Endpoint trilinear estimate for the fully conjugated cubic nonlinearity

Determine whether the spatial Bourgain-space trilinear estimate for the fully conjugated cubic nonlinearity N_3(u,u,u)=\bar u\,\bar u\,\bar u associated with the 2m-th order nonlinear Schrödinger equation on the half-line holds at the endpoint regularity s=s_{m,3}=-(2m-1)/3.

Background

The paper proves trilinear Bourgain-space estimates for N_3 only when s>s_{m,3}=-(2m-1)/3. The proof requires a positive gap δ=b-b' between the temporal regularity exponents, but this gap tends to zero as s approaches s_{m,3}, preventing the argument from reaching the endpoint. Consequently, endpoint well-posedness for the N_3 initial-boundary value problem is also not established by the paper.

References

However, in order to achieve this lower regularity, we still have \delta\to 0 as s\to s_{m,3}. Thus, the endpoint case for k=3 is not yet attainable in our current approach. Whether estimate 1.16 holds for k=3 at s=s_{m,3} remains an interesting open problem.

— Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line  (2610.02747 - Xi et al., 2 Oct 2026) in Remark 1.5(ii), Section 1

It is known that for k=1, the estimate is optimal; for k=0,2,3, the question remains open. In particular, for k=3, the endpoint case s=s_{m,3} is still unresolved.

— Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line  (2610.02747 - Xi et al., 2 Oct 2026) in Remark 1.7, Section 1