Explain the regularity gap in bilinear estimates for the sixth-order Boussinesq equation

Determine the reason for the gap between the currently known regularity thresholds in bilinear estimates for the sixth-order Boussinesq equation, analogous to the gap between the regularity indices -5/8 and -3/4 in the corresponding KdV estimates, and clarify whether the approach based on mixed-norm estimates can identify the optimal regularity index.

Background

The paper develops trilinear estimates for the cubic sixth-order Boussinesq equation in Bourgain-type spaces and obtains local well-posedness for regularity s > -1/2 for both the initial-boundary value problem and the initial value problem. The authors discuss two broad approaches to multilinear estimates: Tao’s multiplier-norm method and methods based on mixed Lp_x Lq_t estimates.

They note that the mixed-norm approach can be applied to bilinear estimates and may require less case analysis than the multiplier-norm method, but may not reveal the optimal regularity threshold. In the KdV literature, the known thresholds -5/8 and -3/4 leave a gap; the authors state that an analogous gap occurs for bilinear estimates of the sixth-order Boussinesq equation and explicitly leave the reason for that gap unresolved.

References

However, it seems that this approach can not directly indicate the optimal conclusion of index $s$. According to Kenig's early work on KdV equation, reader can observe that there still exists a gap between $-\frac{5}{8}$ and $-\frac{3}{4}$ . The similar situation also happens to bilinear estimate of SOBE. The problem for the reason remains open.

— Initial and initial-boundary value problems for a cubic sixth-order Boussinesq equation  (2610.02748 - Xie et al., 2 Oct 2026) in Section 1, Introduction, concluding discussion before the paper organization paragraph