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

Abstract

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 \(P=x+χ\), \(L_{\rm per}P_α=0\). 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.