In mathematical physics (for instance hydrodynamics), one important issue lies in the modeling of phenomena at different scales, in particular emphasizing macroscopic and microscopic descriptions. In this context, but also in various scientific fields such as life science, chemistry, oncology, economics, or complex systems, irregular phenomena arise. This project, which gathers mathematicians from Brazil and from France, puts emphasis on stochastic and deterministic analysis of irregular phenomena.

We are motivated by irregular physical phenomena (e.g. turbulence, complex systems) which are difficult to formulate and study mathematically. Usually those phenomena are modeled either macroscopically (via PDEs) or microscopically via stochastic differential equations. The originality of the project is threefold.

• The systematic study of the irregularity, which is mathematically expressed by the irregularity of the noise, or of
  the coefficients, and the presence of memory. The mathematical tools will be PDEs with singular coefficients,
  path-dependent PDEs, (mean-field) SDEs with memory and singular drifts.

• We will investigate simultaneously the two modeling (microscopic-macroscopic) approaches. From the mathematical point of view, this will be articulated by the use of analytical and probabilistic tools. From the modeling point of view, this will help to select more realistic models which avoid the ill-posedness issues.

• We will propose efficient and original approximation schemes, taking advantage of the two modeling features. Those two aspects are complementary: the mathematical model formulation inspires numerical approximation and (viceversa) the approximations provide new ideas for the formulation of the models.

Examples of models that will be investigated:

• Keller-Segel equations, Burgers equation, 2d Navier-Stokes equation and their stochastic representations.

• Quasilinear PDEs of self-organized criticality.

• Rough PDEs.

• Path-dependent PDEs with irregular coefficients.

• 2d stochastic Navier-Stokes with fractional noise.

• 3d alpha-Vlasov-Navier-Stokes equation.

• Random dynamical systems perturbed by rough paths.