In this talk, I will discuss very recent developments on the global regularity phenomenon of the logarithmically supercritical PDE, initiated by Terence Tao in 2007. The idea is that typically by obtaining the bound such as

\[\partial_{t} \lVert f(t) \rVert_{H}^{2} \leq A(t) \lVert f(t) \rVert_{H}^{2}\]

one may deduce the global bound of \(\lVert f(t) \rVert_{H}^{2}\) by Gronwall’s inequality if \(A(t)\) is locally integrable, and \(f(0) \in H\). It is well-known from elementary ODE theory that one cannot possibly hope to deduce a similar result if the power of \(\lVert f(t)\rVert_{H}\) on the right hand side is bigger than 2, even by an arbitrary small amount (this leads to the notion of criticality threshold in PDE). However, Tao realized that although such an exponential worsening is not allowed, a logarithmic worsening is in fact allowed; that is,

\[\partial_{t} \lVert f(t) \rVert_{H}^{2} \leq A(t) \lVert f(t) \rVert_{H}^{2} \ln (e+ \lVert f(t) \rVert_{H}^{2})\]

still leads to the desired global bound of \(f\) in \(H\) under the same conditions on \(A(t), f(0)\).

For example, for a wave equation with power nonlinearity, it was well-known that there is a certain threshold power depending on dimension such that for any power equal to or less than this threshold, the global regularity could be proven but not if the power is higher. Tao multiplied this nonlinear power term with logarithmic function and still proved the global well-posedness.

Similarly for the Navier-Stokes equations, it was well-known that if one replaces the Laplacian in the dissipation with a fractional Laplacian, there is a certain threshold power of this fractional Laplacian depending on dimension such that for any power equal to or more than this threshold, the global regularity could be proven but not if the power is lower. Tao divided the dissipation term on the Fourier side by a logarithmic function and still proved the global well-posedness.