The Bardos-Tartar Conjecture for the Navier-Stokes Equations
Abstract
We consider the incompressible viscous Navier-Stokes equations in dimension d=2,3 with periodic boundary conditions and time-independent external forcing. When d=2, the Bardos-Tartar conjecture asks whether the set of solutions at any given positive time is dense in the Hilbert subspace of L2 consisting of mean-zero divergence-free vector fields, with respect to the strong L2 norm. We prove the Bardos-Tartar conjecture and, more generally, its analogue for the hyperviscous Navier-Stokes equations studied by Guo-Titi. In addition, for d=2,3 we show that any given initial condition is either backwards-extendable (i) for all values of viscosity ν>0, or (ii) only for viscosities that belong to a Lebesgue-null subset of ℝ+. When (i) holds, we show the initial condition and forcing must be real-analytic (in fact, both belong to D(eσA) for some σ>0, where A is the Stokes operator). Furthermore, we show that (i) is extremely unstable under perturbations. In particular, given any initial condition we show that two Fourier modes can be perturbed so that (ii) holds.
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BibTeX
@misc{barnes2026bardostartar,
title = {{The Bardos-Tartar Conjecture for the Navier-Stokes Equations}},
author = {Douglas Barnes and Samuel Liew and Edriss Titi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21530},
eprint = {2610.21530},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21530},
}
RIS
TY - UNPB
TI - The Bardos-Tartar Conjecture for the Navier-Stokes Equations
AU - Douglas Barnes
AU - Samuel Liew
AU - Edriss Titi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21530
UR - https://arxiv.org/abs/2610.21530
AB - We consider the incompressible viscous Navier-Stokes equations in dimension d=2,3 with periodic boundary conditions and time-independent external forcing. When d=2, the Bardos-Tartar conjecture asks whether the set of solutions at any given positive time is dense in the Hilbert subspace of L^2 consisting of mean-zero divergence-free vector fields, with respect to the strong L^2 norm. We prove the Bardos-Tartar conjecture and, more generally, its analogue for the hyperviscous Navier-Stokes equations studied by Guo-Titi. In addition, for d=2,3 we show that any given initial condition is either backwards-extendable (i) for all values of viscosity ν>0, or (ii) only for viscosities that belong to a Lebesgue-null subset of ℝ_+. When (i) holds, we show the initial condition and forcing must be real-analytic (in fact, both belong to D(e^σA) for some σ>0, where A is the Stokes operator). Furthermore, we show that (i) is extremely unstable under perturbations. In particular, given any initial condition we show that two Fourier modes can be perturbed so that (ii) holds.
ER -