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McKean’s conjecture for the Landau–Coulomb equation

Let ftf_t be a sufficiently regular solution of the spatially homogeneous Landau equation with Coulomb interaction, and let H(ft)=∫R3ft(v)log⁡ft(v) dvH(f_t)=\int_{\mathbb{R}^3} f_t(v)\log f_t(v)\,\mathrm{d}v be its entropy. McKean's conjecture asserts that the entropy dissipation D(ft)=−ddtH(ft)D(f_t)=-\frac{\mathrm{d}}{\mathrm{d}t}H(f_t) is monotone nonincreasing along every solution: ddtD(ft)≤0\frac{\mathrm{d}}{\mathrm{d}t}D(f_t)\leq 0 for all t≥0t\geq 0. Equivalently, d2dt2H(ft)≥0\frac{\mathrm{d}^2}{\mathrm{d}t^2}H(f_t)\geq 0 for all t≥0t\geq 0.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Entropy convexity formulation of McKean's conjecture

    For every sufficiently regular solution ftf_t of the spatially homogeneous Landau equation with Coulomb interaction, the entropy H(ft)=∫R3ft(v)log⁡ft(v) dvH(f_t)=\int_{\mathbb{R}^3} f_t(v)\log f_t(v)\,\mathrm{d}v is convex in time: d2dt2H(ft)≥0\frac{\mathrm{d}^2}{\mathrm{d}t^2}H(f_t)\geq 0 for all t≥0t\geq 0.

    source: A counterexample to McKean's conjecture for the Landau-Coulomb equation

References

Primary source

arXiv

Additional references

kinetic-theorymathematical-physicspartial-differential-equations
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 preprint reports a counterexample that would overturn McKean’s entropy principle, but the result has not been refereed.

McKean’s conjecture proposed a convexity principle for entropy dissipation in the Landau–Coulomb dynamics, with a corresponding assertion for the Boltzmann equation. The latest preprint claims this principle fails in general.

September 2026 counterexample

On September 3, 2026, an arXiv preprint constructed near-equilibrium counterexamples to monotone entropy dissipation, thereby claiming to disprove McKean’s conjecture and the associated Boltzmann-equation assertion. The result is reported as an unrefereed preprint and remains unverified.

Current status (as of September 2026): McKean’s conjecture is claimed false by a near-equilibrium counterexample, while independent verification of the unrefereed preprint remains outstanding.

Sources

Solutions 0

No solutions have been posted yet.