reference request - Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary condition is present? That is, given a smooth generalized solution of

$-nu bigtriangleup v + bigtriangledown q = f$ on $Omega subset mathbb{R}^3$

$bigtriangledown cdot v = 0$ on $Omega$

$S(v,q) = 0$ on $partial Omega$ where $S_i(v,q) =q n_i - nu sum_{j=1}^3 (partial_i v_j + partial_j v_i)n_j$ for $i=1,2,3$

how can it be shown that the zero-traction boundary condition is met? It's not difficult to show that the first two equations are satisfied on $Omega$ and using the relevant Green's formula one can obtain

$int_{partial Omega} S(v,q) cdot phi = 0$

for all solenoidal $phi in H^1$. However, I can't quite figure out why this necessarily leads to $S(v,q)=0$.

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