A mathematical preprint reports a uniform regularity bound at a critical endpoint for a class of periodic viscous Hamilton-Jacobi equations, under a condition called uniform equi-integrability. The condition limits how much of a source term can be packed into very small sets. A constructed counterexample in the same analysis shows that a bounded source norm alone does not support the same endpoint conclusion.
The question is posed in the superquadratic regime, meaning the equation's gradient term has a growth exponent above two, and in dimensions of at least two. The theorem considers a bounded family of source terms at a critical integrability level, together with periodic solutions whose average is zero and whose second derivatives already belong to the stated Sobolev class. Those assumptions define the result's scope: it is a theorem about normalized strong solutions, not a claim about an unspecified solution class.
The condition that sets the boundary
Uniform equi-integrability is stronger than a simple bound on each source's critical norm. It comes with a common modulus of control: as the size of the set being tested shrinks, the allowed contribution tends to zero. This is the part of the hypothesis that addresses concentration at the critical scale. The paper's counterexample tests the boundary by keeping the critical norms bounded while dropping this uniform condition.
Under the stated assumptions, the theorem gives one uniform bound for both the Hessian, the collection of second spatial derivatives, and the equation's nonlinear gradient term in the critical integrability space. The constant is controlled by the dimension, the growth exponent, the source-norm bound and the equi-integrability modulus. The estimate therefore applies across the solution family covered by the theorem.
A proof built by zooming in
To establish the endpoint estimate, the proof uses a two-stage blow-up. In this method, a potential failure is magnified by rescaling, so the analysis can examine behavior at smaller and smaller distances. The argument then combines small-drift regularity, which upgrades weak local compactness to strong local compactness, with Liouville rigidity. In this setting, the Liouville statement says that a globally Hölder, locally regular limit solving the homogeneous rescaled equation must be constant.
Before that rigidity step, the blow-up analysis establishes bounded local energy on every fixed ball for the rescaled sequence. When the first rescaling coefficient is sufficiently small, the rescaled solutions converge strongly on bounded regions in the relevant second-derivative space, and their gradients converge strongly in the corresponding gradient space. This compactness result supplies the local limit needed by the Liouville argument.
The theorem's other key output is a uniform small-scale statement about the solution family. Its Hölder quotient, a measure of how much the solution changes relative to the distance between two points, tends to zero as the distance scale shrinks. Because the statement is uniform over the family, it controls the common small-scale behavior of all solutions covered by the theorem, rather than describing only one solution at a time.
The counterexample sets a clear limit. In that construction, source terms remain bounded in the critical space but fail the uniform equi-integrability condition, and maximal regularity does not hold. The analytical message is narrower: boundedness in the critical space alone is insufficient for the endpoint estimate. The paper does not extend the conclusion to arbitrary critical-space-bounded sources.
The mean-field-games application
The analysis also presents an endpoint result for a defocusing mean-field-games system. Under the theorem's stated assumptions, it asserts a unique variational solution and gives integrability conditions for the density coupling, the derivative of the Hamiltonian and the flux. This application is stated for dimensions of at least four and for a potential with two continuous derivatives, making its scope narrower than the main regularity theorem.
The variational route starts with a functional that has a unique minimizer over the admissible pairs of population density and flux. A duality argument identifies that minimizing flux with the feedback generated by the Hamiltonian, closing the coupled system. In the paper's formulation, the flux is therefore linked analytically to the density and the gradient of the value function.
How far the claim reaches
The assumptions remain central to how the findings should be read. The main theorem is stated for bounded, uniformly equi-integrable source families and zero-average strong solutions in the specified Sobolev class. The mean-field-games theorem stays within its listed dimensional, coupling and potential assumptions. The work is an arXiv version-two preprint dated 1 Sep 2026. Its acknowledgments record helpful discussions and assistance with references, but the supplied material gives no financial-support statement.
Paper data and sources
Original title: Maximal Regularity of Superquadratic Hamilton-Jacobi Equations: The Endpoint Case in Lions' Conjecture
Authors: Fanze Kong
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text