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Consider functions v:R^d->R^d (d is dimension 2 or 3 and v is a velocity field) and v in X=H_0^1 (Sobolev space). I would like to prove the inequality ||div(v)||_X <= d^(1/2)||grad(v)||.

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Since $v \in H_0^1$ there will be a case with $\|\nabla \cdot v\|_{H_0^1} = \infty$ (since the norm needs more regularity than $v$ may have).

The best you can hope for, in general, is $\| \nabla \cdot v \| \leq C\|\nabla v \|$, both norms being in $L^2$, which is true with $C=1$.

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But how to prove this estimate? I have seen in books C=\sqrt{d}. –  withoutname Oct 22 '12 at 8:33
    
Let $\| \cdot \|$ be the $\ell_1$ norm on $\mathbb R^d$. Then the right hand side is the left hand side plus some extra nonnegative terms. Is $\| \cdot \|$ is the standard norm then you will probably get the $\sqrt{d}$ factor, but this is a basic exercise in the equivalence of Euclidean norms. –  dcs24 Oct 29 '12 at 10:13
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