A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization
We develop a two-dimensional structural local-defect theory for scalar non-divergence advection--diffusion operators \(Lu=-a:D^2u+b\cdot\nabla u\), where \(a=a^{\mathrm{per}}+a^e\) and \(b=b^{\mathrm{per}}+b^e\). The periodic coefficients and the bounded local defects are uniformly H\"older continuous, the interpolating matrices \(a_t=a^{\mathrm{per}}+t a^e\) are symmetric and uniformly elliptic, and \(a^e\in L^r(\mathbb{R}^2)\), \(b^e\in L^s(\mathbb{R}^2)\), where \(1 0\), the defects \(B^e\) and \(A^e:=ma-B-(m^{\mathrm{per}}a^{\mathrm{per}}-B^{\mathrm{per}})\) belong to \(L^{M^*}\cap L^\infty\cap C_{\mathrm{unif}}^{0,\beta}\) and vanish uniformly at infinity. This supplies the missing two-dimensional structural reduction in the scalar regular non-endpoint regime.