The scalar Oseen operator −Δ+∂/∂x1-\Delta + {\partial}/{\partial x_1} in R2\mathbb{R}^2

summary:This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in $L^{p}$ theory