By Josef Bemelmans, Giovanni P. Galdi, Mads Kyed

The authors examine the unconstrained (free) movement of an elastic good $\mathcal B$ in a Navier-Stokes liquid $\mathcal L$ occupying the complete area outdoors $\mathcal B$, below the belief consistent physique strength $\mathfrak b$ is performing on $\mathcal B$. extra in particular, the authors have an interest within the regular movement of the coupled approach $\{\mathcal B,\mathcal L\}$, this means that there exists a body with recognize to which the correct governing equations own a time-independent resolution. The authors end up the life of one of these body, supplied a few smallness regulations are imposed at the actual parameters, and the reference configuration of $\mathcal B$ satisfies appropriate geometric houses

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Extra resources for On the steady motion of a coupled system solid-liquid

Example text

5. In fact, when establishing properties that do not require more than C 1 -regularity of the boundary (C 1,α to be precise), we may choose to work with the equations in the current conﬁguration and thereby avoid the perturbation terms occurring when the equations are written in the reference domain. 10) is uniquely solvable in the class of weak solutions deﬁned above. 1, we need to work in a setting where the solutions to the ﬂuid equations have higher order regularity near the boundary ∂Ω. It will therefore be necessary not only to show the existence a weak solution z, but also to establish the existence of a pressure term π so that (z, π) is a strong solution near the boundary.

In fact, V is a unique weak solution in W01,2 (Eσεu )3 . 35) applied to the diﬀerence V − V that V − V = 0. 21). 17) in the distributional sense. 39) εu Eα 1 −1 π ˆ |Jεu | dy = 0. We thereby also obtain uniqueness of π ˆ. 39). We have also established the bound |ˆ z |1,2 ≤ |V |1,2 + |WR |1,2 ≤ |V |1,2 + c36 R− 2 ≤ c37 (ε2 RR2 + R− 2 ). 1 1 We emphasize that c37 does not depend on σ. We need a similar bound, in an appropriate norm, on π ˆ . 40) |E | εu ⎪ E α 1 ⎪ α1 ⎪ ⎪ ⎩ |s| ≤ c εu . π ˆ 1,2 38 2,Eα1 52 5.

11. 9. 39) T(z, π) − Tu (Z, Π) 1−1/s,s,∂Ω ≤ C8 with C8 = C8 (s, t). 5. THE STOKES PROBLEM 31 Proof. Put (w, q) := (z − Z, π − Π). Then (w, q) ∈ YSs,t (E) and satisﬁes ⎧ T T ⎪ ⎨ Δw − ∇q = div ∇Z(Fu Au − I) − Π(Au − I) in E, in E, div w = (Au − I)T : ∇Z ⎪ ⎩ w = v∗ − v˜∗ on ∂Ω. 40) ≤ c2 v˜∗ + Π 1,s,ER0 ) u 2,p,Ω + 2−1/s,s u 2−1/s,s 2,p,Ω + v∗ − v˜∗ 2−1/s,s , with c2 = c2 (s, t). 41) ≤ c4 T(z − Z, π − Π) ≤ c5 z−Z +( Z ≤ c6 2,s,ER0 2,s,ER0 (w, q) 1,s,ER0 + π−Π + Π s,t YS,R (E) 0 1,s,ER0 + T(Z, Π) − Tu (Z, Π) 1,s,ER0 2,s,ER0 ) + v˜∗ 1,s,ER0 u 2,p,Ω 2−1/s,s,∂Ω u 2,p,Ω with c6 = c6 (s, t).