In Majda and Bertozzi, Vorticity and Incompressible Flow on page 32, the orthogonal decomposition of a vector field $v\in L^2(\Bbb R^n)\cap C^\infty(\Bbb R^n)$ is proved, namely $$v=w+\nabla q, \quad w\bot_{L^2} \nabla q,$$ with both $w$ and $\nabla q$ in $L^2\cap C^\infty$. The proof is stated ...
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