= Coercivity from a drift with nonpositive weak divergence
{title2=$\int_\Omega(b\cdot\nabla u)u\geq0$}
On a bounded open set, suppose $b\in L^\infty$ has <weak divergence> in $L^1_{\mathrm{loc}}$ with $\operatorname{div}b\leq0$. For $\varphi\in C_0^\infty$, testing the divergence identity with $\varphi^2/2$ gives $\int(b\cdot\nabla\varphi)\varphi=-\tfrac12\int(\operatorname{div}b)\varphi^2\geq0$. The left side is continuous in the <zero-boundary Sobolev space> $H_0^1$, by the <Poincaré inequality> and boundedness of $b$. Density therefore extends its nonnegativity to every $u\in H_0^1$. One need not assume that $(\operatorname{div}b)u^2$ is integrable. It follows that $a(u,v)=\int\nabla u\cdot\nabla v+\int(b\cdot\nabla u)v$ is a <coercive bilinear form> in the gradient norm, giving unique weak Dirichlet solvability by the <Lax-Milgram theorem>.
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