Non-Osgood Moduli of Continuity

• Non-Osgood moduli of continuity are functions with finite integrals near zero that fail the Osgood condition, characterizing sub-Lipschitz regularity in dynamics.
• They play a pivotal role in demonstrating nonuniqueness in ODE trajectories and PDE weak solutions by using methods like parallelization and fixed-point iterations.
• Examples such as r|ln r|^(-α) illustrate how logarithmic corrections lead to breaches in uniqueness, informing studies on well-posedness in transport and elliptic equations.

A modulus of continuity is a function $\omega:[0,\infty)\rightarrow[0,\infty)$ that is continuous, strictly increasing, concave, and satisfies $\omega(0) = 0$. Such functions quantify the regularity of functions or vector fields, generalizing classical notions as Lipschitz or Hölder continuity. The Osgood condition for a modulus of continuity, $\int_0^{\epsilon} \frac{dr}{\omega(r)} = \infty$ for some $\epsilon>0$, is a sharp threshold for the uniqueness of solutions to certain ODEs and PDEs: when $\omega$ fails this condition (i.e., $\int_0^{\epsilon}\frac{dr}{\omega(r)} <\infty$), it is termed a "non-Osgood modulus." These sub-Osgood rates of continuity play a central role in understanding the limits of uniqueness and regularity in nonlinear PDEs and transport phenomena.

1. The Osgood Condition and Its Sharpness

The Osgood condition arose in the theory of ordinary differential equations as the precise integrability barrier required to guarantee uniqueness for initial value problems with velocity fields of a given regularity. For an ODE of the form $\dot\gamma(t) = v(t,\gamma(t))$ with vector field $v$ continuous with spatial modulus $\omega$, a Grönwall-type argument applies if and only if $\omega$ is Osgood, i.e., when

$\int_{0^+} \frac{ds}{\omega(s)} = \infty.$

If this integral converges, the modulus is non-Osgood, and the ODE may admit multiple solutions starting from the same initial datum. This threshold is sharp: whenever the Osgood condition fails, it is possible to construct counterexamples of nonuniqueness with divergence-free vector fields in $C_t C_x^\omega$, both for ODE trajectories and for the continuity equation governing transported measures. This was established in a precise and constructive manner for arbitrary non-Osgood moduli (Colombo et al., 17 Jan 2026).

2. Characterization and Examples of Non-Osgood Moduli

Non-Osgood moduli are those for which the aforementioned integral is finite. Such functions decay sufficiently rapidly near zero. Canonical examples are logarithmic corrections or iterated logarithms multiplying a linear modulus:

• $\omega(r) = r|\ln r|^{-\alpha}$, $\alpha>1$
• $\omega(r) = r(\ln_+ r)^{1+\epsilon}$ for $\epsilon>0$
• $\omega(r) = r\ln_+ r(\ln_+\ln_+ r)^{1+\epsilon}$ for $\epsilon>0$
• $\omega(r) = 1/(\ln (1/r))^\ell$, $\ell > 0$

For these cases, the integral $\int_{0^+} dr/\omega(r)$ converges, violating Osgood and demarcating the regime where uniqueness theorems break down (Colombo et al., 17 Jan 2026, Xu, 2022). The table below summarizes illustrative examples:

Modulus $\omega(r)$	Osgood?	Fails Uniqueness
$r$	Yes	No
$r|\ln r|^{-\alpha}$, $\alpha>1$	No	Yes
$1/(\ln(1/r))^\ell$, $\ell>0$	No	Yes

3. Nonuniqueness for ODE and PDE with Non-Osgood Moduli

For any non-Osgood modulus $\omega$, there exist divergence-free velocity fields $v \in C([0,T];C^\omega(\mathbb{R}^d))$ for $d\geq 2$ such that the flow map associated to

$\dot\gamma(t) = v(t, \gamma(t)),\quad \gamma(0)=x,$

is nonunique on a set of positive (even full, within a cube) Lebesgue measure. This nonuniqueness is not limited to exceptional or singular initial points, but occurs constructively for almost every $x$ in a large set. Furthermore, the continuity equation

$$\partial_t \mu + \operatorname{div}(\mu v) = 0, \quad \mu(0) = \bar\mu \ll \mathscr{L}^d$$

admits at least two distinct weak solutions in $L^\infty_t L^1_x$, each remaining absolutely continuous with respect to Lebesgue measure for a.e. $t$, but differing for positive-time slices (Colombo et al., 17 Jan 2026). These results show that the Osgood threshold is both necessary and sufficient for the well-posedness of both trajectories and densities under only $\omega$-continuity.

4. Constructive Methods: Parallelization and Fixed-Point Frameworks

Key advances have stemmed from new techniques for building velocity fields and solutions at the critical regularity:

• Parallelization: At every time $t$, "building block" vector fields are superposed at infinitely many dyadic spatial scales. Each scale consists of velocity components supported on a nested Cantor-like family of cubes, evolving in both location and size. This method ensures that bad regularity is present at every scale and everywhere inside the support, a crucial enabler for constructing maximal nonuniqueness for all non-Osgood moduli (Colombo et al., 17 Jan 2026).
• Fixed-Point Iteration in Velocity and Density Spaces: The fixed-point machinery operates on sequences of velocity fields ($\mathcal{B}$) and measures ($\mathcal{T}$), each component compactly supported at given scales and defined with respect to explicit norms, e.g., $$\|B\|_\mathcal{B} = \sup_{k,t} W(\ell_k) \|B_t^k\|_{C^\omega}$$ for suitable weight $W$. The contraction property of the iteration maps in these norms facilitates the construction of anomalous flows and densities as unique fixed points, exploiting the scale separation built into the parallelization (Colombo et al., 17 Jan 2026).

5. Non-Osgood Modulus in Elliptic Problems: Regularity Outcomes

In the setting of singular elliptic PDEs, the regularity of weak solutions can also be dictated by non-Osgood moduli. For example, solutions $p$ to

$-\operatorname{div}[(I + m\otimes m)\nabla p] = S$

under minimal integrability (e.g., $m \in (L^4)^2$, $S\in L^q$, $q>1$) are continuous but may have modulus $\omega(r) = 1/[\ln(1/r)]^\ell$, which is non-Osgood for any $\ell>0$. The modulus can thus be arbitrarily poor among non-Osgood classes, in contrast to degenerate elliptic equations where the allowable modulus is even weaker (i.e., only up to a small power of the logarithm). The proof employs annular decompositions, comparison estimates, and barrier techniques tailored to the matrix structure and integrability constraints (Xu, 2022).

Without further structural assumptions (e.g., $m \in \mathrm{BMO}$), such solutions lack Hölder continuity entirely. The presence of a non-Osgood modulus of continuity ensures some regularization and excludes wild oscillations, but cannot guarantee the stronger uniqueness or regularity results that would follow from Osgood moduli.

6. Implications and Broader Context

Non-Osgood moduli mark the sharp limit of the regularizing effect conferred by continuity of coefficients or vector fields in PDE and ODE contexts:

• For ODEs and the continuity equation, the failure of the Osgood criterion is both necessary and sufficient for the possibility of nonuniqueness in physically relevant classes—i.e., positive-measure sets of trajectories and $L^1$ densities (Colombo et al., 17 Jan 2026).
• In elliptic PDEs, non-Osgood moduli provide a minimal regularity level consistent with very weak assumptions, but do not generally ensure strong regularity (e.g., Hölder) or improved qualitative properties (Xu, 2022).
• The techniques developed—scalewise parallelization and scale-sensitive fixed-point spaces—provide a flexible framework capable of capturing all non-Osgood cases and yield new benchmarks for the limits of well-posedness with minimal assumptions.

A plausible implication is that the presence of a non-Osgood modulus of continuity should be viewed as a precise threshold both for loss of classical uniqueness and for the minimal continuity outcome possible under highly degenerate or weakly regular scenarios.

• "Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields" (Colombo et al., 17 Jan 2026)
• "Modulus of continuity of weak solutions to a class of singular elliptic equations" (Xu, 2022)