We study weighted $L^2$ Hessian estimates under diffusion marginals, allowing the equation's principal matrix to be chosen independently. For regular uniformly elliptic coefficients, bounded reference drift and compactly supported doubling initial laws, the optimal large-damping constant on the full interval is the maximum of initial, flat and propagation contributions. Finiteness yields weighted Sobolev well-posedness and a contraction criterion for nonlinear perturbations. An explicit operator formula determines the initial contribution; propagation bounds become exact under uniform directional limits or asymptotic isotropy at infinity. Under geometric assumptions including nonnegative Ricci curvature, propagation is determined by terminal momenta of action-minimizing paths. For reference covariance $a$, matching principal matrix $a/2$ and Hessian normalization $a^{1/2}D^2u\,a^{1/2}$, every initial law gives a finite estimate. Without an additional time weight, the limiting constant lies between $2$ and $2\sqrt2$: finite atomic laws attain the upper bound, while nondegenerate Gaussian laws and their finite mixtures attain the lower bound.
We establish the asymptotic-preserving property of the exponential Euler method for stochastic reaction-diffusion-advection equations in $\mathbb R^2$ under fast Hamiltonian advection with multiple critical points. In the fast-advection limit, the dynamics reduces to a stochastic partial differential equation on a nonc...
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 $d\geq 2$, $\gamma>2$, 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, unif...
We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally H\"older $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this visco...
For anisotropic diffusion with mixed derivatives, standard compact coordinate-aligned stencils and some local finite-volume or finite-element constructions can lose nonnegative coefficients unless suitable coefficient or mesh conditions are imposed. Here nonnegative coefficients are generated directly from conditional...
We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth p...
We study convergence rates when periodic homogenization and vanishing viscosity occur at the same scale in Hamilton--Jacobi equations whose momentum Hessians are positive definite at every point. For bounded Lipschitz initial data and a class of smooth Hamiltonians, we establish an optimal rate $O(\varepsilon|\log\vare...
Kai-Zhao Qin· 0 citations
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