Uniqueness of Rough Continuity Equations

Definitions and mathematical statement

The supplied record asks about the continuity equation \[ \partial_t u+\operatorname{div}_x(bu)=0 \] in spatial dimension $d$, with a bounded, divergence-free vector field $b\in W^{1,p}$ and a density $u\in L^q$, in the strict exponent regime \[ 1<\frac1p+\frac1q<1+\frac1d. \] The record does not specify the spatial domain, time interval, temporal regularity, initial data, boundary conditions, or the precise meanings of these function-space memberships. Consequently, the following describes the mathematical target but does not supply a fully specified initial-value problem.

To explain the distributional equation locally, write $I$ for an unspecified open time interval and $\Omega\subseteq{\mathbb{R}}^d$ for an unspecified open spatial domain. These are placeholders, not a choice of the intended global setting. Provided $u,bu\in L^1_{\mathrm{loc}}(I\times\Omega)$, the distributional formulation is \[ \int_I\int_\Omega \left(u\,\partial_t\varphi+bu\cdot\nabla_x\varphi\right)\,{\,\mathrm{d}} x\,{\,\mathrm{d}} t=0 \qquad\text{for every }\varphi\in C_c^\infty(I\times\Omega). \] Divergence-free means $\operatorname{div}_x b=0$ in distributions. The usual spatial interpretation of $W^{1,p}$ requires the vector field and its first weak spatial derivatives to belong to $L^p$; the record does not say whether this membership is global or local, or how it depends on time. Likewise, the temporal and spatial scope of $u\in L^q$ is unspecified.

Question as supplied. Throughout the displayed strict exponent regime, does every bounded divergence-free vector field in the intended $W^{1,p}$ class have uniqueness of distributional solutions in the intended $L^q$ class, and is every such solution Lagrangian?

Here uniqueness would require that any two admissible solutions for the same vector field, with the same prescribed initial data and any required boundary data, agree almost everywhere. No such data or trace class is supplied. Lagrangianity refers to representation by transport along trajectories of $b$: a trajectory $X(\cdot,x)$ is absolutely continuous and satisfies \[ \frac{{\,\mathrm{d}}}{{\,\mathrm{d}} t}X(t,x)=b(t,X(t,x)) \quad\text{for almost every }t. \] For a divergence-free field, a characteristic representation would express preservation of the transported density along these trajectories, with the appropriate almost-everywhere and initial-trace interpretation. The source does not specify whether it intends a regular Lagrangian flow, a representation by measures on trajectories, or another precise formulation. No particular flow notion or missing hypothesis is imposed here.

Short English statement

For bounded divergence-free Sobolev vector fields, the supplied question asks whether distributional solutions of the continuity equation are unique and transported by trajectories whenever $1<1/p+1/q<1+1/d$. The domain, time-dependent function spaces, initial and boundary data, and meaning of Lagrangianity remain unspecified. The supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004147$, title Uniqueness of Rough Continuity Equations, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2019.
• Supplied report attribution: Nonlinear Hyperbolic Problems: Modeling, Analysis, and Numerics (2020), Oberwolfach Reports, pp. 1419--1497, DOI 10.4171/owr/2019/24. Supplied source link: https://doi.org/10.4171/owr/2019/24. The background attributes the collection to workshop open-problem discussions. No specific page, contribution, or numbered passage supporting this formulation is supplied.
• Supplied literature reference, labelled Caravenna--Crippa: https://arxiv.org/abs/1608.04324.
• Supplied literature reference, labelled Comptes Rendus version: https://doi.org/10.1016/j.crma.2016.10.009.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports uniqueness and Lagrangianity under several hypotheses, including some lack-of-integrability regimes, while describing the full stated distributional claim as unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves the target. These are supplied provenance claims, not mathematical verification by this draft.