Estimates for Green functions of Stokes systems in two dimensional domains

Abstract

We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the L 1 -mean oscillations of the coefficients.2010 Mathematics Subject Classification. 76D07, 35R05, 35J08.

Green functions for pressure of Stokes systems

“…In the theorem below, we extend the results in [8] and [9] by showing the pointwise bounds (7.4) and (7.5) under the stronger assumption that the coefficients are of Dini mean oscillation in a domain having a C 1,Dini boundary. The corresponding results for the case with d = 2 was proved in [7]. Moreover, for any x, y ∈ Ω with x = y, we have…”

“…, we refer the reader to [8,9,7]. In [8], the authors established the existence and pointwise bound of the Green function on a bounded C 1 domain when d ≥ 3 and A αβ have vanishing mean oscillations (VMO).…”

“…The corresponding results were obtained in [9] in the whole space and a half space when A αβ are merely measurable in one direction and have small mean oscillations in the other directions (partially BMO). In [7], the authors constructed the Green function in a two dimensional domain when A αβ are measurable and bounded. They also considered pointwise bounds of the Green function and its derivatives.…”

Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

“…Boundary Domain Integral Equations appear naturally when applying the Boundary Integral Method to boundary value problems with variable coefficient. These class of boundary value problems has a wide range of applications in Physics or Engineering, such as, heat transfer in non-homogeneous media [28], motion of laminar fluids with variable viscosity [5], or even in the acoustic scattering by inhomogeneous anisotropic obstacle [6].…”

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

“…Boundary Domain Integral Equations appear naturally when applying the Boundary Integral Method to boundary value problems with variable coefficient. This class of boundary value problems has a wide range of applications in Physics or Engineering, such as, heat transfer in non-homogeneous media [27], motion of laminar fluids with variable viscosity [5], or even in the acoustic scattering by inhomogeneous anisotropic obstacle [6].…”