One Language, Many Phenomena: Partial Differential Equations Across Science

by Leonardo Abbrescia, Gong Chen, Ronghua Pan, Andrzej Święch, Tobias Ried, and Chongchun Zeng

From collective decision-making and quantum materials to atmospheric instability, shock waves, and solitary waves, researchers use partial differential equations to uncover the mechanisms behind stability, singularity, and change.

Why does a fluid at rest suddenly begin to overturn? How can a smooth sound wave sharpen into a shock? Why can some nonlinear waves travel long distances without changing shape, while others slowly radiate away? And how can the decisions of millions of interacting individuals be captured mathematically? These questions come from very different parts of science, but they share a common language: partial differential equations, or PDEs.

PDEs appear throughout physics, probability, optimization, geometry, materials science, and engineering. That breadth is reflected in the work of PDE researchers in Georgia Tech's School of Mathematics, whose interests range from optimal control and transport to fluid dynamics and nonlinear waves. What connects this research is not one particular equation or technique. The same fundamental questions appear again and again: What remains stable or becomes unstable? How do local interactions produce large-scale behavior? What happens when smoothness breaks down? And what is the long-time fate of a nonlinear system?

When Classical Solutions are Not Enough

Many nonlinear PDEs can develop solutions that are not smooth enough for the usual definition of a derivative to apply. Andrzej Święch studies equations where this difficulty is central. An important tool in his work is the theory of viscosity solutions, which makes it possible to define and analyze solutions even when classical differentiability is unavailable. Instead of differentiating a nonsmooth solution directly, the theory compares it locally with smooth test functions.

Viscosity solutions have become particularly important in optimal control, where one asks how a system should be steered to minimize a cost or maximize a reward. Święch helped develop the theory of Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces associated with optimal control of stochastic PDEs. He also contributed to the theory of L^p-viscosity solutions for equations with merely measurable terms, establishing maximum principles and regularity estimates that extend the theory to much rougher settings, and has worked on nonlocal equations in finite and infinite dimensions.

More recently, his research has focused on Mean Field Games and Mean Field Control, which describe systems containing very large numbers of interacting decision-makers, such as investors, pedestrians, energy consumers, or agents in learning systems. Rather than tracking every agent separately, mean field theory studies the problem for a representative agent, together with the evolution of the entire population distribution based on optimal strategies.

With Wilfrid Gangbo, Święch gave one of the early existence results for the master equation of first-order Mean Field Games. With other collaborators, he established existence and uniqueness of solutions for broad classes of mean field game systems in Hilbert spaces and convergence from finite-particle control problems to limiting equations on spaces of probability measures. Earlier work with Gangbo on multimarginal optimal transport provides a direct bridge to the next theme.

Finding Structure by Zooming In

Optimal transport asks for the most efficient way to transform one distribution of mass into another. One can picture reshaping a pile of sand while paying for every grain moved and every distance traveled. Today this idea connects PDEs with probability, geometry, materials science, statistics, and machine learning. But the optimal transport rule need not be smooth: neighboring particles can sometimes be sent in very different directions, producing singularities in the transport map.

Tobias Ried studies where such spatial singularities can occur. Together with collaborators, Ried uses a scale-by-scale approach: a complicated transport map is compared locally with a simpler harmonic approximation, and then one zooms in. If the approximation improves at smaller scales, the map is smooth there; where it fails, singular behavior may remain. With Gero Friesecke, this led to the first partial regularity result for transport governed by repulsive Coulomb forces, motivated by strongly correlated electrons. Related work studies Wasserstein barycenters - nonlinear averages of probability distributions.

Ried's work also connects with spectral theory, where eigenvalues of Schrödinger operators describe quantum bound states. Ried and collaborators developed a harmonic-analysis method that improved a bound on quantum bound states that had stood for more than forty years, with later work sharpening related estimates. In materials, he has studied magnetic-domain branching and how microscopic disorder influences large-scale patterns.

Across these different problems, the same philosophy appears: complex global structures can sometimes be understood by identifying mechanisms that repeat from scale to scale.

The same search for structure extends from space to evolution in time. In fluid dynamics, for example, one asks how an apparently quiet state becomes unstable - or even singular.

When a Quiet Fluid Becomes Unstable

Warm air rises and dense material sinks, but determining exactly how a fluid becomes unstable can be surprisingly difficult. One of Ronghua Pan's recent research directions is convective instability, the study of how fluids under gravity begin to move vertically. A classical example is the Rayleigh-Taylor instability: if a heavier fluid lies above a lighter one, dense fluid moves downward while lighter fluid rises, eventually producing mixing.

Real fluids and atmospheres, however, also involve temperature and heat diffusion. Once these effects are included, the simple rule "heavy fluid above light fluid is unstable" is no longer enough. This leads to instability criteria involving entropy. In recent work, Pan and Qin gave a nonlinear mathematical justification of such a criterion in a regime without heat diffusion. When diffusion is included, the problem becomes harder; Pan and collaborators have shown that when heat diffusion is sufficiently small, a strengthened entropy condition guarantees instability.

The broader question is one that appears throughout nonlinear PDE: Which mechanism determines the transition from stability to instability? Sometimes instability produces growing motion while the solution remains smooth. In other systems, it continues until smoothness itself breaks down.

When Smooth Waves Become Shocks

The equations of compressible fluids can produce shock waves, rarefactions, and implosions. One of the remarkable facts of fluid dynamics is that a shock - a sudden jump in pressure and density - can emerge from completely smooth initial conditions. In the nineteenth century, Bernhard Riemann showed in one dimension that sound waves can compress until their gradients become infinite. Understanding how a smooth solution reaches this singular state, and how a shock subsequently forms and propagates, is known as the shock development problem.

In multiple spatial dimensions, the problem becomes much more difficult. A major breakthrough came from Demetrios Christodoulou, who gave a geometric description of shock formation for fluids with zero vorticity and constant entropy. Leonardo Abbrescia's work also addresses the more general situation, where vorticity and changing entropy are present. These quantities propagate differently from sound and interact strongly with it near a forming shock.

Abbrescia and Jared Speck developed a framework that controls this nonlinear coupling and tracks the interaction of sound, vorticity, and entropy up to shock formation, substantially broadening the mathematical theory of multidimensional shocks. An important challenge remains: after the first singularity forms, how does a genuine discontinuous shock emerge and continue to propagate? In full generality, this is still open.

Fluid equations therefore illustrate one possible fate of nonlinear waves: gradients grow, energy concentrates, and smoothness fails. Dispersive equations often display the opposite tendency - waves spread apart. Yet even there, nonlinear effects can create remarkably persistent structures.

Waves that Refuse to Disappear

In dispersive wave equations, different frequencies travel at different speeds, so a generic wave tends to spread and weaken in amplitude. Yet many nonlinear dispersive equations also support solitons - localized waves that can travel while largely retaining their shape. One of Gong Chen's research themes is the interaction between dispersion and these coherent structures.

His research combines harmonic analysis, spectral theory, and dynamical systems to investigate whether solitons are stable, how they interact, and what remains after a nonlinear wave evolves for a very long time. A central issue is soliton resolution: the conjecture predicts that a general solution eventually separates into a finite collection of solitons together with radiation that disperses toward infinity.

An important model is the sine-Gordon equation, which has applications ranging from crystal dislocations and Josephson junctions to field theory. In work with Liu and Lu, Chen established soliton resolution for the sine-Gordon equation using its special integrable structure. Later, with Jonas Lührmann, he developed a different approach to studying kink stability based on Fourier analysis and spectral theory, designed to apply beyond integrable models.

A particularly interesting feature of sine-Gordon is the existence of breathers - localized waves that oscillate periodically rather than simply traveling away. Their presence shows that nonlinear waves can store energy in unexpectedly persistent forms. Current work with students and postdoctoral researchers extends these ideas to systems with internal oscillations and to configurations involving multiple solitons and kinks. This leads to a natural question: in more general nonlinear wave systems, are there localized nonlinear waves that can oscillate forever?

When Almost Eternal is Not Eternal

Exact breathers exist in special integrable equations such as sine-Gordon and in some nonlinear Schrödinger models. In more general nonlinear wave equations, however, physicists and mathematicians expect them to be exceptional. The reason is radiation: a localized oscillation can interact weakly with extended wave modes and gradually lose energy. The leakage may be so small that ordinary perturbation theory cannot detect it, yet over very long times an exponentially small effect can determine whether an exact breather exists at all.

Chongchun Zeng's recent work studies precisely these beyond-all-orders effects. The problem can be reformulated in a spatial-dynamics framework. An exact breather corresponds to a so-called homoclinic orbit, an intersection between stable and unstable manifolds in an infinite-dimensional phase space. Even an exponentially small splitting between those manifolds can destroy the breather.

The analysis identifies a Stokes constant that detects this splitting. For generic nonlinearities, it is nonzero, ruling out small exact breathers except possibly in special resonant regimes. The result helps explain why many numerically observed breather-like structures may persist for extremely long times without being truly permanent. Similar exponentially small mechanisms also appear in quantum tunneling, celestial mechanics, and other systems where tiny effects ultimately determine the outcome.

Different Equations, Recurring Questions

There is no single equation that defines this PDE research group. Its members study collective decision-making, optimal transport, quantum bound states, fluid instability, shock formation, solitary waves, and extremely long-lived oscillations. The works above also illustrate the range of mathematical tools involved, from viscosity solutions and harmonic approximation to geometric methods, spectral theory, Fourier analysis, and dynamical systems.

What connects the group is not one equation and not one technique, but a recurring set of mathematical themes. Seen this way, the breadth of PDE is not a collection of unrelated applications: structure, stability, scale, singularity, and long-time dynamics recur across very different parts of science. From populations and probability distributions to electrons, fluids, shocks, and solitary waves, partial differential equations provide a common framework for understanding how complex systems change.