= Solution
A <weak solution> belongs to the <zero-boundary Sobolev space> $H_0^1(\Omega)$ and satisfies
$$
\boxed{\int_\Omega\nabla u\cdot\nabla\varphi\,dx+\int_\Omega(b\cdot\nabla u)\varphi\,dx=\int_\Omega f\varphi\,dx\quad(\varphi\in H_0^1(\Omega)).}
$$
The boundedness of $b$ and the <Cauchy-Schwarz inequality> make the drift term integrable. This <weak formulation> moves one derivative from the <Laplacian> onto the <test function>, while retaining the first-order drift term.
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