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

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.

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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.

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Correspondence to Tuoc Phan.

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T. Phan’s research is partially supported by the Simons Foundation, Grant #354889.

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Phan, T. Well-posedness for the Navier–Stokes equations in critical mixed-norm Lebesgue spaces. J. Evol. Equ. 20, 553–576 (2020). https://doi.org/10.1007/s00028-019-00529-1

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Keywords

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

Mathematics Subject Classification

  • 35Q30
  • 76D05
  • 76D03
  • 76N10