= Osgood uniqueness criterion
{c}
{title2=$\int_{0^+}\frac{dr}{\mu(r)}=\infty$}
If a continuous nondecreasing modulus $\mu$ satisfies this divergence condition, an estimate $|b(t,x)-b(t,y)|\leq A(t)\mu(|x-y|)$ with locally integrable $A$ gives uniqueness of <characteristic curves>. Apply a regularized integral comparison to their separation. For the <log-Lipschitz modulus>, $\delta'(t)\leq A(t)\delta(1-\log\delta)$ gives $\delta(t)\leq\exp(1-(1-\log\delta(0))\exp(-\int_0^tA))$ while $\delta\leq1$. Zero initial separation stays zero. Unlike the usual linear <Gronwall inequality>, this criterion allows non-Lipschitz vector fields.
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