标量非散度对流扩散均匀化的二维结构局部缺陷理论
A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization
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中文总结 AI 辅助
建立二维非端点局部缺陷理论，针对含Holder周期背景和局部缺陷的标量非散度对流扩散算子。核心方法是用周期调和坐标处理周期漂移，主要贡献是给出相关估计、校正器、不变测度及散度形式约简。
中文摘要 AI 辅助
我们为标量非散度对流扩散算子\(Lu = -a:D^2u + b\cdot\nabla u\)（其中\(a = a^{\rm per} + a^{\rm e}\)，\(b = b^{\rm per} + b^{\rm e}\)）建立了二维非端点局部缺陷理论，具有Holder周期背景和满足特定条件的Holder局部缺陷。主要估计是在\(1 < q < 2\)范围内对\(L_t = -a_t:D^2 + b_t\cdot\nabla\)的全空间界。二维的困难通过周期调和坐标\(P = x + \chi\)，\(L_{\rm per}P_\alpha = 0\)解决。在这些变量中， blow - down方程有小的局部\(L^2\)漂移，这产生了有限能量Liouville定理并完成了延拓论证。相同的坐标将不变测度源简化为\(H + \operatorname{div} Q\)形式的平面Hodge问题，并且Piola拉回给出了最终的散度形式代表\(mLu = -\operatorname{div}((ma - B)\nabla u)\)。因此，中心估计、校正器、不变测度和散度形式约简在标量正则非端点 regime 中成立。
英文摘要
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.
