Existence of Incompressible and Immiscible Flows in Critical Function Spaces on Bounded Domains

  • Myong-Hwan Ri
  • Ping ZhangEmail author


We study global existence and uniqueness of solutions to inhomogeneous incompressible Navier–Stokes equations on bounded domains of \(\mathbb {R}^n, n\ge 2\), with initial velocity in the Besov space \(B^0_{q,\infty }(\Omega )\), \(q\ge n\), and piecewise constant initial density. Existence of solutions is proved when \(B^0_{n,\infty }\)-norm of initial velocity and initial density difference are small, and for uniqueness we require that \(q>n\). The proof of existence of solutions is done via an iterative scheme based on maximal \(L^\infty _\gamma \)-regularity of the Stokes operator in little Nicolskii spaces and on solvability for transport equations in the spaces of pointwise multipliers for little Nicolskii spaces, while the proof of uniqueness is done via a Lagrangian approach using the result of an time-evolutionary Stokes system with nonzero divergence obtained in this paper.


Existence Uniqueness Inhomogeneous Navier–Stokes equations Immiscible flow Divergence problem 

Mathematics Subject Classification

35Q30 35B35 76D03 76D07 76E99 



P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Institute of MathematicsState Academy of SciencesPyongyangDemocratic People’s Republic of Korea
  2. 2.Academy of Mathematics and Systems Science and Hua Loo-Keng Center for Mathematical SciencesChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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