Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields

TL;DR

The paper establishes the sharpness of the Osgood criterion for the continuity/transport equation under divergence-free vector fields by constructing, for any non-Osgood modulus ω, a velocity field v ∈ C([0,T];C^ω(ℝ^d)) that induces non-unique flow maps on a set of positive-measure initial data and yields two distinct measure-valued solutions μ^1, μ^2 from the same initial datum. The core methodology combines a novel parallelization across multiple nested spatial scales with a fixed-point framework that couples a multiscale velocity field to an evolving density, producing anomalous solutions in L^1(ℝ^d) a.e. in time. The construction first demonstrates non-uniqueness of trajectories on a full-measure subset via a Cantor-set-driven base field and time-reversal, then strengthens this to non-uniqueness of transported densities through a two-layer fixed-point argument that yields two distinct solutions to the continuity equation starting from the same initial data. The results significantly advance our understanding of well-posedness thresholds for transport-type PDEs in incompressible flows and open avenues for applying parallelization-inspired multiscale techniques in convex integration frameworks and related fluid-dynamics problems.

Abstract

For any modulus of continuity $ω$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^ω_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial conditions $E$ of positive Lebesgue measure. In fact, the set $E$ has full measure inside a cube where the construction is supported. Moreover, we also construct a divergence-free velocity field $v \in C_{t}C^ω_x$ for which the associated continuity equation admits two distinct solutions $μ^1$ and $μ^2$ which are absolutely continuous with respect to Lebesgue measure for almost every time, and start from the same initial datum $\bar μ\ll \mathscr{L}^{d}$. Our construction introduces two novel ideas: (i) We introduce the notion of "parallelization", where at each time, the velocity field consists of simultaneous motion across multiple nested spatial scales. This differs from most explicit constructions in the literature on mixing or anomalous dissipation, where the velocity on different scales acts at separate times. This is crucial to cover the whole class of non-Osgood moduli of continuity. (ii) Inspired by a recent work of Bruè, Colombo and Kumar, we develop a new fixed-point framework that naturally incorporates the parallelization mechanism. This framework allows us to construct anomalous solutions of the continuity equation that belong to $L^1(\mathbb{R}^d)$ a.e. in time.

Figures (4)

• Figure 1: illustrates the positions and sizes of Cantor cubes of different generations as they are translated in space by the velocity field $b$. The picture resembles Figure 2 in K2023. However, there is one crucial difference. In our construction, Cantor cubes from all generations move simultaneously, rather than sequentially. As a result, the size of a Cantor cube at a given generation must expand in time to accommodate the increasing separation of the smaller cubes nested within it. This is clearly visible in the figure. Throughout the construction, the ratio of sizes between successive generations is maintained at $1/4$. When the $n$th generation cubes complete their translation at time $t = t_n$, their size becomes one half of the size of the dyadic cube of the same generation.
• Figure 2: depicts the superposition of building block velocity fields that translate Cantor cubes of two successive generations. For clarity, only two levels of superposition are shown. In the actual construction, this superposition extends across infinitely many generations. The figure emphasize two essential properties of the construction. (i) Even under superposition, each inner Cantor cube is subject only to a uniform velocity field and therefore translates rigidly. (ii) As the separation between inner cubes grows during the evolution, the size of the enclosing cube must increase to maintain the nested structure of the Cantor set. Consequently, the outer cube expands as it translates.
• Figure 3: Time decomposition of the interval $[0, 1]$ into subintervals of length $2 \tau^k_m$ for $m \in \mathbb{N}$. At time $t = 0$, the density is concentrated on cubes of side length $\ell_{k+1}$. By the time $t = 1$, the density is uniformly distributed inside a cube of side length $\ell_k$. In the first half of the interval of length $2 \tau^k_1$, the "breaking" process takes place. At time $t = \hat{t}^{k, f}_{N^k_2}$, the density concentrates on $2^{N^k_2 d}$ cubes. These cubes are not yet uniformly distributed. Instead, they are contained in several generations of nested fictitious cubes (shown in yellow in Panel 2), in an arrangement analogous to the Cantor-type construction from the previous section. In the second half of the interval, by time $t=t^{k,f}_{N_2^k}$, the small cubes are translated in parallel and arranged uniformly on a dyadic grid (as illustrated in Panel 3). The same "breaking" and "parallel translation" procedure is repeated on each interval of length $2\tau_m^k$. \ref{['fig: time series 2']} illustrates the corresponding subdivision of a generic $2\tau_m^k$ interval.
• Figure 4: shows an expanded view of an interval of length $2 \tau^k_m$ from \ref{['fig: time series 1']}. It is divided into two halves. The first half consists of consecutive subintervals of the form $[\hat{t}^{k,s}_n, \hat{t}^{k,s}_n)$, while the second half consists of nested subintervals of the same form $[\hat{t}^{k,s}_n, \hat{t}^{k,s}_n)$ for $n \in \{N^k_m +1, \cdots N^k_{m+1}\}$.

Theorems & Definitions (22)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 3.1