Navier-Stokes equations in a curved thin domain. III: Thin-film limit. (English) Zbl 1468.35112
Adv. Differ. Equ. 25, No. 9-10, 457-626 (2020); erratum ibid. 28, No. 3-4, 341-346 (2023).
Summary: We consider the Navier-Stokes equations with Navier’s slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions, we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of the average of a strong solution to the bulk equations towards a weak solution to the limit equations by showing estimates for the difference between them. In some special case, our limit equations agree with the Navier-Stokes equations on a Riemannian manifold in which the viscous term contains the Ricci curvature. This is the first result on a rigorous derivation of the surface Navier-Stokes equations on a general closed surface by the thin-film limit.
For Parts I and II, see [the author, “Navier-Stokes equations in a curved thin domain. I: Uniform estimates for the Stokes operator”, Preprint, arXiv:2002.06343; J. Math. Fluid Mech. 23, No. 1, Paper No. 7, 60 p. (2021; Zbl 1458.35305)].
For Parts I and II, see [the author, “Navier-Stokes equations in a curved thin domain. I: Uniform estimates for the Stokes operator”, Preprint, arXiv:2002.06343; J. Math. Fluid Mech. 23, No. 1, Paper No. 7, 60 p. (2021; Zbl 1458.35305)].
MSC:
| 35Q30 | Navier-Stokes equations |
| 76A20 | Thin fluid films |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B25 | Singular perturbations in context of PDEs |
| 35R01 | PDEs on manifolds |
Citations:
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