Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces

@misc{fukuda2026asymptotic,
  title = {{Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces}},
  author = {Ikki Fukuda and Ryosuke Nakasato},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30902},
  eprint = {2609.30902},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30902},
}

TY  - UNPB
TI  - Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces
AU  - Ikki Fukuda
AU  - Ryosuke Nakasato
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30902
UR  - https://arxiv.org/abs/2609.30902
AB  - We consider the large time asymptotic behavior of solutions to the initial value problem for the incompressible Navier-Stokes equations in the whole space ℝ^d (d≥ 2). When the initial velocity belongs to L^1(ℝ^d), it is well-known that its spatial integral vanishes as a consequence of the divergence-free compatibility condition. Taking this property into account, Carpio (1996) and Fujigaki-Miyakawa (2001) derived higher-order asymptotic formulas for the strong solutions. In this paper, we investigate the large time behavior of solutions when the initial velocity belongs to the Fourier-Herz space L̂^1(ℝ^d), which provides a broader framework than L^1(ℝ^d) and imposes an additional condition in the low-frequency region. In particular, in the two-dimensional case, we show that an asymptotic profile different from those obtained by Carpio and Fujigaki-Miyakawa arises.
ER  -