The optimal upper and lower bounds for the 3D generalized Navier-Stokes equations. (Chinese. English summary) Zbl 07801530

Wang, Wenjuan; Jia, Yan

Acta Math. Appl. Sin. 46, No. 3, 440-456 (2023).

Summary: This paper concerns the decay rates of weak solutions to the 3D generalized Navier-Stokes equations with fractional Laplacian dissipation \(\Lambda^{2 \alpha} u\). It is proved that if the weak solution \(u(x,t)\) to the 3D generalized Navier-Stokes equations lies in the regular class \[ \nabla u \in L^p (0,\infty; B_{q,\infty}^0 (R^3)),\; \frac{2 \alpha}{p} + \frac{3}{q} = 2 \alpha,\; \frac{3}{ 2 \alpha} < q < \infty,\; \frac{1}{2} \leq \alpha \leq 1, \] the large initial perturbation \(w_0 \in L^2 (\mathbb{R}^3)\) satisfies \[ \int_{\mathcal{S}^2} |\hat{w_0}(r\omega)|^2 \mathrm{d} \omega = C r^{2\alpha \gamma -3} + o (r^{2\alpha \gamma -3}) (r \to 0), \frac{10}{\alpha} -8 \leq \gamma \leq \frac{25}{2 \alpha} -10, \] then every weak solution \(v(x,t)\) of the perturbed system converges algebraically to \(u(x,t)\) with the optimal upper and lower bounds \[C_1 (1+t)^{- \frac{\gamma}{2}} \leq \|v(t) - u(t)\|_{L^2} \leq C_2 (1+t)^{-\frac{\gamma}{2}},\] for large \(t>1\).

MSC:

35Q35	PDEs in connection with fluid mechanics
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

generalized Navier-Stokes equations; optimal decay rates; Besov space

References:

[1]	Lions J L. Quelques Méthods de Résolution des Problémes aux Limites NonlinWaires. Dunod, Paris, 1969 · Zbl 0189.40603
[2]	Wu J H. Generalized MHD equations. Journal of Differential Equations, 2003, 195: 284-312 · Zbl 1057.35040
[3]	Wu J H. The generalized incompressible Navier-Stokes equations in Besov spaces. Dynamics of Partial Differential Equations, 2004, 1: 381-400 · Zbl 1075.35043
[4]	Wu H W, Fan J S. Weak-strong uniqueness for the generalized Navier-Stokes equations. Applied Mathematics Letters, 2012, 25: 423-428 · Zbl 1239.35118
[5]	Chen Z M. Analytic semigroup approach to generalized Navier-Stokes flows in Besov spaces. Journal of Mathematical Fluid Mechanics, 2017, 19: 709-724 · Zbl 1386.35017
[6]	Caffarelli L, Kohn R, Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Communications on Pure and Applied Mathematics, 1982, 35: 771-831 · Zbl 0509.35067
[7]	Dong B Q, Jia Y. Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations. Nonlinear Analysis-real World Applications, 2016, 30: 41-58 · Zbl 1342.35214
[8]	Dong B Q, Zhang Z F. On the weak-strong uniqueness of Koch-Tataru’s solution for the Navier-Stokes equations. Journal of Differential Equations, 2014, 256: 2406-2422 · Zbl 1285.35064
[9]	Teman R. The Navier-Stokes Equations. North-Holland, Amsterdam, 1977 · Zbl 0383.35057
[10]	Leray J. Sur le mouvement d’un liquide visqueux remplissant l’espace. Acta Mathematica, 1934, 63: 193-248 · JFM 60.0726.05
[11]	Schonbek M E. L 2 decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 1985, 88: 209-222 · Zbl 0602.76031
[12]	Schonbek M E. Large time behavior of solutions to the Navier-Stokes equations. Communications in Partial Differential Equations, 1986, 11: 733-763 · Zbl 0607.35071
[13]	Ponce G, Racke R, Sideris T C, Titi E S. Global stability of large solutions to the 3D Navier-Stokes equations. Communications in Mathematical Physics, 1994, 159: 329-341 · Zbl 0795.35082
[14]	Secchi P. L 2 stability for weak solutions of the Navier-Stokes equations in R 3 . Indiana University Mathematics Journal, 1987, 36: 685-691 · Zbl 0635.35076
[15]	Zhou Y. Asymptotic stability to the 3D Navier-Stokes equations. Communications in Partial Differ-ential Equations, 2005, 30: 323-333 · Zbl 1142.35548
[16]	Jia Y, Xie Q Q, Wang W J. The optimal upper and lower bounds of convergence rates for the 3D Navier-Stokes equations under large initial perturbation. Journal of Mathematical Analysis and Applications, 2018, 459: 437-452 · Zbl 1457.35029
[17]	Dong B Q, Chen Z M. On the weak-strong uniqueness of the dissipative surface quasi-geostrophic equation. Nonlinearity, 2012, 25: 1513-1524 · Zbl 1241.35158
[18]	Chemin J Y. Perfect Incompressible Fluids. Oxford Lecture Series in Mathematics and its Applica-tions, London, 1998 · Zbl 0927.76002
[19]	Bony J M. Calcul symbolique et propagation des singularités pour leséquations aux dérivées partielles non linéaires. Annales Scientifiques de I’ École Normale Supérieure, 1981, 14: 209-246 · Zbl 0495.35024
[20]	Oliver M, Titi E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in R n . Journal of Functional Analysis, 2000, 172: 1-18 · Zbl 0960.35081