An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity

@misc{liang2026anisotropic,
  title = {{An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity}},
  author = {Siyu Liang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11488},
  eprint = {2609.11488},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11488},
}

TY  - UNPB
TI  - An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity
AU  - Siyu Liang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11488
UR  - https://arxiv.org/abs/2609.11488
AB  - We consider distributional weak solutions u∈ L^∞_tL^2_x of the two-dimensional Navier--Stokes equations with horizontal viscosity on 𝕋^2 or ℝ^2, without assuming ∂_1u∈ L^2_tL^2_x. We show that the Lions integrability condition u∈ L^4_tL^4_x together with Onsager-critical regularity in the nondissipative direction, u∈ L^3_tB^1/3,v_3,∞ with the Besov regularity imposed only in the vertical direction, implies ∂_1u∈ L^2_tL^2_x, with a quantitative bound on the horizontal dissipation. Thus the dissipative regularity is a consequence of the equation and need not be part of the definition of the solution. If moreover u∈ L^3_tB^1/3,v_3,c_0, then u belongs to the isotropic critical Onsager space L^3_tB^1/3_3,c_0, satisfies the energy equality, and is continuous in L^2. Here c_0 indicates that the corresponding dyadic Besov sequence tends to zero at high frequencies. The proof combines absorption of the horizontal energy flux into the dissipation with a one-dimensional commutator estimate for the vertical energy flux.
ER  -