The workshop aims at presenting fundamental theoretical advances and the most promising current research directions in the analysis of evolutionary partial differential equations and control theory. A central focus will be on challenging problems arising in the theory of conservation laws, fluid dynamics, and optimal control, together with their applications to related areas of mathematics, physics, and applied sciences.
The event also celebrates the 70th birthday of Alberto Bressan, honoring his outstanding scientific achievements and his lasting impact on several areas of modern analysis.
The scientific program is deeply inspired by the seminal contributions of Alberto Bressan, whose work has played a decisive role in shaping modern research in hyperbolic conservation laws and control theory. By solving long-standing open problems, introducing entirely new analytical frameworks, and opening unexpected connections with other branches of analysis, his ideas have profoundly influenced the development of the field and continue to generate new and vibrant research directions.
Over the last decades, the mathematical theory of hyperbolic conservation laws and fluid dynamics—both in their classical and non-classical formulations—has undergone remarkable progress. These advances have unlocked a wide range of further developments, spanning from foundational theoretical results to novel applications and interdisciplinary connections. Topics include, among others: mixing phenomena in incompressible fluids and the well-posedness of transport equations for nearly incompressible vector fields; convex integration techniques for the Euler and magnetohydrodynamics equations; energy conservation and dissipation mechanisms for the Navier–Stokes equations; stability and instability of shear flows for hydrostatic Euler equations; transport of vortex points and vortex sheets; singular limits in fluid models; surface waves at plasma–vacuum interfaces; compactness, regularity, and Lagrangian representations of solutions to hyperbolic conservation laws, possibly in the presence of local or non-local source terms; and well-posedness issues arising in granular flow models.
Starting from pioneering works, in recent years it has become increasingly clear that tools from geometric measure theory and the calculus of variations can be successfully employed to address fundamental questions in the analysis of multidimensional hyperbolic systems, including well-posedness, fine structure, and regularity properties of solutions. Conversely, problems originating in nonlinear hyperbolic PDEs have stimulated the development of new techniques in geometric measure theory and variational analysis, most notably convex integration methods, which have reshaped our understanding of flexibility, rigidity, and non-uniqueness phenomena.
A major unifying theme of the workshop is the study of regularity and singularities of solutions that can be interpreted as minimizers or critical points of suitable variational functionals, following a pioneering vision initiated by De Giorgi. Particular emphasis will be placed on the structure of dissipation and defect measures, which encode concentration and oscillation effects and play a key role both in the mathematical theory and in applications to physics. Progress in this direction has the potential to lead to major breakthroughs in the understanding of turbulence, energy transfer, and uniqueness versus non-uniqueness of weak solutions.
Another important aspect concerns free boundary problems, especially in models involving phase transitions, traffic and pedestrian flows, and other systems where interfaces evolve dynamically. Understanding the structure and regularity of free boundaries is crucial for addressing questions of existence, uniqueness, and long-time behavior of solutions, and naturally connects with spectral optimization problems and geometric variational methods.
Control theory represents a further central pillar of the workshop. It provides powerful mathematical tools to address pressing real-world challenges, such as urban traffic and network management, smart cities, distribution
systems (waterways, gas pipelines), cost-efficient production lines, and the control of invasive species or fire propagation. From a theoretical perspective, this area poses deep and largely unexplored mathematical questions, including the differential structure of control problems, Pontryagin-type necessary conditions, controllability of ODEs and evolution equations, and Lagrangian and variational approaches to optimality. These topics are closely intertwined with the analysis of PDE-constrained measures, singularities, and free boundaries.
Overall, the workshop highlights strong conceptual links between evolutionary PDEs, control theory, geometric measure theory, and variational methods. In this sense, it naturally reflects the scientific vision of the ERC project “Regularity and Singularity of Solutions to Geometric Variational Problems” devoted to the regularity of minimizers and critical points, the singular structure of PDE-constrained measures, and the analysis of free boundaries in geometric and spectral optimization problems.