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  "body": "\n## Statement\n\nLet $D \\subseteq \\R \\times \\R^n$ be open and let $f : D \\to \\R^n$ be continuous.\nLet $\\omega : [0, \\infty) \\to [0, \\infty)$ be continuous and non-decreasing, with $\\omega(0) = 0$ and $\\omega(r) \u003e 0$ for $r \u003e 0$, and with\n$$\\int_0^{\\varepsilon} \\frac{dr}{\\omega(r)} = \\infty \\quad \\text{for some } \\varepsilon \u003e 0.$$\nAssume that $\\norm{f(t, x) - f(t, y)} \\le \\omega(\\norm{x - y})$ whenever $(t, x), (t, y) \\in D$.\nLet $I$ be an interval and let $\\gamma_1, \\gamma_2 : I \\to \\R^n$ be [[solutions|04SW7WECS8]] of $x' = f(t, x)$ whose graphs lie in $D$.\nIf $\\gamma_1(t_0) = \\gamma_2(t_0)$ for one $t_0 \\in I$, then $\\gamma_1 = \\gamma_2$ on $I$.\n\n## Remarks\n\nProof 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 $\\delta \u003e 0$.\nThe [[Bihari–LaSalle inequality|9KMJSKNZ72]] with $c = \\delta$ gives $\\int_\\delta^{u(t)} dr / \\omega(r) \\le t - t_0$ when $u(t) \u003e \\delta$.\nIf $u(t) \u003e 0$, let $\\delta \\to 0$: the left side tends to $\\infty$, a contradiction.\nThe Lipschitz case is $\\omega(r) = K r$.\nA weaker modulus that still works is $\\omega(r) = r \\ln(1/r)$ for $0 \u003c r \\le 1/e$, extended by the constant $1/e$ for $r \u003e 1/e$; here $\\int_0^{1/e} dr / (r \\ln(1/r))$ diverges.\nThe 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 $t \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nUniqueness of solutions usually comes from a Lipschitz condition on the right side.\nOsgood's criterion shows that a slightly weaker modulus of continuity is enough.\nWhat matters is whether a solution of $u' = \\omega(u)$ can leave $0$ in finite time; when $\\int_0 dr/\\omega(r)$ diverges, it cannot.\n\n### L2 human\n\nThe criterion extends [[uniqueness for Lipschitz vector fields|025PKW8AMR]], and it is sharp in the sense of the example $2\\sqrt{\\abs{x}}$.\nLog-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.\n",
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