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 $C=1$. 

