JunZhi Cui | Searcharxiv

arXiv2011

Existence of Rosseland equation

The global boundness, existence and uniqueness are presented for the kind of Rosseland equation with a small parameter. This problem comes from conduction-radiation coupled heat transfer in the composites; it's with coefficients of high order growth and mixed boundary conditions. A linearized map is constructed by fixing the function variables in the coefficients and the right-hand side. The solution to the linearized problem is uniformly bounded based on De Giorgi iteration; it is bounded in the Hölder space from a Sobolev-Campanato estimate. This linearized map is compact and continuous so that there exists a fixed point. All of these estimates are independent of the small parameter. At the end, the uniqueness of the solution holds if there is a big zero-order term and the solution's gradient is bounded. This existence theorem can be extended to the nonlinear parabolic problem.

math-ph↗

arXiv2011

Regularity of the correctors and local gradient estimate of the homogenization for the elliptic equation: linear periodic case

$C^α$ and $W^{1,\infty}$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation with piecewise smooth coefficients. If the data are smooth enough, the error of the first-order expansion for piecewise smooth coefficients is locally $O(ε)$ in the Hölder norm; it is locally $O(ε)$ in $W^{1,\infty}$ when coefficients are Lipschitz continuous. It can be partly extended to the nonlinear parabolic equation.

math.AP↗