Optimal uniform-in-time spatial regularity for endpoint-trace density

Determine the largest exponent p for which endpoint-trace density for weak solutions of the three-dimensional incompressible Navier–Stokes equations on the torus can hold with \(\nabla u\in C_tL_x^p\), and determine whether optimization, localization, or a different averaging mechanism yields a fixed gain above \(6/5\).

Background

The paper proves endpoint-trace density for the explicit exponent pˉ=6/5+5×10−5\bar p=6/5+5\times10^{-5} using moving Hill vortices. The authors emphasize that the resulting exponent is not claimed to be optimal and ask whether the method or a structurally different construction can achieve a larger exponent. The scale-invariant Navier–Stokes exponent p=3/2p=3/2 is identified as a substantially more ambitious target.

References

Determine the largest exponent p for which the endpoint-trace density in eq:intro-trace-density can hold with \nabla u\in C_tL_xp. The ceiling in eq:discussion-pmax concerns only the chosen parameters. Can optimization, localization, or a different averaging mechanism give a fixed gain above 6/5? The scale-invariant value is p=3/2, for which W{1,3/2}(T3)\hookrightarrow L3(T3), but the present estimates do not approach it.

eq:intro-trace-density:

{(u(⋅,0),u(⋅,1)):u∈Spˉ}‾ L2×L2=Lσ2(T3)×Lσ2(T3).\overline{ \left\{ \bigl(u(\cdot,0),u(\cdot,1)\bigr): u\in\mathcal S_{\bar p} \right\} }^{\,L^2\times L^2} = L^2_\sigma(T^3)\times L^2_\sigma(T^3).

eq:discussion-pmax:

pˉ<pmax⁡:=253−γ2(μ+γ)=2401220009≈1.20005997.\bar p < p_{\max} := \frac{2}{ \frac53-\frac{\gamma}{2(\mu+\gamma)} } =\frac{24012}{20009} \approx 1.20005997.

— Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices  (2608.20068 - Nguyen et al., 20 Aug 2026) in Section 8, paragraph “Optimal uniform-in-time spatial regularity”

Can the construction prescribe E(t)=\frac12|u(\cdot,t)|_{L2}2 while retaining \nabla u\in C_tL_x{\bar p}, or give flexibility among energy-compatible endpoints in the Leray--Hopf or suitable class? Arbitrary endpoint density is incompatible with nonincreasing energy, and the present estimates give neither L_t2H_x1 control nor a local energy inequality.

— Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices  (2608.20068 - Nguyen et al., 20 Aug 2026) in Section 8, paragraph “Energy profiles and admissibility”

Whether a vortex-ring profile can meet these requirements and improve the exponent is open.

— Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices  (2608.20068 - Nguyen et al., 20 Aug 2026) in Section 8, paragraph “Coherent vortices beyond the Hill profile”