From 2ecb89470380589a32179cbff22a898b47bce199 Mon Sep 17 00:00:00 2001
From: Fabian Gabel <fabian.gabel@tuhh.de>
Date: Mon, 19 Apr 2021 13:58:32 +0200
Subject: [PATCH] add files for navier-stokes on lipschitz domains

---
 .gitlab-ci.yml          |  1 +
 topics/navier-stokes.md | 40 ++++++++++++++++++++++++++++++++++++++++
 2 files changed, 41 insertions(+)
 create mode 100644 topics/navier-stokes.md

diff --git a/.gitlab-ci.yml b/.gitlab-ci.yml
index 73572b0..311cee4 100644
--- a/.gitlab-ci.yml
+++ b/.gitlab-ci.yml
@@ -27,6 +27,7 @@ pandoc:
     # build research topics
     - pandoc --standalone build/topic-template.md -o build/topic-template.html  --mathjax
     - pandoc --standalone build/aperiodSchr.md -o build/aperiodSchr.html  --mathjax
+    - pandoc --standalone build/navier-stokes.md -o build/navier-stokes.html  --mathjax
     # build index
     - pandoc --standalone build/index.md -o build/index.html --mathjax
     # build static working group pages
diff --git a/topics/navier-stokes.md b/topics/navier-stokes.md
new file mode 100644
index 0000000..30df1c2
--- /dev/null
+++ b/topics/navier-stokes.md
@@ -0,0 +1,40 @@
+# Stokes Operator on Lipschitz Domains
+
+### Working Groups: aa
+
+### Collaborators (MAT): fgabel 
+
+### Collaborators (External): [Patrick Tolksdorf](https://www.funktionalanalysis.mathematik.uni-mainz.de/patrick-tolksdorf/)
+
+## Description
+
+
+In the solution theory for nonlinear partial differential equations, an integral part of the solution process is often to develop a semigroup theory for the linearization of the equation.
+In the case of the famous *Navier-Stokes equations* which for a given domain $\Omega \subseteq \mathbb{R}^d$, $d \geq 2$, describe the behavior of a Newtonian fluid over time, the linearization is given by the *Stokes equations*
+
+$$
+  \partial_t u - \Delta u + \nabla \pi = 0 \quad\text{in } \Omega\,, \;t > 0\,, \quad
+  \operatorname{div}(u) = 0 \quad\text{in } \Omega\,,\; t > 0\,,  
+$$
+
+$$
+  u(0) = a \text{ in } \Omega\,, 
+  u = 0 \text{ on } \partial\Omega\,,\; t > 0\,,
+$$
+
+where $u \colon \mathbb{R}^+ \times \Omega \to \mathbb{R}^d$ stands for the velocity field and $\pi \colon \mathbb{R}^+ \times \Omega \to \mathbb{R}$ represents the pressure of the fluid.
+The so-called *Stokes semigroup* $(\mathrm{e}^{-tA})_{t \geq 0}$ describes the evolution of the velocity $u$ and the *Stokes operator* $A$ corresponds to the term ''$-\Delta u + \nabla \pi$'' in the Stokes equations. 
+
+Having a semigroup makes it possible to look for *mild solutions* to the Navier-Stokes equations using a variation of constants formula to construct an iteration method.
+This approach was introduced by Fujita and Kato [1] and builds mainly on resolvent estimates for the Stokes operator $A$ and the analyticity property of the Stokes semigroup.
+
+## References
+
+[1] Fujita, H. and Kato, T. On the Navier-Stokes initial value problem I. Archive for Rational Mechanics and Analysis 16(1964), 269–315.
+
+[2] Tolksdorf, P. On the Lp-theory of the Navier-Stokes equations on Lipschitz domains. PhD thesis, Technische Universität Darmstadt, 2017.  Available at http://tuprints.ulb.tu-darmstadt.de/5960/.
+
+[3] Gabel, F. On Resolvent Estimates in Lp for the Stokes Operator in Lipschitz Domains. Master thesis, Technische Universität Darmstadt, 2018.
+
+[4] Gabel, F. and Tolksdorf, P. The Stokes operator in two-dimensional bounded Lipschitz domains. *In preparation.*
+
-- 
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