Transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds. (English) Zbl 1464.58007
Summary: The purpose of this paper is to study boundary value problems of transmission type for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in two complementary Lipschitz domains on a compact Riemannian manifold of dimension \({m \in \{2, 3\}}\). We exploit a layer potential method combined with a fixed point theorem in order to show existence and uniqueness results when the given data are suitably small in \(L^2\)-based Sobolev spaces.
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 58C30 | Fixed-point theorems on manifolds |
Keywords:
Navier-Stokes system; Darcy-Forchheimer-Brinkman system; transmission problem; Lipschitz domain; compact Riemannian manifold; layer potential operators; Sobolev spaces; fixed point theoremReferences:
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