Table Of ContentSmooth solutions for motion of a rigid body of general form
in an incompressible perfect fluid
2
1
Yun Wanga, Aibin Zangb,c∗
0
2
n
a
J
a The Department of Mathematics and Statistics, McMaster University,
7
1 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada
] E-mail: [email protected]
P
A b The School of Mathematics and Computer Science, Yichun University,
. Yichun City, Jiangxi Province, 336000, P.R. China
h
t c The Institute of Mathematical Sciences, The Chinese University of Hong Kong,
a
m Shatin, NT, Hong Kong
[
E-mail: [email protected]
1
v
Abstract
3
3
In this paper, we consider the interactions between a rigid body of general form
4
3 andthe incompressible perfectfluid surroundingit. Localwell-posednessin the space
1. C([0,T);Hs) is obtained for the fluid-rigid body system.
0 Keywords: Eulerequations;Fluid-rigidbody interaction;Exteriordomain;Clas-
2
sical solutions
1
: Mathematics Subject Classification(2000): 35Q30; 35Q35
v
i
X
r 1 Introduction
a
In this paper, we investigate the motion of a solid in an incompressible perfect fluid. The
behavior of the fluid is described by the Euler equations. The solid we consider is a rigid
body, whose movement consists of translation and rotation. And its motion conforms to
the Newton’s law. Assume that both the fluid and the rigid body are homogeneous. For
simplicity ofwriting,thedensityofthefluidequals to1. Thedomainoccupiedbythesolid
at the time is O(t), and Ω(t) = R3 \O(t) is the domain occupied by the fluid. Suppose
∗Corresponding Author;
1
O(0) = O and Ω(0) = Ω share a smooth boundary ∂O(or ∂Ω). The equations modeling
the dynamics of the system read(see also [18])
∂u
+(u·∇)u+∇p = f, in Ω(t)×[0,T], (1.1)
∂t
div u= 0, in Ω(t)×[0,T], (1.2)
u·~n = (h′+ω×(x−h(t)))·~n, on ∂Ω(t)×[0,T], (1.3)
lim u(x,t) = u , (1.4)
∞
|x|→∞
mh′′ = p~ndσ+f , in [0,T], (1.5)
rb
Z∂Ω(t)
(Jω)′ = (x−h(t))×p~ndσ+T , in [0,T], (1.6)
rb
Z∂Ω(t)
u(x,0) = u (x) x ∈ Ω, (1.7)
0
h(0) = 0 ∈ R3, h′(0) = l ∈ R3, ω(0) = ω ∈ R3. (1.8)
0 0
Intheabovesystem,uandparethevelocity fieldandthepressureofthefluidrespectively.
f is the external force field applied to the fluid. f and T denote the external force and
rb rb
the external torque for the rigid body respectively. m is the mass, and J is the inertia
matrix moment related to the mass center of the solid. Suppose the density of the rigid
body is ρ, then
m = ρdx = ρdx,
ZO(t) ZO
and
[J(t)] = ρ |x−h(t)|2δ −(x−h(t)) (x−h(t)) dx.
kl kl k l
ZO¯(t)
(cid:2) (cid:3)
Hereh(t)denotes thepositionofthemasscenter oftherigidbodyandδ istheKronecker
kl
symbol. And denote J(0) by J¯. ω(t) is the angular velocity of the rigid body. ~n is the
unit outward normal to ∂Ω(t).
Assume that the center of O is the origin, i.e.,
ydy = 0 ∈ R3.
ZO
Let Q(t) be a rotation matrix associated with the angular velocity ω(t) of the rigid
body, which is the solution of the following initial value problem:
dQ(t)
= A(ω(t))Q(t)
dt (1.9)
Q(0) = Id.
2
Here
0 −ω ω
3 2
A(ω) = ω 0 −ω ,
3 1
−ω ω 0
2 1
and Id is the identity matrix. Then the domain O(t) is given by
O(t) ={Q(t)y+h(t) : y ∈ O}.
For simplicity, we assume that f = 0,u = 0,f =0 and T =0.
∞ rb rb
For the case that the fluid is viscous, there have been many results over the last
two decades. The existence of global weak solutions of the above system was proved by
[4,5, 8,9,10,13,19,20]. If therigid bodyis adisk inR2, T.Takahashiand M.Tucsnak[23]
showed the existence and uniqueness of global strong solutions. Later, P.Cusmille and
T.Takahashi [2] extended the result to general rigid body case in R2. They also proved
the local existence and uniqueness of strong solutions in R3.
For the case that the fluid is inviscid, [17] dealt with this problem first. When the
solid is of C1 and piecewise C2 boundary,and the fluidfills in R2, a uniqueglobal classical
solutionwasobtainedundersomeassumptionontheinitialvorticity in[17]. Aglobalweak
solution was constructed in [25] when the initial data belongs to W1,p, p > 4. Recently,
3
C. Rosier and L. Rosier[18] proved the local existence of Ws,2-strong solutions for d ≥ 2,
s ≥ [d/2]+2 and the solid is a ball. The key idea is to make use of the Kato-Lai theory,
which was originated from [14].
At the same time, the fluid-rigid body system which occupies a bounded domain was
also studied. In particular, [11] proved the existence and uniqueness of strong solution for
the three dimensional case. The approach of [11] follows closely the idea in [1], which is
used to study the classical Euler equations. Their method also applies to the case that
there are several solids in the fluid.
In this paper, we plan to extend the result of [18] to a more general setting. We will
study the case that the solid is smooth and of a general form. As shown by the system
itself, it is a free boundary problem. To deal with the free boundary problem, usually the
first step is to fix the boundary. To fulfill that, [18] made a translation of the coordinate
system. However, this special transformation can only be applied to the case that the
solid is a ball. The solid in this paper is of general form, hence a different change of
coordinates from [18] should be introduced. As said before, the motion of the solid is a
rigid body movement, so a natural idea is to use a coordinate transformation consisting
of the translation and rotation of the solid. We tried in this way. As will be shown in
section 2, after the transformation, the new equivalent system has some term difficult to
control. So we gave up the idea and then applied a new transformation which coincides
3
with the movement of the solid in its neighborhood and becomes identity when far away
from it. The concrete form of the transformation will be given in section 2. In fact, this
kind of transformation has been used by [12, 2, 6].
For the new equivalent system after the transformation, we will use the Kato-Lai
theory[14] to construct a sequence of approximate solutions and prove the limit is the
solution required. The Kato-Lai theory is a Galerkin method in spirit. One can construct
the Galerkin approximation by himself or herself. We usethe theory here directly to avoid
unnecessary details. By the way, the method we apply here can also be used to deal with
the several-solids case after some minor modification.
2 Transformed equations and main result
As noted above, to fix the boundary,one method is to use the coordinates transformation.
Asshownin[16,17],adirectwayisthefollowingoneconsistingoftranslationandrotation,
x = Q(t)y+h(t), u¯(y,t) = Q(t)Tu(Q(t)y+h(t),t),
t
p¯(y,t) = p(Q(t)y+h(t),t), h¯(t) = Q(s)Th′(s)ds,
Z0
J¯= J(0), ω¯(y,t) = Q(t)Tω,
where Q(t) is given in section 1 and Q(t)T is the transpose of Q(t).
After the transformation, an equivalent system is obtained as follows:
∂u¯
+[(u¯−h¯′−ω¯ ×y)·∇]u¯+ω¯ ×u¯+∇p¯= 0, in Ω×[0,T], (2.1)
∂t
div u¯ = 0, in Ω×[0,T], (2.2)
u¯·~n = (h¯′+ω¯ ×y)·~n, on ∂Ω×[0,T], (2.3)
mh¯′′ = p¯~ndσ−mω¯(t)×h¯′(t), in [0,T], (2.4)
Z∂Ω
J¯ω¯′ = y×p¯~ndσ+ J¯ω¯(t) ×ω¯(t), in [0,T], (2.5)
Z∂Ω
(cid:0) (cid:1)
u¯(y,0) = u , y ∈ Ω, (2.6)
0
¯h(0) = 0, h¯′(0) = l , ω¯(0) = ω . (2.7)
0 0
The new problem is a fixed boundary problem now. However, there is a term [ω¯×y)·
∇)]u¯, whose coefficient become unbounded at large spatial distance. For the 2D case, the
difficulty was overcome in [17] by assuming that u belongs to some weighted space. The
0
method there dependsstrongly on the fact that the vorticity satisfies a transportequation
in 2D. However, the fact does not hold any more in 3D. To avoid this difficulty, we will
4
use another change of variables. The new transformation coincides with Q(t)y +h(t) in
a neighborhood of the solid and becomes identity when far away from it. In fact, this
transformation was initially applied by Inoue and Wakimoto [12], later by Dintelmann
etc.[6] and Cusmille and Takahashi [2]
More precisely, for a pair of continuous vector-valued functions (l(t),ω(t)), let
t
h(t) = l(s)ds, t ∈ [0,T], (2.8)
Z0
and
V(x,t) = l(t)+ω(t)×(x−h(t)), (x,t) ∈ R3×[0,T], (2.9)
which is a rigid body movement.
Choose a smooth function ξ : R3 → R with compact support such that ξ(y) = 1 in a
neighborhood of O¯, and set
ψ(x,t) = ξ Q(t)T(x−h(t)) ,
(cid:0) (cid:1)
where Q(t) is given by (1.9).
Then introduce the functions W and Λ,
1 |x−h(t)|2
W(x,t) = l(t)×(x−h(t))+ ω, (2.10)
2 2
Λ(x,t) = ψV +∇ψ×W. (2.11)
It is easy to check that Λ satisfies the following two lemmas(or refer to [2, 6]).
Remark 2.1. In what follows you will see the function Λ will produce a transformation
which coincides with Q(t)y +h(t) in the neighborhood of the solid and becomes identity
when far away from it. The reason why we use Λinstead of ψV isthat we need the function
to be of divergence free. The function W is introduced to eliminate the divergence of ψV.
Lemma 2.2. (1) Λ(x,t) = 0, if x is far away from O(t);
(2) Λ(x,t) = l(t)+ω(t)×(x−h(t)) in O(t)×[0,T];
(3) div Λ= 0 in R3×[0,T];
(4) For all t ∈ [0,T],Λ(·,t) is a C∞(R3,R3) function. Moreover, for every s ∈ N,
kΛ(·,t)kWs,∞(R3) ≤ C(s,T)(|l(t)|+|ω(t)|);
(5) For all x ∈ R3, the function Λ(x,·) is in C0([0,T];R3), provided that l,ω ∈C0[0,T].
5
Next, consider the vector field X(y,t) which satisfies
∂X(y,t)
= Λ(X(y,t),t), t ∈ (0,T],
∂t (2.12)
X(y,0) = y ∈ R3.
Lemma 2.3. For every y∈ R3, the initial-value problem (2.12) admits a unique solution
X(y,·) : [0,T] → R3, which is a C1 function on [0,T]. Moreover, the solution has the
following properties:
(1) For all t ∈ [0,T],the mapping y 7→ X(y,t) is a C∞ diffeomorphism from O onto
O(t) and from Ω onto Ω(t).
(2) Denote by Y(·,t) the inverse of X(·,t). Then for every x ∈ R3 the mapping t 7→
Y(x,t) is C1-continuous and satisfies the following initial value problem,
−1
∂Y(x,t) ∂X(Y(x,t),t)
= − Λ(x,t), t ∈ (0,T],
∂t ∂y (2.13)
(cid:20) (cid:21)
Y(x,0) = x ∈ R3.
(3) For everyx,y ∈ R3 and for every t ∈ [0,T], the determinants of the Jacobian
matrices J of X(y,t) and J of Y(x,t) both equal to 1, i.e.,
X Y
det(J (y,t)) = det(J (x,t)) = 1. (2.14)
X Y
Let
x = X(y,t), v(y,t) = J (X(y,t),t)u(X(y,t),t),
Y
(2.15)
q(y,t) =p(X(y,t),t), H(t)= Q(t)Th(t), L(t) = Q(t)Tl(t),
and R(t) is the vector-valued function satisfying
A(R(t)) = Q(t)TA(ω(t))Q(t). (2.16)
Denote
3
∂Y ∂Y
G(y,t) = gij(y,t) = i(X(y,t),t) j(X(y,t),t) ,
∂xk ∂xk !
k=1
(cid:0) (cid:1) X
gij = 3 ∂Xk(y,t)∂Xk(y,t), (2.17)
Xk=1 ∂yi ∂yj
3
1 ∂g ∂g ∂g
Γki,j(y,t) = 2Xl=1gkl(cid:26)∂yijl + ∂yjil − ∂yilj(cid:27).
6
Now one can transformthe original system (1.1)-(1.8) into the following system, which
is a fixed boundary problem, (see [2, 15])
∂v
+Mv+Nv+G·∇q = 0, in Ω×[0,T], (2.18)
∂t
div v = 0, in Ω×[0,T], (2.19)
v(y,t)·~n = (L(t)+R(t)×y)·~n, on ∂Ω×[0,T], (2.20)
mL′(t) = q~ndσ−mR(t)×L(t), in [0,T], (2.21)
Z∂Ω
J¯R′(t) = y×q~ndσ+J¯R(t)×R(t), in [0,T], (2.22)
Z∂Ω
v(y,0) = u (y), y ∈ Ω, (2.23)
0
H(0) = 0, L(0) = l , R(0) = ω . (2.24)
0 0
where
3 ∂Y ∂v 3 ∂Y ∂Y ∂2X
(Mv) = j i + Γi k + i k v ;
i ∂t ∂y j,k ∂t ∂x ∂t∂y j
j=1 j j,k=1(cid:26) k j(cid:27)
X X
(Nv)i = 3 vj∂∂yvi + 3 Γij,kvjvk; (2.25)
j
Xj=1 jX,k=1
3
∂q
(G·∇q)i = gij .
∂y
Xj=1 j
Now the main result in this paper reads
Theorem 2.4. Suppose that s > 5, u ∈ Hs(Ω) and u ·~n = (l +ω ×y)·~n on ∂Ω.
2 0 0 0 0
Then there exist some T > 0 and a unique solution (v,q,L,R) of (2.18)-(2.24) such that
0
v ∈ C([0,T ];Hs(Ω)), ∇q ∈ C([0,T ];Hs−1(Ω)),
0 0
and
L,R ∈ C1([0,T ];R3).
0
Such a solution is unique up to an arbitrary function of t which may be added to q.
Furthermore, T does not depend on s.
0
Remark 2.5. Following similar proof in section 7 for the uniqueness of the solution, we
can get the stability of the solution. The method is standard as for the classical Euler
equations, so we omit the details.
Remark 2.6. There are still many questions for the fluid-rigid body system. For example,
whether the strong solution blows up or not? If there are many solids in the fluid, is there
7
any collision between these solids? Which comes first, the collision or the blow up of
k∇vkL∞?
For readers’ convenience, we list the outline of the following sections. Section 3 will
be devoted to some notations or definitions for the function spaces and some lemmas to
be used in the proof of the main result. The preliminaries contain the decomposition of
the function spaces associated with this particular problem, the Kato-Lai theory for the
construction of approximate solutions, and the estimates for the coordinate transforma-
tion. In section 4, we will give the a priori estimates for the new pressure term ∇q in
(2.18), under the assumption that kvk is bounded. These estimates will be used
Hs(Ω)∩X˜
for the uniform estimates for the approximate solutions, which are constructed in section
5. Moreover, some uniform estimates for these solutions are derived at the same time.
The convergence of these solutions is studied in section 6. And the limit is exactly the
solution required. In the last section, section 7, we prove the uniqueness of the solution.
Making use of the uniqueness result, the continuity of the solution with respect to time t
in some space H is also proved.
s
3 Preliminaries
Before stating our main result, we’d like to introduce some notations and definitions.
Suppose S is a domain in R3. L2(S) is the space of L2-integrable functions with the
standard inner product (·,·) . By the way, we will not distinguish the scalar function
L2(S)
spaces and the corresponding vector-valued function spaces.
Suppose s is a nonnegative integer, then
Hs(S)= {u ∈ L2(S) : Dαu ∈L2(S), ∀ α ∈ N3,s.t., |α| ≤ s},
with the inner product
(u,v) = (Dαu,Dαv) .
Hs(S) L2(S)
|α|≤s
X
The homogeneous Sobolev space
D1,2(S) = {u ∈ L1 (S) : ∇u∈ L2(S)},
loc
with the seminorm
|u| = k∇uk .
D1,2(S) L2(S)
If one identifies the two functions u ,u ∈ D1,2(S) whenever |u − u | = 0, i.e.,
1 2 1 2 D1,2(S)
u and u differ by a constant, the quotient space D˙1,2(S), with the norm |·| , is
1 2 D1,2(S)
8
derived. In the following text, without any confusion, we do not distinguish the elements
in D1,2(S) and D˙1,2(S) very strictly.
Let B (y) denote the ball centered at y and with the radius R. Ω := Ω∩B (0). Let
R R R
ρ = m , where |O| stands for the volume of O. Hence ρ is the density of the solid. Let
|O|
X˜ = L2(R3) be endowed with the inner product,
(u,v) = u(y)·v(y)dy +ρ u(y)·v(y)dy.
X˜
ZΩ ZO
Define
X˜ = {u ∈ X˜ : div u= 0 in R3, ∃ l,ω ∈ R3,s.t., u= l+ω×y in O},
∗
which is a closed subspace of X˜.
Remark 3.1. For every u∈ X˜ , and suppose that u= l+ω×y on O. In fact, l and ω
∗
are unique vectors satisfying the above relation, and
1
l = ρudy, ω = −J¯−1 (u×y)dy.
m
ZO ZO
The fact has been proved, see [3] or [26]. In what follows, we will denote the vectors l, ω
associated with u∈ X˜ by l , ω .
∗ u u
Let H = {u∈ X˜ : u| ∈ Hs(Ω)} be endowed with the scalar product
s Ω
(u,v) = (u,v) +ρ(u,v) .
Hs Hs(Ω) L2(O)
V isthespaceoffunctionsv ∈ H suchthatv| belongstoD(A),whereAistheellipticop-
s s Ω
eratorAf = (−1)|α|∂2αf withNeumannboundaryconditions,andD(A) ⊆ H2s(Ω).
|α|≤s
V is endowed with the scalar product
s P
(u,v) = (u,v) +ρ(u,v) .
Vs H2s(Ω) L2(O)
As in [18], we introduce a bilinear form on V ×X˜:
s
hv,ui = (−1)|α|∂2αv,u +ρ(v,u) .
L2(O)
|Xα|≤s L2(Ω)
Since the main idea in this paper is the Kato-Lai theory, so we’d like to give a brief
description of the theory, which is cited from [18]. For more details, please refer to [14].
LetV,H,X bethreerealseparableBanach spaces. We say that{V,H,X} is anadmissible
triplet if the following conditions hold.
9
• V ⊂ H ⊂ X, and the inclusions are dense and continuous.
1
• H is a Hilbert space, with inner product (·,·) and norm k·k = (·,·)2 .
H H H
• There is a continuous, nondegenerate bilinear form on V ×X, denoted by h·,·i, such
that
hv,ui = (v,u) for all v ∈ V,u ∈ H.
H
Recall that the bilinear form hv,ui is continuous and nondegenerate when
|hv,ui| ≤ Ckvk kuk for some constant C > 0; (3.1)
V X
hv,ui = 0 for all u∈ X implies v = 0; (3.2)
hv,ui = 0 for all v ∈ V implies u= 0. (3.3)
A map A :[0,T]×H → X is said to be sequentially weakly continuous if A(t ,v )⇀
n n
A(t,v) in X whenever t → t and v ⇀ v in H. We denote by C ([0,T];H) the space of
n n w
sequentially weakly continuous functions from [0,T] to H, and by C1([0,T];X) the space
w
of the functions u∈ W1,∞(0,T;X) such that du ∈ C ([0,T],X).
dt w
We are concerned with the Cauchy problem
dv
+A(t,v) = 0, with v(0) = v . (3.4)
0
dt
The Kato-Lai existence result for abstract evolution equations is as follows.
Theorem 3.2. Let {V,H,X} be an admissible triplet. Let A be a sequentially weakly
continuous map from [0,T]×H into X such that
hv,A(t,v)i ≥ −β kvk2 for t ∈[0,T], v ∈ V,
H
(cid:0) (cid:1)
where β(r) ≥ 0 is a continuous nondecreasing function of r ≥ 0. For any v ∈ H, consider
0
the ODE γ′(t) = β(γ(t)), γ(0) = kv k2 ,. Suppose the maximal time of existence of γ is
0 H
T , then for (3.4), there exists a solution v of (3.4) in the class
0
v ∈C ([0,T ];H)∩C1([0,T ];X).
w 0 w 0
Moreover, one has
kv(t)k2 ≤ γ(t), t ∈[0,T ],
H 0
In fact, it was proved in [18] that the triplet {X˜,H ,V } is admissible for any smooth
s s
boundary.
The following lemma gives a decomposition of L2(R3). In particular, the second part
is a replacement of Proposition 3.1 of [18].
10
