Advertisement

Well-posedness for the Navier–Stokes equations in critical mixed-norm Lebesgue spaces

  • Tuoc PhanEmail author
Article
  • 14 Downloads

Abstract

We study the Cauchy problem in n-dimensional space for the system of Navier–Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong, and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper demonstrate the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Young’s inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz–Leray projection, and the boundedness of the Riesz transform are developed in mixed-norm Lebesgue spaces. These analysis results are topics of independent interests, and they are potentially useful in other problems.

Keywords

Local well-posedness Global well-posedness Navier–Stokes equations Mixed-norm Lebesgue spaces 

Mathematics Subject Classification

35Q30 76D05 76D03 76N10 

Notes

Acknowledgements

The author wishes to thank anonymous referees for their important remarks and suggestions that significantly improve the manuscript. The author also would like to thank professor Lorenzo Brandolese (Institut Camille Jordan, Université Lyon 1) and professor Nam Le (Indiana University) for their interests and valuable comments.

References

  1. 1.
    H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011.Google Scholar
  2. 2.
    J. Bourgain, N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal. 255 (2008), 2233–2247.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. Brandolese, F. Vigneron, New asymptotic profiles of nonstationary solutions of the Navier-Stokes system. J. Math. Pures Appl. (9) 88 (2007), no. 1, 64–86.Google Scholar
  4. 4.
    M. Cannone, Ondelettes, paraproduits et Navier-Stokes. Diderot Editeur, Paris, 1995.zbMATHGoogle Scholar
  5. 5.
    M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam. 13 (1997), 515–541.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Cannone, F. Planchon, On the non-stationary Navier-Stokes equations with an external force. Adv. Differential Equations 4 (1999), no. 5, 697–730.MathSciNetzbMATHGoogle Scholar
  7. 7.
    D. V. Cruz-Uribe, J. M. Martell, and C. Pérez. Weights, extrapolation and the theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
  8. 8.
    H. Dong, D. Kim, On \(L_p\) -estimates for elliptic and parabolic equations with \(A_p\) weights. Trans. Amer. Math. Soc. 370 (2018), no. 7, 5081–5130.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. Dong, N.V. Krylov, Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces, arXiv:1806.00077.
  10. 10.
    H. Dong, T. Phan, Mixed norm \(L_p\) -estimates for non-stationary Stokes systems with singular VMO coefficients and applications, arXiv:1805.04143.
  11. 11.
    T. Kato, H. Fujita, On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32 1962 243–260.MathSciNetzbMATHGoogle Scholar
  12. 12.
    H. Fujita, T. Kato, On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 1964, 269–315.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. García-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matemática, 104. North-Holland Publishing Co., Amsterdam, 1985.Google Scholar
  14. 14.
    Y. Giga, T. Miyakawa, Solutions in \(L^r\) of the Navier-Stokes initial value problem. Arch. Rational Mech. Anal. 89 (1985), no. 3, 267–281.Google Scholar
  15. 15.
    Y. Giga and T. Miyakawa, Navier-Stokes flow in \({\mathbb{R}}^3\) with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577–618.Google Scholar
  16. 16.
    Y. Giga, Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations 62 (1986), no. 2, 186–212.Google Scholar
  17. 17.
    T. Kato, Strong \(L^p\) -solutions of the Navier-Stokes equation in \({\mathbb{R}}^m\) , with applications to weak solutions. Math. Z. 187 (1984), no. 4, 471–480.Google Scholar
  18. 18.
    T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat. (N.S.) 22 (1992), 127–155.Google Scholar
  19. 19.
    H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001), no. 1, 22–35.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    H. Kozono, M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    H. Kozono, M. Yamazaki, The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation, Indiana Univ. Math. J. 44 (1995), 1307–1335.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    N.V. Krylov, Rubio de Francia extrapolation theorem and related topics in the theory of elliptic and parabolic equations. A survey, arXiv:1901.00549.
  23. 23.
    N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal. 250 (2007), no. 2, 521–558.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. R. Lemarié-Rieusset, The Navier-Stokes problem in the 21st century. CRC Press, Boca Raton, FL, 2016.CrossRefzbMATHGoogle Scholar
  25. 25.
    J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l’Hydrodynamique, J. Math. Pures Appl. 9 (1933), 1–82.zbMATHGoogle Scholar
  26. 26.
    J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Y. Meyer, Wavelets, Paraproducts and Navier-Stokes Equations. Current Developments in Mathematics, 1996 (Cambridge, MA), Int. Press, Boston, MA, 1997, pp. 105–212.Google Scholar
  28. 28.
    F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \({\mathbb{R}}^3\), Ann. Inst. Henri Poincare, Anal. Non Lineaire 13 (1996), 319–336.Google Scholar
  29. 29.
    T. V. Phan, N. C. Phuc, Stationary Navier-Stokes equations with critically singular external forces: existence and stability results. Adv. Math. 241 (2013), 137–161.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    T. Phan, Liouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue spaces, arXiv:1812.10135.
  31. 31.
    M.E. Taylor, Analysis of Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407–1456.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T.-P. Tsai, Lectures on Navier-Stokes equations. Graduate Studies in Mathematics, 192. American Mathematical Society, Providence, RI, 2018.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

Personalised recommendations