Asymptotic behavior of D-solutions to the steady Navier–Stokes flow in an exterior domain of a half-space

  • Zhengguang Guo
  • Peter WittwerEmail author
  • Yong Zhou


We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an exterior domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.


Steady Navier–Stokes equations Fluid–structure interaction Weak solution 

Mathematics Subject Classification

35Q30 76D05 35B40 35B42 



Zhengguang Guo expresses his gratitude to Peter Wittwer for his hospitality during the stay at the University of Geneva. The authors thank the anonymous referee for his helpful comments on the initial version of this paper.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuaiyin Normal UniversityHuaianChina
  2. 2.Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina
  3. 3.Theoretical Physics DepartmentUniversity of GenevaGenevaSwitzerland
  4. 4.School of Mathematics (Zhuhai)Sun Yat-Sen UniversityZhuhaiChina
  5. 5.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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