Instantaneous blowup of incompressible flow with passive tracer

Abstract: We construct a family of solutions $(u,b)$ of the incompressible flow with a passive tracer for which both $|u(t)|<em>{L<sup>\infty}$ and $|b(t)|</em>{L<sup>\infty}$ blow up at time $T_<em>$. Away from $T_</em>$, 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.

• The paper establishes instantaneous L∞ blowup in the coupled incompressible Navier–Stokes and passive tracer system with sharp Type I bounds.
• The authors use convex integration and an innovative tensor-vector decomposition to achieve compatibility between the velocity field and the passive scalar.
• The analysis provides rigorous asymptotic estimates for both the velocity and tracer, confirming that the addition of a passive scalar does not prevent singularities.

Instantaneous Blowup in Incompressible Flow with Passive Tracer

Introduction and Context

The paper "Instantaneous blowup of incompressible flow with passive tracer" (2604.11769) addresses instantaneous $L^\infty$ blowup for the coupled incompressible Navier–Stokes equation with a linear passive tracer. Specifically, the authors construct solutions $(u, b)$ to the system

$$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$

on the torus $\mathbb T^d$ (mainly $d=2$), exhibiting singular (Type I) blowup at a prescribed time $T_*$ while remaining classical for $t<T_*$. This flow can be viewed as a "Navier–Stokes–tracer" or, in certain 2.5D symmetry reductions, as a specialization of the incompressible magnetohydrodynamics (MHD) equations. The construction is inspired by and adapts convex integration and inverse cascade mechanisms used in recent nonuniqueness and blowup works for Navier–Stokes (Cheskidov et al., 12 Nov 2025), Euler, and MHD [26...].

Main Results

The principal result asserts the existence of solutions to the above system which are smooth prior to some $T_*$, but for which both $\|u(t)\|_{L^\infty}$ and $\|b(t)\|_{L^\infty}$ exhibit lower and upper bounds of the form

$(u, b)$0

and similarly for their derivatives: $(u, b)$1 These are sharp, in the sense of Type I singularities, and hold along sequences $(u, b)$2. The solution, appropriately interpreted in the sense of weak solutions, is weak-* continuous into $(u, b)$3 and satisfies $(u, b)$4-based energy as well as Ladyzhenskaya–Prodi–Serrin-type regularity for all $(u, b)$5 with $(u, b)$6.

A similar result holds for $(u, b)$7 or higher, paralleling developments in the corresponding literature for Navier–Stokes and MHD equations.

Technical Construction

Convex Integration and Inverse Cascade

The core mechanism is a convex integration scheme producing an "inverse cascade," injecting energy from high frequencies to lower ones at rapidly vanishing time intervals, converging to the singularity time $(u, b)$8. The iterative construction is inspired by recent works on instantaneous blowup in Navier–Stokes (Cheskidov et al., 12 Nov 2025), requiring an intricate arrangement of time and spatial frequency scales indexed by $(u, b)$9, and, for each stage, building approximate solutions ("principal profiles") governed by geometric "building blocks."

Passive Tracer Compatibility

The principal innovation required for the flow–passive tracer system is maintaining compatibility between the velocity and scalar at each convex integration stage. Unlike the pure Navier–Stokes case, the amplitudes used to build the high-frequency oscillatory components must generate both:

• the correct velocity stress contributions (via suitable sums over symmetric tensors)
• the appropriate transport terms for the passive scalar, delegated to the principal velocity profiles.

The authors formalize this with a tensor-vector decomposition lemma (Lemma 3), constructed as a generalization of the geometric lemma used for symmetric tensors in convex integration of incompressible Euler/Navier–Stokes ([De Lellis–Székelyhidi, 2009, 2013]). This lemma enables the passage of the high–high to low-mode transfer mechanism simultaneously for the velocity and tracer, which is a genuinely nontrivial extension.

Profile Construction and Estimates

The authors organize the solution as sums $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$0, where $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$1 are principal profiles coordinating the inverse cascade and $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$2 are perturbative correctors, constructed to ensure the exact system and initial data are matched. The principal profiles comprise rapidly decaying oscillatory building blocks localized in thin "pipe" regions, with frequency scales and time intervals chosen according to a precise hierarchy.

The estimates involve a separation into "principal" and correction parts, using detailed semigroup and commutator analysis, and leveraging the monotonicity of the frequency/time hierarchy to control error terms. Notably, the residual forcing in the equation for $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$3 is controlled in $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$4 and critical function spaces.

Gluing and Weak Solutions

To obtain a true solution to the Cauchy problem (with prescribed initial data), the authors "glue" the constructed blowing-up profile with a classical solution up to $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$5, and then use weak-* $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$6 continuity and compactness arguments to justify that the entire solution (with a singularity at $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$7) remains a weak solution globally in time.

Sharp Asymptotics

The asymptotic lower and upper blowup bounds (Type I rates) are realized by carefully tracking the contribution of the main, low-frequency modes generated by the cascade at each stage, using matching scaling arguments and the explicit structure of the building blocks. The lower bound, in particular, is realized along a sequence of times dictated by the cascade scale parameters.

Implications and Connections

Mathematical Fluid Dynamics

This result establishes, for the first time, the possibility of instantaneous blowup in the maximum norm for coupled incompressible flow and passive tracer in the viscous setting, in both two and higher dimensions. The result is sharp in terms of Type I rates and confirms that the addition of a passive scalar does not regularize the system, at least at the level of weak solution theory. The construction demonstrates that convex integration, previously developed for passive and active scalar equations (see e.g. [Modena–Székelyhidi 2018] for transport equations, [FLS2021, FLS2024] for ideal MHD), can be extended to treat the intricate coupling and compatibility required by flows with scalars or reduced MHD in the viscous context.

The work also demonstrates a path toward generalizations involving more complex couplings (e.g., in full MHD or systems with nonlinear transport). It suggests that, at the level of irregular (Onsager or below) solutions, the presence of a coupled scalar does not prevent wild behavior (nonuniqueness, blowup) inherent in the underlying velocity field.

Function Spaces and Regularity Theory

A notable feature is that the weak solution constructed lies in critical function spaces such as $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$8, $$\begin{cases} \partial_t u - \Delta u + \operatorname{div}(u \otimes u) + \nabla p = 0, \ \partial_t b - \Delta b + (u \cdot \nabla) b = 0, \ \operatorname{div} u = 0, \end{cases}$$9 for all $\mathbb T^d$0, and weak-* continuous in $\mathbb T^d$1, matching the threshold regularity traditionally conjectured to be on the "well-posedness" side for Navier–Stokes and MHD. These spaces thus cannot guarantee regularity or uniqueness in this coupled tracer setting.

Broader Directions and AI

This work, as part of the contemporary convex integration expansion, pushes the envelope of what structures are possible within dissipative PDEs, especially as they relate to questions of blowup, regularity, and nonuniqueness in fluid dynamics. For applied mathematics and numerical analysis, the existence of such solutions underlines the challenges in establishing global regularity or uniqueness below the critical threshold, and highlights that models augmented by passive or active scalars must be treated with equal rigor in the analysis of singularity formation.

While not directly related to AI, these insights have implications for computational fluid dynamics, turbulence modeling, and any domain where data-driven or learning based methods might rely on assumptions of regularity or uniqueness in the underlying PDE models.

Conclusion

The paper rigorously establishes the occurrence of instantaneous Type I $\mathbb T^d$2 blowup for incompressible viscous flows with passive tracers, via a sophisticated adaptation of convex integration schemes and an original tensor–vector decomposition guaranteeing structural compatibility of the solution at each iteration. The result clarifies the range of possibilities for weak solutions in fluid–scalar systems and underscores intrinsic limitations of regularity and uniqueness at the critical threshold. The methods and decomposition tools developed are likely to be of foundational importance for further advances in analysis of coupled dissipative PDEs (2604.11769).