In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult.
The context is the following:
- a smooth bounded domain $\Omega \subset \mathbb{R}^3$
- $|\cdot|_s$ denotes the Sobolev norm of the space $W^{s,2}(\Omega)=H^2(\Omega)$ and $|\cdot|_\infty$ the norm in $L^{\infty}(\Omega)$
- $u$ is a vector valued function (the velocity of a fluid)
This has to be one of the many imbedding theorems which should give $$|\nabla u|_\infty \leq C \: |u|_3$$ where $C$ is a constant depending on $\Omega$ alone I suppose.
I'd appreciate if you can give me a reference as well. ThX!
I'm assuming that your function $u$ belongs to $W^{3,2}(\Omega)$. We have the following result that can be found in any good book of Sobolev spaces (for example in Leoni's book or even in Brezi's book, but in the later you have to iterate the estimates that he find only to $W^{1,p}(\Omega)$).
If $\Omega$ is a bounded regular domain, $p\geq 1$, $k>\frac{n}{p}$ then $$W^{k,p}(\Omega)\hookrightarrow C^m(\overline{\Omega}),\ \forall\ 0\leq m<k-\frac{n}{p}$$
In your case: $p=2$, $n=3$ and $k=3$, then $m\in[0,3/2]$, which implies your result.