Let L>0, M∈Rsetminus0, and varepsilon>0. Consider the transport–diffusion equation u_t+M u_x=varepsilon u_xx on (0,T)×(0,L), with a Dirichlet boundary control at x=0 and homogeneous Dirichlet data at x=L. Let T_unif be the infimum of the times T for which the cost of null-controllability remains bounded as varepsilon→0. The Coron–Guerrero conjecture asserts that T_unif=L/M when M>0, and T_unif=2L/|M| when M<0.

Coron–Guerrero uniform null-controllability conjecture

Let $L>0$ , $M\in\mathbb{R}\setminus\{0\}$ , and $\varepsilon>0$ . Consider the transport–diffusion equation $u_t+M u_x=\varepsilon u_{xx}$ on $(0,T)\times(0,L)$ , with a Dirichlet boundary control at $x=0$ and homogeneous Dirichlet data at $x=L$ . Let $T_{\mathrm{unif}}$ be the infimum of the times $T$ for which the cost of null-controllability remains bounded as $\varepsilon\to0$ . The Coron–Guerrero conjecture asserts that $T_{\mathrm{unif}}=L/M$ when $M>0$ , and $T_{\mathrm{unif}}=2L/|M|$ when $M<0$ .

References

Primary source

arXiv

Additional references

Uniform null-controllability times for the Coron--Guerrero problems are 2 and 2 + 2 √2 — arXiv — Kai Koike, Vincent Laheurte

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new paper claims to determine the exact control times, replacing the earlier prediction in both speed regimes, but the result has not been independently verified.

Coron and Guerrero conjectured the minimal time for uniform null-controllability of a one-dimensional convection-diffusion equation as viscosity vanishes. The conjecture distinguishes positive and negative constant transport speeds.

Known results

The original conjecture predicted $T>L/M$ for $M>0$ and $T>2L/|M|$ for $M<0$ .
A conditional result gave exponential decay for $T>L/M$ when $M>0$ and $T>(1+\sqrt{2})L/|M|$ when $M<0$ .
Later work established bounds for variable transport fields but did not settle the constant-coefficient problem.

September 2026 exact-threshold claim

Kai Koike and Vincent Laheurte’s paper Uniform null-controllability times for the Coron--Guerrero problems are 2 and $2+2\sqrt{2}$ claims exact thresholds $2L/M$ for $M>0$ and $(2+2\sqrt{2})L/|M|$ for $M<0$ . It therefore corrects the conjectural prediction; the negative-speed prediction had already been disproved. The claim is unverified in the retrieved material.

Current status (as of September 2026): An arXiv paper claims the conjecture is solved with exact thresholds $2L/M$ for $M>0$ and $(2+2\sqrt{2})L/|M|$ for $M<0$ , but independent verification is not recorded.

Sources

No solutions have been posted yet.