Endpoint L-infinity-in-time L3 regularity criterion

Statement

Let $u$ be an unforced three-dimensional Leray–Hopf solution on $\mathbb R^3\times(0,T)$ . The Escauriaza–Seregin–Šverák endpoint criterion asserts that

u\in L^\infty(0,T;L^3(\mathbb R^3))

implies regularity at positive times, including absence of a singularity at the terminal time $T$ . In particular the velocity is smooth in the interior positive-time region.

The hypothesis is a mixed time-space norm in this regularity criterion.

Reading the notation

The condition is the mixed norm $\operatorname*{ess\,sup}_{0<t<T}\|u(t)\|_{L^3_x}<\infty$ . The original notation $L_{3,\infty}$ refers here to spatial exponent three and time exponent infinity. It does not replace $L^3_x$ by the weak Lorentz space $L^{3,\infty}_x$ . This is an unforced criterion with its own solution hypotheses.

References

Escauriaza, Seregin and Šverák, L3,infinity solutions and backward uniqueness (2003).