An arXiv preprint reports a conditional result for a difficult class of partial differential equations. When the equations' critical source terms are bounded and uniformly equi-integrable, the Hessian, or collection of second derivatives, and the nonlinear gradient term are kept within a uniform critical-space bound across the specified family of solutions.
That condition is the paper's answer to a sharply defined question: can uniform equi-integrability rule out concentration and establish maximal regularity at the endpoint? Here, maximal regularity means that the second-derivative term D^2u and the power of the gradient, |Du|^gamma, remain uniformly controlled in the critical integrability space.
The condition behind the bound
The work considers an arbitrary family U of strong solutions with zero average, with corresponding source terms drawn from a family F. The analysis is set on a torus and treats the problem as a theoretical family of solutions rather than a sample of observed cases.
The theorem requires dimension d at least 2, an exponent gamma greater than 2, and the critical exponent q_c = d(gamma - 1)/gamma. For each source f in F paired with a zero-average strong solution u in W^{2,q_c}(T^d), it bounds the L^{q_c} norm of D^2u together with the L^{q_c} norm of |Du|^gamma by a constant C. That constant depends on the dimension, the exponent, the uniform source-norm bound and the equi-integrability modulus.
In ordinary language, equi-integrability is a no-piling-up condition. It prevents the family of source terms from hiding an increasing share of its critical mass in ever smaller regions as the family changes. The question is whether that extra control is enough to stop concentration from undermining the endpoint estimate.
The proof zooms in twice
The proof follows a two-stage blow-up argument, a way of testing a regularity claim by zooming in on the scales where it might fail. It combines local energy estimates, regularity for equations with a sufficiently small drift, strong compactness of the rescaled solutions and a Liouville rigidity result that restricts possible global limits.
In the first blow-up, the rescaled source g_n disappears locally: on every fixed ball, it converges to zero in the local critical norm. This part of the argument operates under the bounded, uniformly equi-integrable source-family assumption.
The second blow-up addresses the possibility that energy could still gather at a smaller scale. Its energy is uniformly bounded on each fixed ball, and on balls centered in a fixed region it is bounded by a universal constant for all sufficiently large n.
When the first-rescaling coefficient is sufficiently small, a subsequence of the blow-up solutions converges locally strongly in W^{2,q_c}; the gradients also converge locally strongly in L^{gamma q_c}. The powers |Dv_n|^{gamma q_c} form a uniformly equi-integrable family on every fixed ball.
The limiting step uses a Liouville rigidity theorem. It says that any global function v in W^{2,q_c}_{loc}(R^d) and C^{0,alpha_c}(R^d) solving -Delta v + a|Dv|^gamma = 0 with a at least 0 must be constant. A nonconstant concentration profile therefore cannot survive all the way to the global limit.
After the uniform Hölder-vanishing estimate is established, the argument absorbs the Hessian term and obtains the uniform W^{2,q_c} bound and uniform control of the nonlinear gradient term required by the theorem.
A sharp limit to the claim
The boundary of the result is clearest in the paper's counterexample. A constructed sequence keeps its critical source norms uniformly bounded, yet its maximal-regularity quantities diverge as epsilon decreases. The source terms in that sequence are not uniformly equi-integrable.
That contrast shows that a bounded critical norm, by itself, does not establish the endpoint estimate in this construction. The theorem's additional requirement is what distinguishes its controlled source family from the sequence in which the regularity quantities become unbounded.
The conclusion remains tied to the stated setting: zero-average strong solutions on the torus, the specified exponent regime and a bounded, uniformly equi-integrable source family. The author interprets the theorem as an affirmative answer to the superquadratic endpoint problem and presents the two-stage blow-up construction as the proof's novelty.
The paper is listed as arXiv preprint version one, dated 25 August 2026. Funding and conflict-of-interest information are not reported in the supplied document.
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