In this appendix, we will use the letterη for the fluid flow andτ for the solid flow.
To (`, r)∈ C0([0, T];R2×R) we can associateh`,r, θ`,r, u`,rS , τ`,r and F`,r by (3), (4), (5), (6) and (7). We also introduce
ϕ`,r := Ψ(τ`,r),
where Ψ was defined in Lemma 1. We can ensure that τ`,r belongs to the set U of definition of Ψ by choosingT suitably small.
We may omit the dependence on (`, r) on the above objects when there is no ambiguity.
As in Section 3.1 we suppose that∂Ω hasg+ 1 connected components Γ1, . . . ,Γg+1and that Γg+1 is the outer one; and we denote by Γ0= Γ0(t) =∂S(t), bytthe tangent to∂Ω and∂S(t) and we define
γ0i :=
Z
Γi
u0·tdσ fori= 1, . . . , g and γ0:=
Z
∂S0
u0·tdσ.
We will use the following variant of Lemma 1.
Lemma 11. For anyR >0, there existsC >0 such that ifS=τ(S0)forτ∈SE(2)satisfies (45)then any u: Ω\ S →R2 satisfying
divu= 0 in Ω\ S, u·n= 0 on ∂Ω and u·n= (`+rx⊥)·non ∂S, where(`, r)∈R2×R, verifies (setting againΓ0:=∂S):
kukLL(Ω\S)6C
kcurlukL∞(Ω\S)+
g
X
i=0
Z
Γi
u·tdσ
+|`|+|r|
. (106)
Proof. It is a direct consequence of Lemma 1 and Morrey’s estimates. It can also be established directly by following the lines of the proof of Lemma 1.
5.1 With a prescribed solid movement
We first prove the following result, which concerns the Euler system with a prescribed solid movement ofS(t) inside Ω, and gives existence of a solution as long as no collision occurs.
Proposition 4. Let T > 0 and a regular closed connected subset S0 ⊂ Ω and define F0 := Ω\ S0. Consider (`, r)∈C0([0, T];R2×R)such that
for any t∈[0, T], dist τ`,r(t)[S0], ∂Ω
>0. (107)
Consider u0∈C0(F0;R2)satisfying (15)and (16). Then the problem (9)-(10)-(11)-(12)(with S(t) :=
τ`,r(S0)andF(t) := Ω\ S(t)) admits a unique solution
u∈L∞(0, T;LL(F(t)))∩C0([0, T];W1,q(F(t)))], ∀q∈[1,+∞).
Proof of Proposition 4. We use Schauder’s fixed point theorem in order to prove the existence part. Let (`, r) be fixed so that (107) holds. We deduceτ(t),ϕ(t),S(t) andF(t) as previously. We will also use, forT >0, the notation
FT :=∪t∈(0,T){t} × F(t).
We let
C:=n
w∈L∞((0, T)× F0)/kwkL∞((0,T)×F0)6kω0kL∞(F0)
o . We endowC with theL∞(0, T;L3(F0)) topology. Note thatCis closed and convex.
Now we define T =T`,r :C → C as follows. Givenw∈ C, we define ω:FT →R2 by
ω(t, x) =w(ϕ(t)−1(x)), (108)
which belongs toL∞(0, T;L∞(F(t))). Next we defineu:FT →R2 by the following system
curlu=ω in FT, div u= 0 inFT, u·n= 0 on [0, T]×∂Ω,
u(t, x)·n=uS(t, x)·nfort∈[0, T] andx∈∂S(t), R
Γiu·tdσ=γ0i for alli= 1. . . g, R
∂S(t)u·tdσ=γ0,
(109)
withuS defined in (4). According to Lemma 11,ubelongs toL∞(0, T;LL(F(t))).
Consequently we can define the flowη(t, x) associated touin a unique way. This flow sends, for each t, F0 toF(t). Finally, we let
T(w) :=ω0◦η(t,·)−1◦ϕ(t). (110)
It is trivial thatT(C)⊂ C. It remains to prove thatT is continuous and thatT(C) is relatively compact inL∞((0, T);L3(F0)).
Let us begin with the continuity. We consider (wn)∈ CNconverging tow∈ Cfor theL∞(0, T;L3(F0)) norm. We associate un and ηn corresponding to wn in the above construction, and accordingly uand η corresponding to w. Using (109), Sobolev imbeddings and Lemma 1, it is not difficult to see that the velocitiesun converge to the velocityuinL∞(0, T, L∞(F(t))). Also, from Lemma 11 we deduce that for someC >0,
kukL∞(0,T;LL(F(t))), kunkL∞(0,T;LL(F(t)))6C.
This involves the uniform convergence of η−1n to η−1. The convergence of ϕn to ϕ comes from the continuity of Ψ. From this we can deduce that
T(wn)−→ T(w) in L∞((0, T);L3(F0)).
Indeed, if ω0 ∈ C0(F0), this can be straightforwardly deduced from the uniform continuity of ω0 and (110). The general case can be inferred by using the density of C0(F0) in L∞(F0) for the L3(F0) topology.
Now let us prove the relative compactness ofT(C) inL∞(0, T;L3(F0)). This is a consequence of the following lemma.
Lemma 12. Let C >0,α∈(0,1) andω0∈L∞(F0). Then the set A(ω0) :=n
ω0◦ψ(t, x)forψ∈Cα([0, T]× F0;F0) such thatkψkCα6C andψ is measure-preservingo , is relatively compact in L∞(0, T;L3(F0)).
Proof of Lemma 12. We prove the total boundedness of A(ω0). Let us be given ε > 0. There exists ω1∈C0(F0) such that
kω1−ω0kL3(F0)6ε.
Due to the continuity ofω1, it is a direct consequence of Ascoli’s theorem thatA(ω1) is relatively compact inC0([0, T]× F0), and hence inL∞(0, T;L3(F0)). We deduce the existence ofψ1, . . . ,ψN as above such that
A(ω1)⊂B(ω1◦ψ1;ε)∪ · · · ∪B(ω1◦ψN;ε), where the balls are considered in the spaceL∞(0, T;L3(F0)). One sees that
A(ω0)⊂B(ω1◦ψ1; 2ε)∪ · · · ∪B(ω1◦ψN; 2ε), which concludes the proof of the lemma.
Back to the proof of Proposition 4. Using again Lemma 11, we see that we have uniform log-Lipschitz estimates on the velocities uas w ∈ C. This implies uniform H¨older estimates on the flows η that we constructed forw∈ C. So we conclude by Lemma 12 thatT(C) is relatively compact.
Hence we deduce by Schauder’s fixed point theorem that T admits a fixed point. One checks easily that the correspondingufulfills the claims.
Finally, the uniqueness is proved exactly as in Yudovich’s original setting for the fluid alone; alterna-tively, one can use the proof of uniqueness established in Subsection 3.2 with (`1, r1) = (`2, r2).
5.2 Continuous dependence on the solid movement
Now we prove that the solution constructed in Subsection 5.1 depends continuously on the solid movement (`, r).
Precisely, givenT >0 and (`, r) as in Subsection 5.1, we denoteu`,rthe unique corresponding solution ugiven by Proposition 4, andω`,r := curlu`,r the corresponding vorticity. We associate then
˜
u`,r:=u`,r◦ϕ`,r and ˜ω`,r:=ω`,r◦ϕ`,r. We have the following Proposition.
Proposition 5. Let T > 0, (`n, rn) ∈ C0([0, T];R2×R)N and (`, r) ∈ C0([0, T];R2×R) such that (`n, rn)and(`, r)satisfy (107)and
(`n, rn)−→(`, r)in C0([0, T];R2×R) as n→+∞.
Then
˜
u`n,rn−→u˜`,r inC0([0, T]× F0) as n→+∞. (111) Proof of Proposition 5. Following Subsection 5.1, we see that (˜u`n,rn) is relatively compact inC0([0, T]×
F0). Also, the sequence (˜ω`n,rn) is weakly-∗ relatively compact inL∞((0, T)× F0). To prove (111), it is hence sufficient to prove that the unique limit point of the sequence (˜u`n,rn,ω˜`n,rn) in the space C0([0, T]× F0)×[L∞((0, T)× F0)−w∗] is (˜u`,r,ω˜`,r).
Now consider a converging subsequence of (˜u`n,rn,ω˜`n,rn). For notational convenience, we still denote this subsequence (˜un,ω˜n), and call (˜u,ω) the limit. We associate the functions˜ wn,ηn,w,ηcorresponding to (`n, rn,u˜n,ω˜n) and (`, r,u,˜ ω) as in Subsection 5.1.˜
By uniqueness in Proposition 4, to prove
(˜u,ω) = (˜˜ u`,r,ω˜`,r),
is it sufficient to prove that (˜u,ω) corresponds to a solution of Proposition 4 with prescribed solid˜ movement (`, r). For that, we observe that the relation
wn(t,·) =ω0◦ηn(t,·)−1◦ϕ`n,rn(t).
passes to the L∞((0, T)× F0) weak-∗ limit (or to the L∞(0, T;Lp(F0)) one, p∈ [1,∞)) since ˜u`n,rn converges uniformly to ˜u, so the corresponding flows converge uniformly (using again the uniform log-Lipschitz estimates). So we infer
w(t,·) =ω0◦η(t,·)−1◦ϕ`,r(t).
On the other side, it is not difficult to pass to the limit in (109), so that in particular curl
˜u◦(ϕ`,r(t,·)−1)
=w(t,·)◦(ϕ`,r(t,·)−1).
Hence we deduce that (˜u,ω) is indeed a solution of Proposition 4 with solid movement (`, r). The˜ convergence (111) follows.
5.3 Endgame
Let us now proceed to the proof of Theorem 1. Once again we are going to use Schauder’s fixed point theorem.
whereT >0 is chosen such thatCT(1 + diam(S0))6 d3. Observe that this condition yields in particular thatτ`,r satisfies
dist τ`,r(t)(S0), ∂Ω
>d
3 for any t∈[0, T]. (112)
Note thatDis closed and convex.
Now we construct an operator A on D in the following way. To (`, r) ∈ D, we associate, as in Subsection 5.1, Q(t), S(t), F(t) and u as the fixed point of the operator T`,r. We also consider the Kirchhoff potentials Φi as in (55) and the mass matrix Mas in (57). Then, we defineA(`, r) := (˜`,˜r), follows from Ascoli’s theorem. ThatAis continuous follows from Proposition 5 and the convergence for allt, under the assumptions of Proposition 5:
Φ`in,rn◦ϕ`n,rn −→Φ`,ri ◦ϕ`,r in C2(F0) as n→+∞.
This convergence can be deduced from the compactness of the sequence (∇Φ`in,rn) in C1(F0) (due to Lemma 2) and the fact that Φ`,r◦ϕ`n,rn◦(ϕ`,r)−1 converges to a function satisfying the correct system (55) in the limit (use for instance the computations of Subsection 3.2). Therefore we can apply Schauder’s theorem which proves the existence of a fixed point.
To see that a fixed point of the operator A corresponds to a solution of (1)-(14) it is sufficient to observe that, thanks to an integration by parts, the solid equations can be recast as
M
Observe that this reformulation is slightly different from (56) and is obtained by using (9) and an integration by parts instead of (54). This establishes the existence part of Theorem 1.
Finally, that the lifetime can be uniquely limited by a possible encounter of the body with the boundary follows by contraposition, as a positive distance allows to extend the solution for a while, according to the previous arguments.
Remark 4. Mixing the techniques of [16] and the ones of [15], one could prove some results about the regularity in time of the flows associated to the solutions given by Theorem 1. More precisely, consider a solution (`, r, u)given by Theorem 1, then the corresponding fluid velocity field uis log-Lipschitz in the x-variable; consequently there exists a unique flow mapη continuous from R× F0 toF(t)such that
η(t, x) =x+ Z t
0
u(s, η(s, x))ds.
Moreover there existsc >0 such that for anyt, the vector fieldη(t,·) lies in the H¨older space C0,exp(−c|t|kω0kL∞(F0 ))
(F0).
If one assume that the boundaries∂S0and∂ΩareCk+1,ν, withk∈Nandν ∈(0,1), then the flow(τ, η) are Ck from [0, T] to SE(2)×C0,exp(−cTkω0kL∞(F0 ))(F0). If one assume that the boundaries ∂S0 and
∂Ωare Gevrey of order M >1, then the flow (τ, η)are Gevrey of order M + 2 from[0, T] to SE(2)× C0,exp(−cTkω0kL∞(F0 ))(F0). In particular in the case whereM = 1, we see that when the boundaries are real-analytic, then the flows(τ, η)belong to the Gevrey spaceG3.
Acknowledgements. The authors are partially supported by the Agence Nationale de la Recherche, Project CISIFS, grant ANR-09-BLAN-0213-02.
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