1. Introduction & Purposes

We will study the gradient estimates of the Stokes system in several settings: interior, global, local boundary with zero BC and local boundary with Navier BC, in both stationary and time dependent cases. An important distinction is emphasized on whether we assume any pressure bound.

2. Outline & Descriptions

The following is the tentative list of topics:

A.       Interior estimates

(1)     Fundamental solution in whole space

(2)     Interior estimates by cut-off

(3)     Serrin's parasitic solution

(4)     Interior estimates with no pressure condition

B.        Global estimates in a domain (sketch only)

(1)     Maximal regularity

(2)     semigroup theory and resolvent estimates

(3)     half space: Poisson kernel and Green tensor

C.        Boundary estimates, under no-slip BC

(1)     stationary case, estimates for flat and curved boundaries

(2)     non-stationary case, estimates with pressure condition

(3)     non-stationary case, counterexamples with no pressure condition

D.       Boundary estimates, under Navier BC

(1)     stationary case, estimates for flat and curved boundaries

(2)     non-stationary case, estimates with and without pressure condition

(3)     non-stationary case, blowup of second derivative

3. Prerequisites

First year graduate course in PDE, including Laplace and heat equations, Sobolev spaces

4. Registration

https://forms.gle/uXrBmrbV4BVB9zbc9