Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

Hongjie Dong, Doyoon Kim, Tuoc Phan

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1 Citation (Scopus)


We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm (Formula presented.) -estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with (Formula presented.) the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

Original languageEnglish
Pages (from-to)1700-1731
Number of pages32
JournalCommunications in Partial Differential Equations
Issue number8
Publication statusPublished - 2022

Bibliographical note

Funding Information:
H. Dong was partially supported by the NSF under agreement DMS-1600593; D. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1A2C1084683); T. Phan is partially supported by the Simons Foundation, grant #354889.

Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.


  • Time-dependent Stokes system
  • boundary Lebesgue mixed-norm estimates

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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