A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization

Abstract

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 Hölder 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 \(1<r,s<2\). Under the periodic centering condition \(\langle m^{\mathrm{per}}b^{\mathrm{per}}\rangle=0\), global harmonic coordinates remove the periodic drift. After blow-down, the transformed defect drift is small in the critical local \(L^2\) space. Critical-drift compactness then yields a finite-energy Liouville theorem and closes the continuation argument, giving a whole-space estimate for \(1<q<2\) and \(1/q^*=1/q-1/2\). This estimate provides defect correctors and, by duality, a positive invariant density \(m=m^{\mathrm{per}}+m^e\). A planar Hodge construction in harmonic coordinates, followed by a Piola pull-back, produces a skew-symmetric field \(B=B^{\mathrm{per}}+B^e\) such that \(mLu=-\operatorname{div}((ma-B)\nabla u)\). If \(M=\max\{r,s\}\) and \(M^*=2M/(2-M)\), then, for some \(β>0\), the defects \(B^e\) and \(A^e:=ma-B-(m^{\mathrm{per}}a^{\mathrm{per}}-B^{\mathrm{per}})\) belong to \(L^{M^*}\cap L^\infty\cap C_{\mathrm{unif}}^{0,β}\) and vanish uniformly at infinity. This supplies the missing two-dimensional structural reduction in the scalar regular non-endpoint regime.