Non Existence of the Stable 2D Boussinesq Equations in H2 via Density-Vorticity Feedback
Abstract
We establish instantaneous H2 norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile −x2. While previous ill-posedness results for this system relied on L∞-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both H1 Euler ill-posedness of Bourgain \& Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(ρ(t), ω(t))$ originating from data with arbitrarily small H2 × (H1∩ Ḣ−1) norm, which immediately exhibits H2 × H1 norm explosion for all t > 0 while maintaining uniqueness within the energy class.
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BibTeX
@misc{bianchini2026existence,
title = {{Non Existence of the Stable 2D Boussinesq Equations in $H^2$ via Density-Vorticity Feedback}},
author = {Roberta Bianchini and Luis Martínez-Zoroa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38584},
eprint = {2609.38584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38584},
}
RIS
TY - UNPB
TI - Non Existence of the Stable 2D Boussinesq Equations in H^2 via Density-Vorticity Feedback
AU - Roberta Bianchini
AU - Luis Martínez-Zoroa
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38584
UR - https://arxiv.org/abs/2609.38584
AB - We establish instantaneous H^2 norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile −x_2. While previous ill-posedness results for this system relied on L^∞-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both H^1 Euler ill-posedness of Bourgain \& Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(ρ(t), ω(t))$ originating from data with arbitrarily small H^2 × (H^1∩ Ḣ^−1) norm, which immediately exhibits H^2 × H^1 norm explosion for all t > 0 while maintaining uniqueness within the energy class.
ER -