Osgood uniqueness criterion

Remarks

Proof idea for $t \ge t_0$: the function $u = \norm{\gamma_1 - \gamma_2}$ satisfies $u(t) \le \delta + \int_{t_0}^t \omega(u(s))\, ds$ for every $\delta > 0$. The Bihari–LaSalle inequality with $c = \delta$ gives $\int_\delta^{u(t)} dr / \omega(r) \le t - t_0$ when $u(t) > \delta$. If $u(t) > 0$, let $\delta \to 0$: the left side tends to $\infty$, a contradiction. The Lipschitz case is $\omega(r) = K r$. A weaker modulus that still works is $\omega(r) = r \ln(1/r)$ for $0 < r \le 1/e$, extended by the constant $1/e$ for $r > 1/e$; here $\int_0^{1/e} dr / (r \ln(1/r))$ diverges. The divergence hypothesis is needed: $f(x) = 2\sqrt{\abs{x}}$ satisfies the condition with $\omega(r) = 2\sqrt{r}$, whose reciprocal is integrable at $0$, and $x' = 2\sqrt{\abs{x}}$, $x(0) = 0$ has the two solutions $x = 0$ and $x = t^2$ for $t \ge 0$.

Going deeper

For a student

The criterion extends uniqueness for Lipschitz vector fields, and it is sharp in the sense of the example $2\sqrt{\abs{x}}$. Log-Lipschitz moduli such as $r \ln(1/r)$ appear in fluid mechanics, where velocity fields of the two-dimensional Euler equations with bounded vorticity have this regularity and still have unique trajectories.