Asymptotic stability for the 3D Navier-Stokes equations in $L^3$ and nearby spaces
HTML articles powered by AMS MathViewer
- by Zachary Bradshaw and Weinan Wang;
- Proc. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/proc/17301
- Published electronically: June 26, 2025
- PDF | Request permission
Abstract:
We provide a short proof of $L^3$-asymptotic stability around vector fields that are small in weak-$L^3$, including small Landau solutions. We show that asymptotic stability also holds for vector fields in the range of Lnorentz spaces strictly between $L^3$ and weak-$L^3$, as well as in the closure of the test functions in weak-$L^3$. To provide a comprehensive perspective on the matter, we observe that asymptotic stability of Landau solutions does not generally extend to weak-$L^3$ via a counterexample.References
- A. P. Blozinski, On a convolution theorem for $L(p,q)$ spaces, Trans. Amer. Math. Soc. 164 (1972), 255–265. MR 415293, DOI 10.1090/S0002-9947-1972-0415293-1
- Zachary Bradshaw and Tai-Peng Tsai, Forward discretely self-similar solutions of the Navier-Stokes equations II, Ann. Henri Poincaré 18 (2017), no. 3, 1095–1119. MR 3611025, DOI 10.1007/s00023-016-0519-0
- Zachary Bradshaw and Tai-Peng Tsai, Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 45 (2020), no. 9, 1168–1201. MR 4134389, DOI 10.1080/03605302.2020.1761386
- Calixto P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in $L^p$, Trans. Amer. Math. Soc. 318 (1990), no. 1, 179–200. MR 968416, DOI 10.1090/S0002-9947-1990-0968416-0
- Matthias Hieber and Thieu Huy Nguyen, Periodic solutions and their stability to the Navier-Stokes equations on a half space, Discrete Contin. Dyn. Syst. Ser. S 17 (2024), no. 5-6, 1899–1910. MR 4762567, DOI 10.3934/dcdss.2022199
- Grzegorz Karch and Dominika Pilarczyk, Asymptotic stability of Landau solutions to Navier-Stokes system, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 115–131. MR 2835864, DOI 10.1007/s00205-011-0409-z
- Grzegorz Karch, Dominika Pilarczyk, and Maria E. Schonbek, $L^2$-asymptotic stability of singular solutions to the Navier-Stokes system of equations in $\Bbb {R}^3$, J. Math. Pures Appl. (9) 108 (2017), no. 1, 14–40 (English, with English and French summaries). MR 3660767, DOI 10.1016/j.matpur.2016.10.008
- Tosio Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\textbf {R}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR 760047, DOI 10.1007/BF01174182
- P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147, DOI 10.1201/9781420035674
- Yanyan Li, Jingjing Zhang, and Ting Zhang, Asymptotic stability of Landau solutions to Navier-Stokes system under $L^p$-perturbations, J. Math. Fluid Mech. 25 (2023), no. 1, Paper No. 5, 30. MR 4519198, DOI 10.1007/s00021-022-00751-x
- Rainer Mandel, Real interpolation for mixed Lorentz spaces and Minkowski’s inequality, Z. Anal. Anwend. 42 (2023), no. 3-4, 457–469. MR 4699889, DOI 10.4171/zaa/1736
- Yves Meyer, Wavelets, paraproducts, and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), Int. Press, Boston, MA, 1997, pp. 105–212. MR 1724946
- Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
- Tai-Peng Tsai, Lectures on Navier-Stokes equations, Graduate Studies in Mathematics, vol. 192, American Mathematical Society, Providence, RI, 2018. MR 3822765, DOI 10.1090/gsm/192
- Jingjing Zhang and Ting Zhang, Global well-posedness of perturbed Navier-Stokes system around Landau solutions, J. Math. Phys. 64 (2023), no. 1, Paper No. 011516, 7. MR 4541309, DOI 10.1063/5.0087462
- Z. Zhao and X. Zheng, Asymptotic stability of homogeneous solutions to Navier-Stokes equations under ${L}^p$-perturbations, arXiv preprint arXiv:2304.00840, 2023.
Bibliographic Information
- Zachary Bradshaw
- Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
- MR Author ID: 862108
- ORCID: 0009-0009-0622-8333
- Email: zb002@uark.edu
- Weinan Wang
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- MR Author ID: 1314789
- ORCID: 0000-0003-1140-7546
- Email: ww@ou.edu
- Received by editor(s): October 9, 2024
- Received by editor(s) in revised form: February 26, 2025
- Published electronically: June 26, 2025
- Additional Notes: The research of the first author was supported in part by the NSF via grant DMS-2307097 and the Simons Foundation via a TSM grant. The second author was supported in part by the Simons Foundation via a TSM grant (No. 00007730).
- Communicated by: Ariel Barton
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
- MSC (2020): Primary 35Q35
- DOI: https://doi.org/10.1090/proc/17301