Instantaneous blowup of incompressible flow with passive tracer

TL;DR

This paper constructs solutions to incompressible flow with a passive tracer that blow up in finite time, demonstrating a new type of singularity formation while solutions remain smooth before the blowup.

Contribution

It introduces a novel method to produce finite-time blowup solutions in passive scalar-involved incompressible flows, adapting inverse cascade mechanisms.

Findings

Solutions blow up at finite time T_* in both velocity and tracer norms.

Solutions are smooth away from the blowup time.

The method adapts inverse cascade techniques to passive scalar dynamics.

Abstract

We construct a family of solutions $(u,b)$ of the incompressible flow with a passive tracer for which both $\|u(t)\|_{L^\infty}$ and $\|b(t)\|_{L^\infty}$ blow up at time $T_*$. Away from $T_*$, the solutions remain smooth in both space and time. The argument adapts the inverse cascade mechanism from \cite{CDP} to the presence of an advected scalar, but the passive component creates a new compatibility constraint: the iteration must propagate the tracer while preserving the same principal velocity profiles from one stage to the next. We resolve it by introducing a simultaneous decomposition lemma for a symmetric tensor and a vector field.