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PreprintCC BY 4.0arXiv

On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space L3,∞(ℝ3)

  • Ramzi May
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Abstract

We prove uniqueness of solutions to the Navier-Stokes equations in the space Lp([0,T],L̃3,∞(ℝ3)) ∩ C([0,T],B−1,∞∞(ℝ3)), p>2, where L̃3,∞(ℝ3) denotes the closure of the Schwartz space 𝒮(ℝ3) in the Lorentz space L3,∞(ℝ3). Moreover, we prove that any solution u in this class, if it exists, is smooth for positive time, namely, u∈ C∞((0,T]× ℝ3). In addition, we establish precise estimates for the behavior of u(t) as $t\to0^+ $ in various Banach spaces.

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Cite

BibTeXRIS
BibTeX
@misc{may2026uniqueness,
  title = {{On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space $L^3,\infty(\mathbbR^3)$}},
  author = {Ramzi May},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14688},
  eprint = {2609.14688},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14688},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space L^3,∞(ℝ^3)
AU  - Ramzi May
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14688
UR  - https://arxiv.org/abs/2609.14688
AB  - We prove uniqueness of solutions to the Navier-Stokes equations in the space L^p([0,T],L̃^3,∞(ℝ^3)) ∩ C([0,T],B^−1,∞_∞(ℝ^3)), p>2, where L̃^3,∞(ℝ^3) denotes the closure of the Schwartz space 𝒮(ℝ^3) in the Lorentz space L^3,∞(ℝ^3). Moreover, we prove that any solution u in this class, if it exists, is smooth for positive time, namely, u∈ C^∞((0,T]× ℝ^3). In addition, we establish precise estimates for the behavior of u(t) as $t\to0^+ $ in various Banach spaces.
ER  -