The nature of Partial Differential Equations (PDEs) makes them not only interesting mathematical objects in their own right, but also causes them to play important roles in many branches of applied mathematics, science and engineering, such as continuum mechanics, electromagnetism, quantum mechanics, relativity theory, mathematical biology, control theory, finance and economics.
When Steve Dunbar, Mohammad Rammaha and Richard Rebarber were hired in the 1980s and joined senior applied mathematics faculty members David Logan and Tom Shores, it could be said the “modern era” of PDEs at UNL had begun. In 1989, the group expanded again with the additions of Steve Cohn and Glenn Ledder. Their interests reflect the eclectic character of the field. Dunbar, Logan and Ledder were primarily applied mathematicians, Shores a numerical analyst, Rammaha and Cohn analysts, and Rebarber a control theorist.
In 1994, Cohn gave the plenary address at the First International Conference on Inverse and Improperly-Posed Problems in Izmit, Turkey. In 1997, Jennifer Mueller, a student of Shores, became the first UNL mathematics student to win an NSF postdoctoral fellowship. Also in 1997, the second edition of David Logan’s Applied Mathematics was issued; this book, now in its fourth edition, is one of eight that Logan has authored or co-authored during his career. To this day, his monograph Applied Partial Differential Equations is successfully used as the text for the introductory undergraduate PDE course at UNL.
By the late ’90s, the diversity of their interests tended to pull the group apart and the label “PDE group” had become a misnomer. Dunbar, Cohn, Logan, Shores and Ledder pursued their interest in mathematical modeling. Rebarber continued working in control theory for PDEs and subsequently branched out into mathematical biology, though he still occasionally produces papers on control of distributed-parameter systems.
Rammaha kept the PDE seminar going and advised the graduate students who were interested in the analysis and differential equations. Rammaha’s work delves into qualitative analysis of solutions of so-called “hyperbolic” PDEs, which describe wave propagation phenomena, in particular acoustic waves, as well as some aspects of elastodynamics and thin plate theory. Rammaha has supervised six Ph.D. students who are now on either tenured or tenure-track positions at various universities, except for the two most recent graduates who presently hold postdoctoral fellowships.
In 2000, most of the applied group migrated to mathematical biology and the department hired George Avalos. Avalos specializes in the mathematical control theory of partial differential equations and some aspects of numerical analysis. He is particularly interested in coupled systems – those comprised of two or more disparate dynamics, such as thermo-elastic phenomena, or interaction of thin chamber walls with acoustic pressure or fluid flow within. Avalos has since supervised three Ph.D. theses, and his work has been supported by seven NSF grants since 1997.
The arrival of Avalos more clearly defined a group who were interested primarily in PDEs and analysis. In 2004, Petronela Radu came to UNL as a postdoc, and then became an assistant professor in fall of 2005. In her work, Radu resolved some open problems concerning solvability of nonlinear wave equations (see more on Radu on page 11). The hiring of Radu brought the additional benefit of her husband, Mikil Foss, to the department. Foss provided the much-needed expertise to the PDE group in the hitherto unrepresented areas of calculus of variations and elliptic PDEs, which address many important questions in continuum mechanics. A few years ago, Foss and Radu became interested in the new and rapidly developing theory of peridynamics, which deals with non-local models that account for “long-distance” interactions between particles within a body. Among its many applications, nonlocal theory has been successfully used to mathematically describe dynamic fractures. Now, several of their graduate students are working on various questions pertaining to nonlocal models in diffusion and wave propagation.
A few years after the arrival of Radu and Foss, Daniel Toundykov repeated Radu’s feat: he came as a postdoc and stayed as an assistant professor. Daniel’s specialty is in the control of PDEs with emphasis on nonlinear models describing mechanical vibrations and acoustics. His recent work also provides analytic results concerning electromagnetic fields and certain models of hydrodynamics. Daniel’s research has been supported by two grants from the NSF (one of them joint with Avalos) and was recognized by the UNL Edgerton Junior Faculty Award.
With Shores’ retirement in 2010, the department lost its sole full-time numerical expert. In 2014, after an extensive search directed by Avalos, the department hired Adam Larios (see page 12). As a graduate student, Larios worked as a researcher at Los Alamos National Lab, and collaborated with its Climate Ocean Sea-Ice Modeling group and the Computer, Computational, and Statistical Sciences group. Larios investigates problems in fluid dynamics, turbulence, geophysics, phase-field dynamics, and fluid-structure interaction.
Presently the PDE group exchanges ideas and explores new topics in the PDE & Applied Analysis seminar led by Cohn. Larios kicked off the PDE seminar this fall with a series of lectures on the Navier-Stokes equations, which are the subject of a “Million Dollar Millennium Prize” from the Clay Mathematics Institute.