发表机构
University of Washington; Wuhan University of Technology（华盛顿大学; 武汉理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对超二次抛物Hamilton-Jacobi方程，通过双尺度爆破和Liouville定理克服临界梯度能量集中，建立最大L^q_c正则性，并构造有界强解及强收敛性。
AI 中文摘要
我们在 $\u674d^d\times(0,T)$ 上建立了有界强解 $u_t-\u0394u+|Du|^\u03b3=f$ 的内部最大 $L^{q_c}$-正则性，其中 $d\u2265 2$，$\u03b3>2$，且 $q_c=(d+2)(\u03b3-1)/\u03b3$。这些估计对一致有界的解族是均匀的，其源项在 $L^{q_c}$ 的有界、一致等度可积子集上取值。主要困难在于临界梯度能量可能发生集中。我们通过一个双尺度爆破论证克服了这一困难：第一尺度产生一个小的哈密顿量系数，而第二尺度的能量归一化产生强端点紧性，从而允许抛物Liouville定理排除集中。所得的均匀little-Hölder估计，结合临界Gagliardo--Nirenberg插值，允许在抛物Calderón--Zygmund估计中吸收非线性项。作为应用，我们为 $f\u2208 L^{q_c}$ 和连续初值 $u_0$ 构造了有界强解，并证明了光滑逼近在 $W^{2,1}_{q_c}$ 中的子序列强内部收敛。
英文摘要
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-Δu+|Du|^γ=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $γ>2$, 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 $f\in L^{q_c}$ and continuous initial data $u_0$, and prove subsequential strong interior convergence of smooth approximations in $W^{2,1}_{q_c}$.
