You have probably heard the expressions sonic boom and breaking the sound barrier. But have you ever wondered how these phenomena are explained mathematically?
A sonic boom is a thunder-like shock wave produced when an aircraft travels faster than the speed of sound, whereas breaking the sound barrier refers to the challenge of exceeding the speed of sound while minimizing or avoiding the generation of such shock waves. From a mathematical perspective, the airflow around an aircraft is described by the Navier-Stokes equations, which govern the motion of viscous, compressible fluids. As an aircraft moves through the air, it generates pressure waves. As its speed increases, these waves become compressed and may eventually merge into a single shock wave, fundamentally changing the flow dynamics.
One of the main challenges in the mathematical analysis of the Navier-Stokes equations is that their solutions may develop singularities. In practice, this means that physical quantities such as velocity, density, or pressure can undergo abrupt, nearly instantaneous changes associated with shock formation, or that certain variables – most commonly the density – may become unbounded in finite time. Motivated by these challenges, our team launched the FluidVarVisc project to investigate whether introducing a density-dependent viscosity term into the momentum equation could help overcome some of these difficulties. In particular, we ask whether such a viscosity term can be sufficiently strong to prevent finite-time blow-up or whether it can exert a regularizing effect that suppresses the formation of shocks. Answering these questions is not only of theoretical interest but also has important implications for the development of stable and reliable numerical methods.
The study of singularity formation is not limited to aerodynamics. Similar mathematical phenomena arise in astrophysical models, where gravitational forces may lead to the concentration of matter and the formation of singularities. As part of our research, we also study the isothermal Euler–Poisson system, which serves as a mathematical model for the gravitational collapse of a self-gravitating Newtonian star. Initially smooth matter distributions may collapse and form localized regions of extremely high density. Gravity pulls the matter toward the center, while the pressure generated within the gas acts against gravitational attraction. Collapse occurs when the pressure gradient is no longer sufficient to balance the gravitational force.
To describe the behavior of solutions near the center of gravity, we employ the concept of shadow waves to approximate solutions with unbounded density under the assumption of spherical symmetry. If the solution is further assumed to be self-similar, it can be shown that all solution components remain bounded and continuous away from the center.
Another direction of our research considers the same system with the Van der Waals equation of state, which incorporates molecular interactions absent from the ideal gas law. In this model, the equation of state explicitly accounts for a maximal physically attainable density by allowing the pressure to become unbounded as the density approaches this limit. Although the density therefore remains bounded, the model still exhibits rich and physically relevant dynamics and has important applications in astrophysics and cosmology, including the modeling of neutron stars.
Although sonic booms and gravitational collapse appear to belong to entirely different worlds, they are governed by closely related mathematical principles. Understanding how singularities form and how they can be prevented lies at the heart of both problems and is one of the central goals of our work.
Sanja Ružičić, Member of FluidVarVisc project, Faculty of Sciences, University of Novi Sad