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Global well-posedness for fractional Navier-Stokes equations in variable exponent Fourier-Besov-Morrey spaces

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Abstract

In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space \({\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{s\left( \cdot \right)}\left( {{\mathbb{R}^3}} \right)\) with \(s\left( \cdot \right) = 4 - 2\alpha - {3 \over {p\left( \cdot \right)}}\). We prove global well-posedness result with small initial data belonging to \({\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{4 - 2\alpha - {3 \over {p\left( \cdot \right)}}}\left( {{\mathbb{R}^3}} \right)\) The result of this paper extends some recent work.

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References

  1. d’Humieres D, Lallemand P, Frisch U. Lattice gas models for 3D hydrodynamics. Europhysics Letters, 1986, 2(4): 291

    Article  Google Scholar 

  2. Leray J. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica, 1934, 63: 193–248

    Article  MathSciNet  MATH  Google Scholar 

  3. Fujita H, Kato T. On the Navier-Stokes initial value problem. I. Archive for Rational Mechanics and Analysis, 1964, 16(4): 269–315

    Article  MathSciNet  MATH  Google Scholar 

  4. Kato T, Fujita H. On the nonstationary Navier-Stokes system. Rendiconti del Seminario Matematico della Universit di Padova, 1962, 32: 243–260

    MathSciNet  MATH  Google Scholar 

  5. Kato T. StrongL p-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions. Mathematische Zeitschrift, 1984, 187(4): 471–480

    Article  MathSciNet  Google Scholar 

  6. Cannone M. Paraproduits et Navier-Stokes Diderot Editeur. Arts et Sciences, 1995

  7. Wan R, Jia H. Global well-posedness for the 3D Navier-Sokes equations with a large component of vorticity. Journal of Mathematical Analysis and Applications, 2019, 469(2): 504–524

    Article  MathSciNet  MATH  Google Scholar 

  8. Giga Y, Miyakawa T. Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces. Communications in Partial Differential Equations, 1989, 14(5): 577–618

    Article  MathSciNet  MATH  Google Scholar 

  9. Kato T. Strong solutions of the Navier-Stokes equation in Morrey spaces. Boletim da Sociedade Brasileira de Matemtica-Bulletin/Brazilian Mathematical Society, 1992, 22(2): 127–155

    Article  MathSciNet  MATH  Google Scholar 

  10. Taylor M E. Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Communications in Partial Differential Equations, 1992, 17(9/10): 1407–1456

    Article  MathSciNet  MATH  Google Scholar 

  11. Koch H, Tataru D. Well-posedness for the Navier-Stokes equations. Advances in Mathematics, 2001, 157(1): 22–35

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourgain J, Pavlovic N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. Journal of Functional Analysis, 2008, 255(9): 2233–2247

    Article  MathSciNet  MATH  Google Scholar 

  13. Biswas A, Swanson D. Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted p initial data. Indiana University Mathematics Journal, 2007, 56(3): 1157–1188

    Article  MathSciNet  MATH  Google Scholar 

  14. Konieczny P, Yoneda T. On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations. Journal of Differential Equations, 2011, 250(10): 3859–3873

    Article  MathSciNet  MATH  Google Scholar 

  15. Lei Z, Lin F. Global mild solutions of Navier-Stokes equations. Communications on Pure and Applied Mathematics, 2011, 64(9): 1297–1304

    Article  MathSciNet  MATH  Google Scholar 

  16. Lions J L. Quelques mthodes de rsolution des problemes aux limites non linaires. 1969

  17. Wu J. Generalized MHD equations. Journal of Differential Equations, 2003, 195(2): 284–312

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu J. Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Communications in Mathematical Physics, 2006, 263(3): 803–831

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang Y, Xiao J. Well/ill-posedness for the dissipative Navier-Stokes system in generalized Carleson measure spaces. Advances in Nonlinear Analysis, 2017, 8(1): 203–224

    Article  MathSciNet  MATH  Google Scholar 

  20. Li P, Xiao J, Yang Q. Global mild solutions to modified Navier-Stokes equations with small initial data in critical Besov-Q spaces. Electronic Journal of Differential Equations, 2014, 2014(185): 1–37

    MathSciNet  MATH  Google Scholar 

  21. Dong H, Li D. Optimal local smoothing and analyticity rate estimates for the generalized Navier-Stokes equations. Communications in Mathematical Sciences, 2009, 7(1): 67–80

    Article  MathSciNet  MATH  Google Scholar 

  22. Xiao J. Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited. Dyn Partial Differ Equ, 2014, 11(2): 167–181

    Article  MathSciNet  MATH  Google Scholar 

  23. Li P, Zhai Z. Well-posedness and regularity of generalized Navier-Stokes equations in some critical Q-spaces. Journal of Functional Analysis, 2010, 259(10): 2457–2519

    Article  MathSciNet  MATH  Google Scholar 

  24. Yu X, Zhai Z. Well-posedness for fractional Navier-Stokes equations in critical spaces close to \(\dot B_{\infty ,\infty }^{ - \left( {2\beta - 1} \right)}\left( {{\mathbb{R}^n}} \right)\). Math Method Appl Sci, 2012, 35(6): 676–683

    MathSciNet  Google Scholar 

  25. Deng C, Yao X. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in \(\dot F_{\alpha /3 - l}^{ - \alpha ,r}\). Dynamical Systems, 2014, 34(2): 437–459

    MathSciNet  Google Scholar 

  26. Yu X, Zhai Z. Well-posedness for fractional Navier-Stokes equations in the largest critical spaces. Comm Pure Appl Anal, 2012, 11(5): 1809–1823

    Article  MathSciNet  MATH  Google Scholar 

  27. El Baraka A, Toumlilin M. Global well-posedness for fractional Navier-Stokes equations in critical Fourier-Besov-Morrey spaces. Moroccan Journal of Pure and Applied Analysis, 2017, 3(1): 1–13

    Article  MATH  Google Scholar 

  28. Orlicz W. ber konjugierte exponentenfolgen. Studia Mathematica, 1931, 3(1): 200–211

    Article  MATH  Google Scholar 

  29. Orlicz W. ber eine gewisse Klasse von Rumen vom Typus B. Bull Int Acad Pol Ser A, 1932, 8(9): 207–220

    Google Scholar 

  30. Musielak J. Modular spaces//Orlicz Spaces and Modular Spaces. Berlin, Heidelberg: Springer, 1983: 1–32

    Chapter  MATH  Google Scholar 

  31. Nakano H. Topology and Linear Topological Spaces III. Maruzen Company, 1951

  32. Kovik O, Rkosnk J. On spaces Lp(x) and Wk,p(x). Czechoslovak Mathematical Journal, 1991, 41(4): 592–618

    Article  MathSciNet  Google Scholar 

  33. Cruz-Unbe D. The Hardy-Littlewood maximal operator on variable-Lp spaces//Seminar of Mathematical Analysis: Proceedings, Universities of Malaga and Seville (Spain), September 2002–February 2003. 2003, 64(147). Universidad de Sevilla

  34. Diening L. Maximal function on generalized Lebesgue spaces Lp(x). Univ Math Fak, 2002

  35. Cruz-Uribe D, Diening L, Hst P. The maximal operator on weighted variable Lebesgue spaces. Fractional Calculus and Applied Analysis, 2011, 14(3): 361–374

    Article  MathSciNet  MATH  Google Scholar 

  36. Cruz-Uribe D V, Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer Science Business Media, 2013

  37. Acerbi E, Mingione G. Regularity results for stationary electro-rheological fluids. Archive for Rational Mechanics and Analysis, 2002, 164(3): 213–259

    Article  MathSciNet  MATH  Google Scholar 

  38. Ruzicka M. Electrorheological Fluids: Modeling and Mathematical Theory. Springer Science and Business Media, 2000

  39. Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics, 2006 66(4): 1383–1406

    Article  MathSciNet  MATH  Google Scholar 

  40. Fan X. Global C1, Cα regularity for variable exponent elliptic equations in divergence form. Journal of Differential Equations, 2007, 235(2): 397–417

    Article  MathSciNet  Google Scholar 

  41. Luxenberg W. Banach Function Spaces. Assen, 1955

  42. Almeida A, Caetano A. Variable exponent Besov-Morrey spaces. J Fourier Anal Appl, 2020, 26 (1): Art 5

  43. Ru S, Abidin M Z. Global well-posedness of the incompressible fractional Navier-Stokes equations in Fourier-Besov spaces with variable exponents. Computers & Mathematics with Applications, 2019, 77(4): 1082–1090

    Article  MathSciNet  MATH  Google Scholar 

  44. El Baraka A, Toumlilin M. Global well-posedness and decay results for 3D generalized magneto-hydrodynamic equations in critical Fourier-Besov-Morrey spaces. Electronic Journal of Differential Equations, 2017, 65: 1–20

    MathSciNet  MATH  Google Scholar 

  45. Almeida A, Hst P. Besov spaces with variable smoothness and integrability. Journal of Functional Analysis, 2010, 258(5): 1628–1655

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Dr. Ru Shaolei for useful discussions and helpful suggestions on this article.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China

    Muhammad Zainul Abidin & Jiecheng Chen

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  1. Muhammad Zainul Abidin
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  2. Jiecheng Chen
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Correspondence to Jiecheng Chen.

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The research was supported by NSFC (11671363, 11701519).

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Abidin, M.Z., Chen, J. Global well-posedness for fractional Navier-Stokes equations in variable exponent Fourier-Besov-Morrey spaces. Acta Math Sci 41, 164–176 (2021). https://doi.org/10.1007/s10473-021-0109-1

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  • DOI: https://doi.org/10.1007/s10473-021-0109-1

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