The mechanism responsible for blow-up is well-understood for many hyperbolic conservation laws. Indeed, for a whole class of problems including the Burgers equation and many aggregation-diffusion equations such as the 1D parabolic-elliptic Keller-Segel system, the time and nature of the blow-up can be estimated by ODE arguments. It is, however, a much more delicate question to understand the small-scale behaviour of the viscous layers appearing in (classic or fractional) parabolic regularisations of these conservation laws.
Here we give sharp estimates for Sobolev norms and for a class of small-scale quantities such as increments and energy spectrum (which are relevant for the theory of turbulence), for solutions of these conservation laws. Moreover, many of our results can be generalised for perturbations of the viscous conservation laws by random additive noise, and some of them admit a simpler formulation in this case. To our best knowledge, these are the only sharp results of this type for small-scale behaviour of solutions of nonlinear PDEs.
The work on the aggregation-diffusion equations is an ongoing collaboration with Piotr Biler and Grzegorz Karch (Wroclaw) and Philippe Laurençot (Toulouse).[-]