Klainerman's critical well-posedness conjecture for basic field theories

The systems under consideration are basic field theories with critical well-posedness exponent $s_c$, meaning that the $dot H^{s_c} \times dot H^{s_c-1}$ norm of the initial data is invariant under the natural scaling. Klainerman's critical well-posedness conjecture. (1) For all basic field theories, the initial value problem is locally well posed for initial data in $H^s \times H^{s-1}$ with $s>s_c$. (2) The basic field theories are weakly globally well posed for all initial data with small $H^{s_c} \times H^{s_c-1}$ norm. (3) The basic field theories are ill posed for initial data in $H^s \times H^{s-1}$ with $s<s_c$. Here weak well-posedness allows the solutions to fail to depend smoothly or analytically on the data. This is presented as a formulation taken from Klainerman and concerns the expected sharp regularity thresholds for local, small-data global, and ill-posed regimes.