Interpolation spaces and weighted pseudo almost automorphic solutions to parabolic equations and applications to fluid dynamics

Nguyen, Thieu Huy ; Vu, Thi Ngoc Ha ; Le, The Sac ; Pham, Truong Xuan

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 935-955

Abstract (VO)

We investigate the existence, uniqueness and polynomial stability of the weighted pseudo almost automorphic solutions to a class of linear and semilinear parabolic evolution equations. The necessary tools here are interpolation spaces and interpolation theorems which help to prove the boundedness of solution operators in appropriate spaces for linear equations. Then for the semilinear equations the fixed point arguments are used to obtain the existence and stability of the weighted pseudo almost automorphic solutions. Lastly, our abstract results are applied to the Navier-Stokes equations (NSE) on some different circumstances such as the NSE on exterior domains, around rotating obstacles, and in Besov spaces.

Détail

DOI : 10.21136/CMJ.2022.0002-21

Classification : 35B15, 35B35, 35Q30, 76D05
Keywords: linear evolution equation; semilinear evolution equation; almost automorphic function; weighted pseudo almost automorphic function and solution; interpolation space

Bibliographie

[1] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343. Springer, Berlin (2011). | DOI | MR | JFM

[2] Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Grundlehren der mathematischen Wissenschaften 223. Springer, Berlin (1976). | DOI | MR | JFM

[3] Blot, J., Mophou, G. M., N'Guérékata, G. M., Pennequin, D.: Weighted pseudo almost automorphic functions and applications to abstract differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A 71 (2009), 903-909. | DOI | MR | JFM

[4] Bochner, S.: Curvature and Betti numbers in real and complex vector bundles. Univ. Politec. Torino, Rend. Sem. Mat. 15 (1956), 225-253. | MR | JFM

[5] Bochner, S.: Uniform convergence of monotone sequences of functions. Proc. Natl. Acad. Sci. USA 47 (1961), 582-585. | DOI | MR | JFM

[6] Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48 (1962), 2039-2043. | DOI | MR | JFM

[7] Borchers, W., Miyakawa, T.: On stability of exterior stationary Navier-Stokes flows. Acta Math. 174 (1995), 311-382. | DOI | MR | JFM

[8] Chávez, A., Castillo, S., Pinto, M.: Discontinuous almost automorphic functions and almost automorphic solutions of differential equations with piecewise constant arguments. Electron. J. Differ. Equ. 2014 (2014), Article ID 56, 13 pages. | MR | JFM

[9] Diagana, T.: Weighted pseudo-almost periodic solutions to some differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A 68 (2008), 2250-2260. | DOI | MR | JFM

[10] Ezzinbi, K., Fatajou, S., N'Guérékata, G. M.: Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 1641-1647. | DOI | MR | JFM

[11] Farwig, R., Hishida, T.: Stationary Navier-Stokes flow around a rotating obstacle. Funkc. Ekvacioj, Ser. Int. 50 (2007), 371-403. | DOI | MR | JFM

[12] Geissert, M., Heck, H., Hieber, M.: $L^p$-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596 (2006), 45-62. | DOI | MR | JFM

[13] Geissert, M., Hieber, M., Nguyen, T. H.: A general approach to time periodic incompressible viscous fluid flow problems. Arch. Ration. Mech. Anal. 220 (2016), 1095-1118. | DOI | MR | JFM

[14] Ha, V. T. N., Huy, N. T., Sac, L. T., Xuan, P. T.: Almost automorphic solutions to evolution equations in interpolation spaces and applications. Int. J. Evol. Equ. 11 (2018), 501-516.

[15] Hieber, M., Nguyen, T. H., Seyfert, A.: On periodic and almost periodic solutions to incompressible viscous fluid flow problems on the whole line. Mathematics for Nonlinear Phenomena: Analysis and Computation Springer Proceedings in Mathematics & Statistics 215. Springer, Cham (2017), 51-81. | DOI | MR | JFM

[16] Hishida, T., Shibata, Y.: $L_p - L_q$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193 (2009), 339-421. | DOI | MR | JFM

[17] Minh, N. V., Dat, T. T.: On the almost automorphy of bounded solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl. 326 (2007), 165-178. | DOI | MR | JFM

[18] Minh, N. V., Naito, T., Nguerekata, G.: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134 (2006), 3257-3266. | DOI | MR | JFM

[19] N'Guérékata, G. M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic, New York (2001). | DOI | MR | JFM

[20] N'Guérékata, G. M.: Topics in Almost Automorphy. Springer, New York (2005). | DOI | MR | JFM

[21] Nguyen, T. H.: Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle. Arch. Ration. Mech. Anal. 213 (2014), 689-703. | DOI | MR | JFM

[22] Nguyen, T. H., Trinh, V. D., Vu, T. N. H., Vu, T. M.: Boundedness, almost periodicity and stability of certain Navier-Stokes flows in unbounded domains. J. Differ. Equations 263 (2017), 8979-9002. | DOI | MR | JFM

[23] Nguyen, T. H., Vu, T. N. H., Xuan, P. T.: Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Commun. Pure Appl. Anal. 15 (2016), 2103-2116. | DOI | MR | JFM

[24] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18. North Holland, Amsterdam (1978). | MR | JFM

[25] Xiao, J.-T., Liang, J., Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76 (2008), 518-524. | DOI | MR | JFM

[26] Yamazaki, M.: The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force. Math. Ann. 317 (2000), 635-675. | DOI | MR | JFM