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On the stability of solutions to the 3-D anisotropic Navier-Stokes equations in the critical space

  • Nicolas Burq
  • Ning Liu
  • Ping Zhang
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Abstract

In this paper, we study the continuous dependence of solutions to the three-dimensional incompressible anisotropic Navier-Stokes equations (ANS) with initial data in the critical space Ḃ0,1/2. We prove that the data-to-solution map is continuous in this setting. Due to the critical regularity of the problem, the classical Bona-Smith argument \cite{BoSm} (see also the abstract result \cite{ABITZ}) does not appear to apply because a key ingredient (the Lipschitz continuity at lower regularity) does not appear to hold. As a consequence, we develop a new direct Fourier-weighted method within the Besov framework.

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Cite

BibTeXRIS
BibTeX
@misc{burq2026stability,
  title = {{On the stability of solutions to the 3-D anisotropic Navier-Stokes equations in the critical space}},
  author = {Nicolas Burq and Ning Liu and Ping Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14304},
  eprint = {2609.14304},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14304},
}
RIS
TY  - UNPB
TI  - On the stability of solutions to the 3-D anisotropic Navier-Stokes equations in the critical space
AU  - Nicolas Burq
AU  - Ning Liu
AU  - Ping Zhang
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14304
UR  - https://arxiv.org/abs/2609.14304
AB  - In this paper, we study the continuous dependence of solutions to the three-dimensional incompressible anisotropic Navier-Stokes equations (ANS) with initial data in the critical space Ḃ^0,1/2. We prove that the data-to-solution map is continuous in this setting. Due to the critical regularity of the problem, the classical Bona-Smith argument \cite{BoSm} (see also the abstract result \cite{ABITZ}) does not appear to apply because a key ingredient (the Lipschitz continuity at lower regularity) does not appear to hold. As a consequence, we develop a new direct Fourier-weighted method within the Besov framework.
ER  -