Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case
Abstract
We establish interior maximal Lqc-regularity for bounded strong solutions of ut−Δu+|Du|γ=f in 𝕋d×(0,T), where d≥ 2, γ>2, and qc=(d+2)(γ−1)/γ. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly equi-integrable subset of Lqc. The main difficulty is the possible concentration of the critical gradient energy. We overcome it through a two-scale blow-up argument: the first scale produces a small Hamiltonian coefficient, while energy normalization at the second scale yields strong endpoint compactness, allowing a parabolic Liouville theorem to rule out concentration. The resulting uniform little-Hölder estimate, combined with critical Gagliardo--Nirenberg interpolation, permits absorption of the nonlinear term in the parabolic Calderón--Zygmund estimate. As an application, we construct bounded strong solutions for f∈ Lqc and continuous initial data u0, and prove subsequential strong interior convergence of smooth approximations in W2,1qc.
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BibTeX
@misc{kong2026maximal,
title = {{Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case}},
author = {Fanze Kong and Xiaoyu Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33798},
eprint = {2609.33798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33798},
}
RIS
TY - UNPB
TI - Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case
AU - Fanze Kong
AU - Xiaoyu Zeng
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33798
UR - https://arxiv.org/abs/2609.33798
AB - We establish interior maximal L^q_c-regularity for bounded strong solutions of u_t−Δu+|Du|^γ=f in 𝕋^d×(0,T), where d≥ 2, γ>2, and q_c=(d+2)(γ−1)/γ. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly equi-integrable subset of L^q_c. The main difficulty is the possible concentration of the critical gradient energy. We overcome it through a two-scale blow-up argument: the first scale produces a small Hamiltonian coefficient, while energy normalization at the second scale yields strong endpoint compactness, allowing a parabolic Liouville theorem to rule out concentration. The resulting uniform little-Hölder estimate, combined with critical Gagliardo--Nirenberg interpolation, permits absorption of the nonlinear term in the parabolic Calderón--Zygmund estimate. As an application, we construct bounded strong solutions for f∈ L^q_c and continuous initial data u_0, and prove subsequential strong interior convergence of smooth approximations in W^2,1_q_c.
ER -