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Existence and Stability of Periodic and Almost Periodic Solutions to the Boussinesq System in Unbounded Domains

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Abstract

In this paper we investigate the existence and stability of periodic solutions (on a half-line ℝ+) and almost periodic solutions on the whole line time-axis ℝ to the Boussinesq system on several classes of unbounded domains of ℝn in the framework of interpolation spaces. For the linear Boussinesq system we combine the Lp — Lq-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions. Then, we prove the existence of periodic solutions by invoking Massera’s principle. We also prove the existence of almost periodic solutions. Then we use the results of the linear Boussinesq system to establish the existence, uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces. Our results cover and extend the previous ones obtained in [13, 34, 38].

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References

  1. de Almeida M F, Ferreira L C F. On the well-posedness and large time behavior for Boussinesq equations in Morrey spaces. Differential Integral Equations, 2011, 24(7/8): 719–742

    MathSciNet  MATH  Google Scholar 

  2. Banquet C, Villamizar-Roa E J. Existence theory for the Boussinesq equation in modulation spaces. Bulletin of the Brazilian Mathematical Society, New Series, 2020, 51: 1057–1082

    Article  MathSciNet  Google Scholar 

  3. Bergh J, Löfström J. Interpolation Spaces. An Introduction. Grundlehren der mathematischen Wissenschaften. Berlin: Springer, 1976

    Google Scholar 

  4. Bogovski M E. Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl Akad Nauk SSSR, 1979, 248(5): 1037–1040

    MathSciNet  Google Scholar 

  5. Borchers W, Sohr H. On the equations rotv = g and divu = f with zero boundary conditions. Hokkaido Math J, 1990, 19(1): 67–87

    Article  MathSciNet  Google Scholar 

  6. Brandolese L, Schonbek M E. Large time decay and growth for solutions of a viscous Boussinesq system. Trans Amer Math Soc, 2012, 364(10): 5057–5090

    Article  MathSciNet  Google Scholar 

  7. Borchers W, Sohr H. On the semigroup of the Stokes operator for exterior domains in Lq-spaces. Math Z, 1987, 196: 415–425

    Article  MathSciNet  Google Scholar 

  8. Borchers W, Miyakawa T. On stability of exterior stationary Navier-Stokes flows. Acta Math, 1995, 174: 311–382

    Article  MathSciNet  Google Scholar 

  9. Cannon J R, DiBenedetto E. The initial value problem for the Boussinesq equations with data in Lp//Rautmann R. Approximation Methods for Navier-Stokes Problems, Lect Notes in Math. 771. Berlin: Springer-Verlag, 1980: 129–144

    Chapter  Google Scholar 

  10. Chen Z-M, Kagei Y, Miyakawa T. Remarks on stability of purely conductive steady states to the exterior Boussinesq problem. Adv Math Sci Appl, 1992, 1/2: 411–430

    MathSciNet  MATH  Google Scholar 

  11. Ferreira L C F, Villamizar-Roa E J. Well-posedness and asymptotic behaviour for the convection problem in ℝn. Nonlinearity, 2006, 19(9): 2169–2191

    Article  MathSciNet  Google Scholar 

  12. Ferreira L C F, Villamizar-Roa E J. Existence of solutions to the convection problem in a pseudomeasure-type space. Proc R Soc Lond Ser A Math Phys Eng Sci, 2008, 464(2096): 1983–1999

    MathSciNet  MATH  Google Scholar 

  13. Ferreira L C F, Villamizar-Roa E J. On the stability problem for the Boussinesq equations in weak-Lpspaces. Commun Pure Appl Anal May, 2010, 9(3): 667–684

    Article  Google Scholar 

  14. Fife P C, Joseph D D. Existence of convective solutions of the generalized Bernard problem which are analytic in their norm. Arch Rational Mech Anal, 1969, 33: 116–138

    Article  MathSciNet  Google Scholar 

  15. Geissert M, Hieber M, Nguyen T H. A general approach to time periodic incompressible viscous fluid flow problems. Arch Rational Mech Anal, 2016, 220: 1095–1118

    Article  MathSciNet  Google Scholar 

  16. Giga Y, Sohr H. On the Stokes operator in exterior domains. J Fac Sci Univ Tokyo, 1989, 36: 313–333

    MathSciNet  Google Scholar 

  17. Ha V T N, Huy N T, Mai V T. Parabolic evolution equations in interpolation spaces: boundedness, stability, and applications. Zeitschrift für Angewandte Mathematik und Physik ZAMP, 2020, 71(39): 1–17

    MathSciNet  MATH  Google Scholar 

  18. Hieber M, Huy N T, 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, 2017: 51–81

  19. Ishige K. Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition. Differential Integral Equations, 2009, 22: 401–410

    MathSciNet  MATH  Google Scholar 

  20. Nguyen T H. Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle. Arch Ration Mech Anal, 2014, 213: 689–703

    Article  MathSciNet  Google Scholar 

  21. Huy N T, Ha V T N, Xuan P T. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Commun Pure Appl Anal, 2016, 15(6): 2103–2116

    Article  MathSciNet  Google Scholar 

  22. Huy N T, Duoc T V, Ha V T N, Mai V T. Boundedness, almost periodicity and stability of certain Navier-Stokes flows in unbounded domains. J Differ Equ, 2017, 263(12): 8979–9002

    Article  MathSciNet  Google Scholar 

  23. Hishida T. Asymptotic behavior and stability of solutions to the exterior convection problem. Nonlinear Anal, 1994, 22: 895–925

    Article  MathSciNet  Google Scholar 

  24. Hishida T. Global existence and exponential stability of convection. J Math Anal Appl, 1995, 196: 699–721

    Article  MathSciNet  Google Scholar 

  25. Hishida T. On a class of stable steady flow to the exterior convection problem. J Differ Equ, 1997, 141(1): 54–85

    Article  MathSciNet  Google Scholar 

  26. Joseph D. Stability of Fluid Motion. Berlin: Springer-Verlag, 1976

    Book  Google Scholar 

  27. Kato T. Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions. Mathematische Zeitschrift, 1984, 187: 471–480

    Article  MathSciNet  Google Scholar 

  28. Karch G, Prioux N. Self-similarity in viscous Boussinesq equations. Proc Amer Math Soc, 2008, 136(3): 879–888

    Article  MathSciNet  Google Scholar 

  29. Kozono H, Yamazaki M. Exterior problem for the stationary Navier-Stokes equations in the Lorentz space. Math Ann, 1998, 310(2): 279–305

    Article  MathSciNet  Google Scholar 

  30. Kozono H, Nakao M. Periodic solution of the Navier-Stokes equations in unbounded domains. Tôhoku Math J, 1996, 48: 33–50

    Article  MathSciNet  Google Scholar 

  31. Lukaszewicz G, Ortega-Torres E E, Rojas-Medar M A. Strong periodic solutions for a class of abstract evolution equations. Nonli Anal, 2003, 54(6): 1045–1056

    Article  MathSciNet  Google Scholar 

  32. Liu X, Li Y. On the stability of global solutions to the 3D Boussinesq system. Nonli Anal, 2014, 95: 580–591

    Article  MathSciNet  Google Scholar 

  33. Morimoto H. Non-stationary Boussinesq equations. Proc Japan Acad Ser A Math Sci, 1991, 67(5): 159–161

    Article  MathSciNet  Google Scholar 

  34. Nakao E. On time-periodic solutions to the Boussinesq equations in exterior domains. J Math Anal Appl, 2020, 482(2): 123537

    Article  MathSciNet  Google Scholar 

  35. Yamazaki M. Solutions in Morrey spaces of the Navier-Stokes equation with time-dependent external force. Funkcial Ekvac, 2000, 43: 419–460

    MathSciNet  MATH  Google Scholar 

  36. Yamazaki M. The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math Ann, 2000, 317: 635–675

    Article  MathSciNet  Google Scholar 

  37. Taylor M. Partial Differential Equations III, Nonlinear Equations. 2nd ed. New York, Dordrecht, Heidelberg, London: Springer, 2011

    Book  Google Scholar 

  38. Villamizar-Roa E J, Rodriguez-Bellido M A, Rojas-medar M A. Periodic solutions in unbounded domains for the Boussinesq system. Acta Mathematica Sinica, English Series, 2010, 26(5): 837–862

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. 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

    Thieu Huy Nguyen, Thi Ngoc Ha Vu & The Sac Le

  2. Department of Mathematics, Faculty of Information Technology, Thuyloi university Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Viet Nam

    Truong Xuan Pham

  3. Thuyloi University, Dai hoc Thuy Loi, 175 Tay Son, Dong Da, Hanoi, Viet Nam

    The Sac Le

Authors
  1. Thieu Huy Nguyen
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  2. Truong Xuan Pham
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  3. Thi Ngoc Ha Vu
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  4. The Sac Le
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Correspondence to Thieu Huy Nguyen.

Additional information

This work was financially supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2021.04. The work of the last author was financially supported by Vietnam Ministry of Education and Training under Project B2022-BKA-06.

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Nguyen, T.H., Pham, T.X., Vu, T.N.H. et al. Existence and Stability of Periodic and Almost Periodic Solutions to the Boussinesq System in Unbounded Domains. Acta Math Sci 42, 1875–1901 (2022). https://doi.org/10.1007/s10473-022-0510-4

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  • DOI: https://doi.org/10.1007/s10473-022-0510-4

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