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Stokes Problems in Irregular Domains with Various Boundary Conditions

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Author

Listed:
  • Sylvie Monniaux

    (CNRS, Centrale Marseille, Aix-Marseille Université)

  • Zhongwei Shen

    (University of Kentucky)

Abstract

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in ℝ 3 $$\mathbb{R}^{3}$$ , such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behavior near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed-point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.

Suggested Citation

  • Sylvie Monniaux & Zhongwei Shen, 2018. "Stokes Problems in Irregular Domains with Various Boundary Conditions," Springer Books, in: Yoshikazu Giga & Antonín Novotný (ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, chapter 4, pages 207-248, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-13344-7_4
    DOI: 10.1007/978-3-319-13344-7_4
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