Sharp conditions for proper blow-up and global existence of maximal regular flows for non-smooth vector fields

The local version of the DiPerna–Lions theory built by Ambrosio, Colombo and Figalli (Archive for Rational Mechanics and Analysis, 2015) produces an almost-everywhere unique maximal regular flow for vector fields satisfying only local integrability, local one-sided divergence bounds, and local well-posedness of the continuity equation, but the behavior of trajectories near their maximal existence time depends on global information: global bounded compression forces proper (non-oscillatory) blow-up, while under purely local divergence bounds trajectories can oscillate between bounded and unbounded norm before a finite maximal time, already for autonomous Sobolev fields in dimensions $d \ge 3$; autonomous planar $BV_{\mathrm{loc}}$ fields are nevertheless protected. Three linked questions remain open: (1) the sharp global hypotheses — necessary and sufficient, or strictly weaker than global bounded compression — that force proper blow-up; (2) the regularity classes and structural assumptions under which local hypotheses alone prevent oscillation, including the anticipated non-autonomous planar $BV_{\mathrm{loc}}$ counterexample and higher-dimensional mechanisms; (3) minimal Eulerian criteria guaranteeing global existence of the flow without the classical growth condition, beyond the case-by-case constructions available for Vlasov-type systems.

Background

The study of the ordinary differential equation $\dot x(t) = \boldsymbol b(t, x(t))$ when the velocity field $\boldsymbol b$ is not smooth originates with the DiPerna–Lions theory (1989), which establishes existence and uniqueness of a flow in the almost-everywhere sense (with respect to Lebesgue measure $\mathcal L^d$) when $\boldsymbol b(t,\cdot)$ is Sobolev or has bounded variation and satisfies suitable global bounds; Ambrosio (2004) extended the theory to $BV$ vector fields. Here a vector field is of bounded variation, written $b \in BV_{\mathrm{loc}}$, if it is locally integrable and its distributional derivative is a locally finite measure; $W^{1,p}_{\mathrm{loc}}$ denotes the class of locally integrable fields whose weak first derivatives lie in $L^p_{\mathrm{loc}}$; and $\operatorname{div} b = \sum_i \partial_{x_i} b_i$ is understood in the sense of distributions. A regular Lagrangian flow is an almost-everywhere-defined flow solving the ODE for $\mathcal L^d$-a.e. initial point and enjoying a bounded compression property, meaning that the push-forward of Lebesgue measure through the flow is controlled by a constant times Lebesgue measure; this substitutes for the volume control that smooth flows with bounded divergence possess automatically. A key structural asymmetry remained: the classical Cauchy–Lipschitz theory is purely local — existence and uniqueness on short time intervals depend only on local properties of $\boldsymbol b$ — whereas DiPerna–Lions existence requires global-in-space growth bounds on $|\boldsymbol b|$, the prototype being $|\boldsymbol b(t,x)|/(1+|x|) \in L^1_t(L^1_x) + L^1_t(L^\infty_x)$, together with global bounds on the divergence.

Ambrosio, Colombo and Figalli (2015) closed this gap by constructing, under only local integrability of $\boldsymbol b$, a local one-sided bound on the distributional divergence, and a local well-posedness assumption for the continuity equation with nonnegative bounded compactly supported data, a unique maximal regular flow $X(t,x)$, defined up to a maximal time $T_{\Omega,X}(x) > 0$ for $\mathcal L^d$-a.e. $x$ and characterized by the escape property $\limsup_{t \uparrow T_{\Omega,X}(x)} V_\Omega(X(t,x)) = \infty$ whenever $T_{\Omega,X}(x) < T$, where $V_\Omega$ is a prescribed continuous confining potential, i.e., a nonnegative function on the domain that blows up at the boundary and measures how deep inside the domain a point sits.

Section 7 of that paper studies what happens near the maximal time and shows that the answer depends on the global nature of the divergence bounds. The blow-up is called proper if the stronger condition $\lim_{t \uparrow T_{\Omega,X}(x)} V_\Omega(X(t,x)) = \infty$ holds, i.e., the trajectory eventually leaves every compact subset of the domain without returning (no oscillations). Theorem 7.1 proves that blow-up is proper as soon as the bounded compression condition is global; this is implied by a global one-sided divergence bound $\operatorname{div} \boldsymbol b(t,\cdot) \ge m(t)$ with $\int_0^T |m(t)|\,dt < \infty$. Without such a global assumption the conclusion fails dramatically: Proposition 7.3 constructs, in every dimension $d \ge 3$, an autonomous field $\boldsymbol b \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d;\mathbb R^d)$ with $\operatorname{div} \boldsymbol b \in L^\infty_{\mathrm{loc}}$ for which a set of initial data of positive measure has maximal time at most $2$ while the trajectories oscillate, with $\liminf_{t \uparrow T_X(x)} |X(t,x)| = 0$ and $\limsup_{t \uparrow T_X(x)} |X(t,x)| = \infty$ (the field travels through a sequence of cylinders with fast-decaying radii connected by handles, traversed faster and faster). The same construction in the plane gives a two-dimensional counterexample, but that field is neither in $W^{1,p}_{\mathrm{loc}}$ for any $p \ge 1$ nor of bounded variation. Proposition 7.4 shows that for autonomous planar $BV_{\mathrm{loc}}$ fields with $\operatorname{div} \boldsymbol b \in L^\infty_{\mathrm{loc}}$ blow-up is necessarily proper: a one-dimensional estimate on circles shows that each annulus can be crossed only finitely many times within finite time. The authors note that it 'looks likely' that a non-autonomous planar counterexample along the lines of Proposition 7.3 exists, so the autonomy hypothesis is expected to be essential even in the planar $BV$ class.

For the complementary question of ruling out blow-up, Theorem 7.6 gives an Eulerian criterion: if the maximal regular flow satisfies the global compression condition and there exists a nonnegative, weakly-* continuous (i.e., continuous for the weak-* topology of $L^\infty$) solution $\rho_t \in L^\infty$ of the continuity equation $\partial_t \rho_t + \operatorname{div}(\boldsymbol b \rho_t) = 0$ with the integrability condition $\int_0^T \int_{\mathbb R^d} |\boldsymbol b(t,x)|\,\rho_t(x)/(1+|x|)\,dx\,dt < \infty$, then $T_X(x) = T$ and the trajectory is absolutely continuous on $[0,T]$ for $\rho_0 \mathcal L^d$-a.e. initial point. Under the classical growth condition on $|\boldsymbol b(t,x)|/(1+|x|)$, such solutions $\rho_t$ can be produced for any nonnegative $\rho_0 \in L^1 \cap L^\infty$. Remark 7.7 records that, when only local divergence bounds hold, that growth condition alone does not prevent blow-up, while the stronger version $|\boldsymbol b(t,x)|/(1+|x|) \in L^1_t(L^\infty_x)$ does. The criterion was designed for applications to the Vlasov–Poisson system (the kinetic equation for a density of particles moving under the self-consistent electric or gravitational field they generate), where the companion paper of the same authors shows that weak solutions are Lagrangian and global weak solutions exist under minimal assumptions on the initial datum. What remains unknown is the sharp border between these positive and negative results.

Problem Statement

Work in the framework of Ambrosio–Colombo–Figalli (2015): $\boldsymbol b : (0,T) \times \Omega \to \mathbb R^d$ is a Borel vector field satisfying local integrability, a local one-sided bound on the distributional divergence, and local well-posedness of the continuity equation with nonnegative bounded compactly supported data, and $X$ is the associated almost-everywhere unique maximal regular flow with maximal time $T_{\Omega,X}(x)$, characterized by $\limsup_{t \uparrow T_{\Omega,X}(x)} V_\Omega(X(t,x)) = \infty$ for $\mathcal L^d$-a.e. $x$ with $T_{\Omega,X}(x) < T$. Determine, with proofs and/or explicit counterexamples:

(1) Sharp conditions for proper blow-up. Characterize the additional global hypotheses on $\boldsymbol b$ under which $\lim_{t \uparrow T_{\Omega,X}(x)} V_\Omega(X(t,x)) = \infty$ for $\mathcal L^d$-a.e. $x$ with $T_{\Omega,X}(x) < T$: find necessary-and-sufficient conditions, or sufficient conditions strictly weaker than the global bounded compression condition (in particular weaker than global one-sided divergence integrability), thereby identifying the quantitative property of the divergence or of the compression that constitutes the true borderline between Theorem 7.1 and the oscillatory example of Proposition 7.3.

(2) Regularity classes protected against oscillation. Determine the regularity classes and structural assumptions on $\boldsymbol b$ under which purely local hypotheses already force proper blow-up. Beyond autonomous planar $BV_{\mathrm{loc}}$ fields with $\operatorname{div} \boldsymbol b \in L^\infty_{\mathrm{loc}}$ (Proposition 7.4): does a non-autonomous planar $BV_{\mathrm{loc}}$ vector field with $\operatorname{div} \boldsymbol b \in L^\infty_{\mathrm{loc}}$ — so that autonomy is the only hypothesis of Proposition 7.4 being removed — exist whose trajectories oscillate on a set of positive measure, as the paper anticipates? Which structural assumptions on $\boldsymbol b$ in dimensions $d \ge 3$ rule out the oscillatory mechanism of Proposition 7.3?

(3) Minimal Eulerian criteria for global existence. Identify minimal hypotheses of Eulerian type — that is, expressed through solutions of the continuity equation rather than through growth bounds on $\boldsymbol b$ — under which $T_X(x) = T$ (here $T_X$ denotes the maximal time in the whole-space case $\Omega = \mathbb R^d$) for $\mathcal L^d$-a.e. $x$ in the absence of the classical growth condition on $|\boldsymbol b(t,x)|/(1+|x|)$: how generally can bounded, weakly-* continuous solutions $\rho_t$ with the required integrability be produced from natural classes of initial data and vector fields within the local theory, beyond the system-specific constructions available for Vlasov-type equations?

These three questions form a single problem: each fixes one of the three possible almost-everywhere behaviors of a maximal regular flow trajectory near its maximal time — proper blow-up, oscillatory escape, or global existence — and together they locate the exact borderline between local and global hypotheses in the DiPerna–Lions theory.

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