Optimal Mixing Rates Under Palinstrophy Constraints

Definitions and mathematical statement

The supplied question concerns passive-scalar mixing under a uniform palinstrophy or Sobolev $H^2$ budget. It does not specify enough data to determine a unique optimization problem. The following notation describes its mathematical content conditionally, without selecting the missing hypotheses.

Let $\Omega$ denote a spatial domain whose dimension, geometry, and boundary conditions remain unspecified. Under the interpretation of passive-scalar transport as transport without diffusion, a scalar $\theta$ and velocity $u$ satisfy \[ \partial_t\theta+u\cdot\nabla\theta=0, \qquad \theta(0,\cdot)=\theta_0. \] The initial datum $\theta_0$, the solution concept, the regularity of $u$, and whether $\nabla\cdot u=0$ is required are not supplied. This equation is therefore a conditional interpretation, not a complete prescribed initial-boundary value problem.

On a Euclidean domain, the full Sobolev norm for a vector field can be written as \[ \|u\|_{H^2(\Omega)}^2 =\sum_{|\alpha|\leq 2}\int_\Omega |D^\alpha u(x)|^2\,{\,\mathrm{d}} x. \] A uniform full $H^2$ budget would take the form \[ \sup_{t\geq0}\|u(t,\cdot)\|_{H^2(\Omega)}\leq B, \qquad 0<B<\infty. \] For comparison, in two spatial dimensions, with $\omega=\partial_1u_2-\partial_2u_1$, the usual palinstrophy quantity is, up to normalization, \[ \mathcal P(u)=\int_\Omega |\nabla\omega|^2\,{\,\mathrm{d}} x. \] A uniform bound on this quantity and a uniform bound on the full $H^2$ norm are distinct formulations unless additional hypotheses justify their relationship. The record does not specify the dimension, normalization, boundary conditions, or control of lower-order components needed to resolve its use of these terms.

For a chosen scalar norm $\|\cdot\|_X$, write \[ M_X(t)=\|\theta(t,\cdot)\|_X. \] Here $X$ is a placeholder for the unspecified mix-norm; the record does not determine whether the scalar must first have its mean removed or satisfy a mean-zero condition.

The requested problem is to determine the sharp asymptotic decay of the specified mix-norm as $t\to\infty$, among scalar evolutions driven by velocities satisfying the intended uniform budget, and to decide whether the optimal decay is polynomial or exponential. A precise quantified statement still requires the domain, boundary conditions, admissible velocities and scalar data, exact budget, and exact mix-norm. It must also specify whether optimization permits a different velocity for each terminal time, requires one velocity to attain the asymptotic rate, and requires bounds uniform over an initial-data class. No choice among these formulations is made here.

Short English statement

Determine the fastest possible long-time mixing of a passively transported scalar when the velocity has a uniform palinstrophy or $H^2$ budget, including whether the sharp decay is polynomial or exponential. The supplied record leaves the domain, mix-norm, admissible data, budget interpretation, and optimization quantifiers unresolved, so it does not yet specify a unique mathematical problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002940$, title "Optimal Mixing Rates Under Palinstrophy Constraints," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2015.
• Supplied source citation: Mathematical Aspects of Hydrodynamics (2016), Oberwolfach Reports, pp. 2125--2187, DOI 10.4171/owr/2015/37. Original OWR report: https://doi.org/10.4171/owr/2015/37. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's 2016 designation, the DOI's 2015 component, and the recorded proposed year are retained without reconciliation.
• Supplied literature source labeled "Crippa and Schulze 2017": https://arxiv.org/abs/1707.01352.
• Supplied literature source labeled "Crippa and Schulze 2023": https://doi.org/10.3934/mine.2023006.
• Literature assessment checked on 2026-08-21, according to the supplied dated curation, not independently verified here: the record labels the problem open and reports known rigorous lower bounds and polynomial mixing constructions for important classes, while reporting that no proof of the exact optimal decay rate for all palinstrophy-constrained flows was located. The supplied background explicitly calls this a dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its accompanying explanation says that notation and publication prose were repaired without changing the mathematical question. This supplied label is not an independent verification of the formulation, literature assessment, or open status.