.03.001}, }
A. I. Vol’pert, The spaces BV and quasilinear equations, Mathematics of the USSR – Sbornik 2 (1986), 87–158.


Figure 1 is courtesy of Maria Venikides.

Figure 2 is courtesy of Marty Golubistky.

Photo of Fariba Fahroo is courtesy of Hennie Farrow.

Photo of Reza Malek-Madani is courtesy of Photo Lab: US Naval Academy.