In the first half, I will describe recent breakthroughs in the research direction of singular stochastic PDEs that appear frequently in mathematical physics. In the second half, I will describe another recent breakthrough technique of convex integration in hydrodynamics and its consequence on turbulence.

First, many equations in mathematical physics were suggested in the form of PDEs with random force, specifically the so-called space-time white noise, e.g., the Kardar-Parisi-Zhang equation in 1986. Due to the singularity of the noise, the solution turns out to be a distribution rather than a function leading to the nonlinear term within the PDE to be ill-defined. Breakthrough techniques of the theory of regularity structures by Hairer and the theory of paracontrolled distributions by Gubinelli, Imkeller, and Perkowski now allow us to understand its (very weak) solution theory.

Second, turbulence occurs in our daily lives. E.g., in airplanes, we are all reminded by flight attendants to keep our seatbelts fastened in preparation for “unexpected turbulence”. Kolmogorov’s zeroth law of turbulence from 1941 was supported by numerical analysis under the name of “anomalous dissipation”. Closely related are the famous Onsager’s conjecture in 1949 and Taylor’s conjecture in 1974. It is only in the last decade that the breakthrough technique of convex integration finally led to mathematically rigorous resolutions to such conjectures.