We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.
P.Auscher, S.Dubois and P.Tchamitchian, On the stability of global solutions to Navier–Stokes equations in the space, J. Math. Pures Appl. (9)83(6) (2004), 673–697. doi:10.1016/j.matpur.2004.01.003.
2.
H.Bahouri, J.Y.Chemin and R.Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011.
3.
C.Bjorland, L.Brandolese, D.Iftimie and M.E.Schonbek, solutions of the steady-state Navier–Stokes equations with rough external forces, Comm. Partial Differential Equations36(2) (2011), 216–246. doi:10.1080/03605302.2010.485286.
4.
C.Bjorland and M.E.Schonbek, Existence and stability of steady-state solutions with finite energy for the Navier–Stokes equations in the whole space, Nonlinearity22(7) (2009), 1615–1637. doi:10.1088/0951-7715/22/7/007.
5.
G.Bourdaud, Realizations of homogeneous Besov and Lizorkin–Triebel spaces, Math. Nachr.286(5–6) (2013), 476–491. doi:10.1002/mana.201100151.
6.
M.Cannone, A generalization of a theorem by Kato on Navier–Stokes equations, Rev. Mat. Iberoamericana.13 (1997), 515–541. doi:10.4171/RMI/229.
7.
M.Cannone and G.Karch, About the regularized Navier–Stokes equations, J. Math. Fluid Mech.7(1) (2005), 1–28. doi:10.1007/s00021-004-0105-y.
8.
R.Danchin and P.B.Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows, Mém. Soc. Math. Fr. (N. S.)143 (2015).
9.
I.Gallagher, D.Iftimie and F.Planchon, Asymptotics and stability for global solutions to the Navier–Stokes equations, Ann. Inst. Fourier (Grenoble)53(5) (2003), 1387–1424. doi:10.5802/aif.1983.
10.
K.Kaneko, H.Kozono and S.Shimizu, Stationary solution to the Navier–Stokes equations in the scaling invariant Besov space and its regularity, Indiana Univ. Math. J.68(3) (2019), 857–880. doi:10.1512/iumj.2019.68.7650.
11.
H.Koch and D.Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math.157 (2001), 22–35. doi:10.1006/aima.2000.1937.
12.
H.Kozono, T.Ogawa and Y.Taniuchi, Navier–Stokes equations in the Besov space near and , Kyushu J. Math.57 (2003), 303–324. doi:10.2206/kyushujm.57.303.
13.
H.Kozono and M.Yamazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations19(5–6) (1994), 959–1014. doi:10.1080/03605309408821042.
14.
H.Kozono and M.Yamazaki, The stability of small stationary solutions in Morrey spaces of the Navier–Stokes equation, Indiana Univ. Math. J.44(4) (1995), 1307–1336. doi:10.1512/iumj.1995.44.2029.
15.
A.Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
16.
S.Machihara and T.Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc.131, 1553–1556. doi:10.1090/S0002-9939-02-06715-1.
17.
Y.Meyer, Wavelet, Paraproduct and Navier–Stokes Equations, Current Developments in Mathematics, International Press, 1996, pp. 105–212.
18.
T.V.Phan and N.C.Phuc, Stationary Navier–Stokes equations with critically singular external forces: Existence and stability results, Adv. Math.241 (2013), 137–161. doi:10.1016/j.aim.2013.01.016.
E.Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl.107 (1985), 16–66. doi:10.1016/0022-247X(85)90353-1.
21.
Y.Sugiyama, Y.Tsutsui and J.L.L.Velázquez, Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl.410(2) (2014), 908–917. doi:10.1016/j.jmaa.2013.08.065.
22.
H.Tsurumi, Well-posedness and ill-posedness problems of the stationary Navier–Stokes equations in scaling invariant Besov spaces, Arch. Ration. Mech. Anal.234(2) (2019), 911–923. doi:10.1007/s00205-019-01404-6.
23.
Y.Tsutsui, Bounded global solutions to a Keller–Segel system with nondiffusive chemical in , J. Evol. Equ.17(2) (2017), 627–640. doi:10.1007/s00028-016-0330-x.
24.
M.Yamazaki, The Navier–Stokes equations in the weak- space with time-dependent external force, Math. Ann.317(4) (2000), 635–675. doi:10.1007/PL00004418.
