Project

Brief description

Nowadays it is widely recognized that the behavior of any physical system at some (spatio-temporal) scale is deeply ruled by laws involving other different scales and these laws are largely unknown. The paradigmatic problem of turbulence in fluids constitutes the most tangible manifestation of this reality, but many other still inexplicable phenomena seem to depend on the interplay through the scales of the governing laws. These problems affect biological materials as well as geophysical materials. The understanding of this issue asks for a collective effort of many scientific areas and mathematics plays a fundamental role in detecting the right conceptual framework needed to push knowledge forward.

In this framework we have selected the following four high impact fields:

a. Singularity formations in nonlocal evolution equations

Peridynamics was initiated by Silling by introducing a non local approach to continuum mechanics and originates from desire to find analytical descriptions of phenomena such as the spontaneous creation of singularities like cracks, damages or defects and their evolutions, as well as the dispersive character of the waves propagation. These phenomena suggest the presence of many scales involved in the underlying physics.

b. Emergence of defects at different scales in discrete systems.

The multi-scale variational analysis of discrete systems is pivotal in the analysis of emergence and motion of defects in materials. Depending on the different regimes of the interaction energy at the discrete level, defects may arise in the continuum limit as a result of energy-concentration. We plan to study the discrete-to-continuum analysis of a selection of dislocation and spin discrete models which share a common feature: each of them encompasses the possibility of simultaneous formation of defects at multiple dimensions, which may mutually interact.

c. Singular limits and stability in incompressible fluid dynamics

We intend to investigate interactive boundary layer theories, where the interaction between the outer and the inner flow is the key mechanism for the vortex sheddings and separation. Moreover we shall study the justification of the Birkhoff-Rott equation as a valid model for the evolution of highly concentrated vorticity data. We want to investigate the structure of the singularities that form during the roll-up process in 3D, which promises to give new insights on vorticity concentration, small scale phenomena and loss of regularity.

d. Nonlinear dissipative structures in reaction-diffusion systems far from equilibrium

The focus of this part of the project is on the effects induced by the presence of nonlinear/cross diffusion terms in reaction-diffusion systems far-from-equilibrium. Topics include localized solutions in bistable systems, sharp interface solutions of singularly perturbed models and analysis of the secondary instabilities leading to spatio-temporal chaotic states.