Abstract
Partial differential equations are ubiquitous in science and describe a stunning variety of physical phenomena, encompassing diverse topics such as fluid-mechanics, electromagnetism and crystalline structures, just to mention a few. While their use is nowadays common in mathematical modelling, their fine analysis still poses major challenges, many of which stem from the lack of smoothness or regularity of the objects involved in the equations.
Of particular relevance is the study of transport-type equations, which describe, among others, the movement and the evolution of fluids. Examples come easily: turbulent air currents trailing a modern jet, waves following boats across a river, blood and lymph coursing through the human body, milk stirred in a coffee mug or even elastic deformations within materials - they are all transport phenomena.
The BRIDGE project aims to further our current understanding of irregular transport phenomena and of the geometrical aspects arising from their mathematical description. The proposed research imposes a combination of techniques borrowed from several areas and promises bridges between Analysis of PDEs and Calculus of Variations, Harmonic Analysis and Geometric Analysis, fostering new exciting avenues of research at the interface of these disciplines.
▾ Team
Paolo Bonicatto (PI)
▾ Themes
Currents in Euclidean spaces provide a flexible way to describe rough or irregular surfaces. They are central in the study of minimal surfaces (e.g., Plateau’s problem) and can be seen as the natural vector-valued counterpart to Schwartz' distributions. A paradigmatic example of a k-current in the d-dimensional Euclidean space, with k≤d, is given by a vector-valued measure with values in the space of k-multivectors in R^d. This viewpoint often simplifies computations and intuition.
In a series of joint papers [3, 5, 6] with Del Nin and Rindler, we have introduced and developed the theory of the Geometric Transport Equation (GTE), which describes how currents evolve under a possibly time-dependent vector field. The (GTE) unifies both the continuity equation for measures (which can be retrieved as the 0-dimensional case of our analysis) and the scalar transport equation (top-dimensional case). In [5] we also studied when a continuous path of currents can be seen as the transport of a current along (the flow of) a vector field. Certain examples, like the flat mountain, show that this is not always possible (not even when the path is absolutely continuous w.r.t. the flat distance).
For 1-dimensional currents, the (GTE) coincides with the (ideal) induction equation from Magnetohydrodynamics and thus models the transport of the magnetic field in a conducting fluid. See [1] for more details.
For further reading:
[1] P. Bonicatto, A general Frobenius' Theorem via the Transport of Currents (2025) arXiv: 2510.21478
[2] P. Bonicatto, F. Rindler, and H. Turnbull, The Space-Time Connectivity Theorem for Normal Currents (2025) arXiv: 2510.08360
[3] P. Bonicatto and G. Del Nin, Well-posedness of the transport of normal currents by time-dependent vector fields (2025) arXiv: 2504.15974
[4] P. Bonicatto and F. Rindler, Homogenization of elasto-plastic evolutions driven by the flow of dislocations (2024) arXiv: 2410.02906
[5] P. Bonicatto, G. Del Nin, and F. Rindler, Transport of currents and geometric Rademacher-type theorems, Trans. Amer. Math. Soc. 378 (2025), p. 4011–4075 online version
[6] P. Bonicatto, G. Del Nin, and F. Rindler, Existence and uniqueness for the transport of currents by Lipschitz vector fields, J. Funct. Anal., 286 (2024) online version
Theme I shows how important it is to better understand the fine and structural properties of “rough’’ surfaces or functions. These can be studied using the modern mathematical frameworks of the spaces BV (functions with bounded variation) or BD (functions with bounded deformation). Research on non-distributional characterisations of these functional spaces has grown significantly over the last twenty years. Several functionals (involving either continuous convolution-type energies or discrete oscillation-type energies) have been proposed, starting from the pioneering works by Bourgain, Brezis and Mironescu, to the more recent ones by Ambrosio, Bourgain, Brezis and Figalli and many others.
In a joint work [9] with Arroyo-Rabasa, we studied convolution-type energies in the spirit of Bourgain–Brezis–Mironescu and extended these results to BD maps. For oscillation-type energies, instead, one wants to relate the total variation of the distributional gradient of an integrable function to its mean oscillation over certain families of cubes. Various functionals of this type have been considered up to date, but they are often bedevilled by the so-called Cantor part of a BV or BD function, as they cannot detect it. To remedy these problems, in recent works [7,8] with Arroyo-Rabasa and Del Nin we studied these functionals using either a different notion of convergence or different families of cubes. Our results apply to all BV functions, including those with a non-trivial Cantor part.
For further reading:
[7] A. Arroyo-Rabasa, P. Bonicatto, and G. Del Nin, Local Poincaré constants and mean oscillation functionals for BV functions, Rev. Mat. Iberoam. (2026) to appear - arXiv: 2402.10794
[8] A. Arroyo-Rabasa, P. Bonicatto, and G. Del Nin, Representation of the total variation as a Γ-limit of BMO-type seminorms, Indiana University Mathematics Journal, 73 (2024), p. 341–365 online version
[9] A. Arroyo-Rabasa and P. Bonicatto, A Bourgain–Brezis–Mironescu representation for functions with bounded deformation, Calc. Var. Partial Differential Equations, 62 (2023) online version
The “flat mountain” singularity constructed in [5].
The swirl of milk in coffee, the deformation of the atomic structure of crystals and the Sun's magnetic field evolving over time: Transport phenomena shape the world around us.
The Geometric Transport Equation (GTE).