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

We establish a two-dimensional non-endpoint local-defect theory for scalar non-divergence advection-diffusion operators $Lu=-a:D^2u+b\cdot\nabla u$, $a=a^{\rm per}+a^{\rm e}$, $b=b^{\rm per}+b^{\rm e}$, with Holder periodic background and Holder local defects satisfying $a_{ij}^{\rm e}\in L^r(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$, $b_i^{\rm e}\in L^s(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$, $1<r,s<2$. The main estimate is a whole-space bound for $L_t=-a_t:D^2+b_t\cdot\nabla$ in the range $1<q<2$, with $q^*$ defined by $1/q^*=1/q-1/2$. The two-dimensional difficulty is that the periodic drift cannot be treated by the high-dimensional argument of Blanc--Le Bris--Lions. We remove it by periodic harmonic coordinates $P=x+\chi$, $L_{\rm per}P_\alpha=0$. In these variables the blow-down equation has a small local $L^2$ drift, which yields a finite-energy Liouville theorem and closes the continuation argument. The same coordinates reduce the invariant-measure source to a planar Hodge problem of the form $H+\operatorname{div} Q$, and a Piola pull-back gives the final divergence-form representative $mLu=-\operatorname{div}((ma-B)\nabla u)$. Thus the central estimate, correctors, invariant measure and divergence-form reduction hold in the scalar regular non-endpoint regime.