Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 935-955, 2022


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

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

Received January 4, 2021.   Published online June 15, 2022.

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


References:
[1] H. Bahouri, J.-Y. Chemin, R. Danchin: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343. Springer, Berlin (2011). DOI 10.1007/978-3-642-16830-7 | MR 2768550 | Zbl 1227.35004
[2] J. Bergh, J. Löfström: Interpolation Spaces: An Introduction. Grundlehren der mathematischen Wissenschaften 223. Springer, Berlin (1976). DOI 10.1007/978-3-642-66451-9 | MR 0482275 | Zbl 0344.46071
[3] J. Blot, G. M. Mophou, G. M. N'Guérékata, D. Pennequin: Weighted pseudo almost automorphic functions and applications to abstract differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A 71 (2009), 903-909. DOI 10.1016/j.na.2008.10.113 | MR 2527511 | Zbl 1177.34077
[4] S. Bochner: Curvature and Betti numbers in real and complex vector bundles. Univ. Politec. Torino, Rend. Sem. Mat. 15 (1956), 225-253. MR 0084160 | Zbl 0072.17301
[5] S. Bochner: Uniform convergence of monotone sequences of functions. Proc. Natl. Acad. Sci. USA 47 (1961), 582-585. DOI 10.1073/pnas.47.4.582 | MR 0126094 | Zbl 0103.05304
[6] S. Bochner: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48 (1962), 2039-2043. DOI 10.1073/pnas.48.12.2039 | MR 0145283 | Zbl 0112.31401
[7] W. Borchers, T. Miyakawa: On stability of exterior stationary Navier-Stokes flows. Acta Math. 174 (1995), 311-382. DOI 10.1007/BF02392469 | MR 1351321 | Zbl 0847.35099
[8] A. Chávez, S. Castillo, M. Pinto: 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 3177565 | Zbl 1301.47057
[9] T. Diagana: Weighted pseudo-almost periodic solutions to some differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A 68 (2008), 2250-2260. DOI 10.1016/j.na.2007.01.054 | MR 2398647 | Zbl 1131.42006
[10] K. Ezzinbi, S. Fatajou, G. M. N'Guérékata: 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 10.1016/j.na.2008.02.039 | MR 2483585 | Zbl 1165.34418
[11] R. Farwig, T. Hishida: Stationary Navier-Stokes flow around a rotating obstacle. Funkc. Ekvacioj, Ser. Int. 50 (2007), 371-403. DOI 10.1619/fesi.50.371 | MR 2381323 | Zbl 1180.35408
[12] M. Geissert, H. Heck, M. Hieber: $L^p$-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596 (2006), 45-62. DOI 10.1515/CRELLE.2006.051 | MR 2254804 | Zbl 1102.76015
[13] M. Geissert, M. Hieber, T. H. Nguyen: A general approach to time periodic incompressible viscous fluid flow problems. Arch. Ration. Mech. Anal. 220 (2016), 1095-1118. DOI 10.1007/s00205-015-0949-8 | MR 3466842 | Zbl 1334.35231
[14] V. T. N. Ha, N. T. Huy, L. T. Sac, P. T. Xuan: Almost automorphic solutions to evolution equations in interpolation spaces and applications. Int. J. Evol. Equ. 11 (2018), 501-516.
[15] M. Hieber, T. H. Nguyen, A. Seyfert: 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 10.1007/978-3-319-66764-5_4 | MR 3746188 | Zbl 1384.35088
[16] T. Hishida, Y. Shibata: $L_p - L_q$ 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 10.1007/s00205-008-0130-8 | MR 2525121 | Zbl 1169.76015
[17] N. V. Minh, T. T. Dat: On the almost automorphy of bounded solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl. 326 (2007), 165-178. DOI 10.1016/j.jmaa.2006.02.079 | MR 2277774 | Zbl 1115.34068
[18] N. V. Minh, T. Naito, G. Nguerekata: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134 (2006), 3257-3266. DOI 10.1090/S0002-9939-06-08528-5 | MR 2231910 | Zbl 1120.34044
[19] G. M. N'Guérékata: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic, New York (2001). DOI 10.1007/978-1-4757-4482-8 | MR 1880351 | Zbl 1001.43001
[20] G. M. N'Guérékata: Topics in Almost Automorphy. Springer, New York (2005). DOI 10.1007/b139078 | MR 2107829 | Zbl 1073.43004
[21] T. H. Nguyen: Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle. Arch. Ration. Mech. Anal. 213 (2014), 689-703. DOI 10.1007/s00205-014-0744-y | MR 3211864 | Zbl 1308.35172
[22] T. H. Nguyen, V. D. Trinh, T. N. H. Vu, T. M. Vu: Boundedness, almost periodicity and stability of certain Navier-Stokes flows in unbounded domains. J. Differ. Equations 263 (2017), 8979-9002. DOI 10.1016/j.jde.2017.08.061 | MR 3710710 | Zbl 1378.35223
[23] T. H. Nguyen, T. N. H. Vu, P. T. Xuan: Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Commun. Pure Appl. Anal. 15 (2016), 2103-2116. DOI 10.3934/cpaa.2016029 | MR 3565934 | Zbl 1349.35272
[24] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18. North Holland, Amsterdam (1978). MR 0503903 | Zbl 0387.46032
[25] J.-T. Xiao, J. Liang, J. Zhang: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76 (2008), 518-524. DOI 10.1007/s00233-007-9011-y | MR 2395199 | Zbl 1154.46023
[26] M. Yamazaki: The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force. Math. Ann. 317 (2000), 635-675. DOI 10.1007/PL00004418 | MR 1777114 | Zbl 0965.35118

Affiliations:   Thieu Huy Nguyen, Thi Ngoc Ha Vu (corresponding author), School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam, e-mail: huy.nguyenthieu@hust.edu.vn, ha.vuthingoc@hust.edu.vn; The Sac Le, Truong Xuan Pham, Thuyloi University, Dai hoc Thuy Loi, 175 Tay Son, Dong Da, Hanoi, Vietnam, e-mail: SacLT@tlu.edu.vn, xuanpt@tlu.edu.vn


 
PDF available at: