Hong–Kwak–Yang open question on the FPU–KdV continuum limit

For the Fermi–Pasta–Ulam system in the standard continuum-limit scaling, determine whether solutions with initial data at $L^2$ regularity converge, in the appropriate norms, on every fixed time interval to the corresponding counter-propagating solutions of the Korteweg–de Vries equation as the lattice mesh size tends to zero. More generally, determine the lowest Sobolev regularity $s$ for which the local-in-time FPU–KdV continuum limit holds; the claimed threshold is $s>-\frac{3}{4}$ .

References

Primary source

arXiv

Additional references

The Fermi-Pasta-Ulam system and the Korteweg-de Vries equation: a low-regularity continuum limit — arXiv — Herbert Koch, Ruoyuan Liu

dispersive-equations mathematical-physics partial-differential-equations

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to settle the low-regularity FPU–KdV continuum-limit question, including arbitrary-time convergence at $L^2$ regularity.

The question concerns whether periodic FPU dynamics converge to KdV at substantially lower regularity than previously known. Earlier work reached only positive Sobolev regularity, while the new preprint claims both arbitrary-time $L^2$ convergence and a stronger local-in-time threshold.

Known results

Kwak and Yang obtained a local-in-time periodic FPU continuum limit for $s>0$ .
For the non-periodic FPU system, earlier work obtained short-time convergence under $s>\frac{3}{4}$ and an almost-global small-amplitude limit.

September 2026 low-regularity preprint

Herbert Koch and Ruoyuan Liu claim an arbitrary-time $L^2$ continuum limit and lower the local-in-time Sobolev threshold to $s>-\frac{3}{4}$ . This is a claimed resolution of the tracked question, but the result has not been independently verified.

Current status (as of September 2026): The previous $s>0$ local-in-time result is superseded by Koch and Liu's unverified preprint claim of arbitrary-time $L^2$ convergence and local-in-time validity for $s>-\frac{3}{4}$ .

Sources

No solutions have been posted yet.