Abstract
We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in L^p-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.
Highlights
Introduction and main resultWe consider the system ρ(∂tu + (u · ∇)u)) − div T (u, q) = 0, div u = 0, Γ(0) = Γ0, u = VΓ, ν 1 ·en eτn T (u, q)ν =φΓ, VΓ(0) = V0, u(0) = u0, ⎫t > 0, t > 0, x x ∈ ∈ Ω(t), Ω(t), ⎪⎪⎪⎪⎬ t t x
We show the existence of strong solutions for small data and give a precise description of the maximal regularity spaces for the unknowns
(b) We remark that the maximal regularity space Eη for η describing the boundary is not a standard space. It is given as an intersection of three Sobolev spaces. This is due to the fact that the symbol of the complete system has an inherent inhomogeneous structure, and the Newton polygon method is the correct tool to show maximal regularity
Summary
We remark that in the formulation of the boundary conditions in lines 3 and 4 of (1.1), one has to take into account that the Kirchhoff plate model is formulated in a Lagrangian setting, whereas for the fluid an Eulerian setting is used. This is discussed in more detail in the beginning of Sect. (b) We remark that the maximal regularity space Eη for η describing the boundary is not a standard space It is given as an intersection of three Sobolev spaces. The corresponding estimates of the nonlinearities had to be derived by more direct methods
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