Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case

Abstract

We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du{|}^{\gamma }=f$ in ${\mathbb{T}}^d\times (0,T)$, where $d\ge 2$, $\gamma >2$, and $q_c=(d+2)(\gamma -1)/\gamma$. 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 $f\in L^{q_c}$ and continuous initial data $u_0$, and prove subsequential strong interior convergence of smooth approximations in $W_{q_c}^{2,1}$.