examples/navier-stokes.md

 Navier-Stokes equation
 ======================
-We consider the incompressible Navier-Stokes equation in a rectangular domain $`\Omega=(0,Lx)\times(0,Ly)`$,
+We consider the incompressible Navier-Stokes equation in a rectangular domain $`\Omega=(0,L_x)\times(0,L_y)`$,
 ```math
-\partial_t\mathbf{u} - \nabla\cdot\big(\nu\mathbf{D}(\mathbf{u})\big) + (\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla p = \mathbf{f},\qquad\text{ in }\Omega \\
+\varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u}) - \nabla\cdot\big(2\nu\mathbf{D}(\mathbf{u})\big) + \nabla p = \mathbf{f},\qquad\text{ in }\Omega \\
 \nabla\cdot\mathbf{u} = 0
 ```
-with velocity $`\mathbf{u}`$ and pressure $`p`$, w.r.t. to boundary conditions $`\mathbf{u}=\mathbf{g}`$ on $`\partial\Omega`$.
+with velocity $`\mathbf{u}`$, pressure $`p`$, dynamic viscosity $`\nu`$, 
+rate-of-strain tensor $`\mathbf{D}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)`$, 
+and external volume force $`\mathbf{f}`$, w.r.t. to boundary conditions $`\mathbf{u}=\mathbf{g}`$ on $`\partial\Omega`$.
+For a coarse overview, see also [Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations).
+
+In a weak variational formulation, we try to find $`\mathbf{u}(t,\cdot)\in V:=H^1(\Omega)^d`$ and $`p(t,\cdot)\in Q:=L^2(\Omega)`$, such that
+```math
+\int_\Omega \varrho(\partial_t\mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u})\cdot\mathbf{v} + \nu\nabla\mathbf{u}:\nabla\mathbf{v} - p\nabla\cdot\mathbf{v}\,\text{d}\mathbf{x} = \int_\Omega \mathbf{f}\cdot\mathbf{v}\,\text{d}\mathbf{x}\qquad\forall \mathbf{v}\in V, \\
+\int_\Omega q\nabla\mathbf{u}\,\text{d}\mathbf{x} = 0\qquad \forall q\in Q.
+```