On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Jorge San Martín 1 , Erica L. Schwindt 2 , and Takéo Takahashi 3
  • 1 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile
  • 2 Departamento de Matemática, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto, 5800 - Río Cuarto, Córdoba, Argentina
  • 3 Inria, Villers-lès-Nancy, F-54600, France; and Université de Lorraine, IECL, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; and CNRS, IECL, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France
Jorge San Martín, Erica L. Schwindt and Takéo Takahashi

Abstract

We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

1 Introduction

This paper is devoted to reconstructing an unknown structure 𝒮 included in a bounded cavity ΩN (N=2,3) filled by a viscous incompressible fluid. More precisely, we aim to obtain some geometrical information on 𝒮 by measurement on the boundary Ω of Ω. Such a geometrical inverse problem is important in several applied areas such as medicine (foreign bodies in the bloodstream), biology (fishes), naval engineering (submarines), etc.

We assume in what follows that 𝒮 is a compact connected subset of Ω with nonempty interior and that =Ω𝒮 is connected.

In the first part of the article, the fluid equations that we consider are the nonstationary Stokes system

𝒖t-𝐝𝐢𝐯𝝈(𝒖,p)=in(0,T)×,
div𝒖=0in(0,T)×,
𝒖=on(0,T)×𝒮,
𝒖=𝒇on(0,T)×Ω,
𝒖(0,)=in.

In the above system, (𝒖,p) are the velocity and the pressure of the fluid. Moreover, we have denoted by 𝝈(𝒖,p) the Cauchy stress tensor, which is defined by the Stokes law as

𝝈(𝒖,p)=-p𝑰N+2𝑫(𝒖),

where 𝑰N is the identity matrix of N(), with N() denoting the space of real square matrices of order N, and where 𝑫(𝒖) is the strain tensor defined by

[𝑫(𝒖)]kl=12(ukxl+ulxk).

To simplify the writing, we take in this paper the kinematic viscosity of the fluid equal to 1.

The idea is to impose a condition 𝒇 in (1.4) and to measure the corresponding Cauchy force

𝝈(𝒖,p)𝒏|(0,T)×Ω

in order to deduce information on the obstacle 𝒮. Here and in all what follows, 𝒏 denotes the unit outer normal to the fluid domain.

We also consider in this paper the following linear fluid–rigid body system:

𝒖t-𝐝𝐢𝐯𝝈(𝒖,p)=in(0,T)×,
div𝒖=0in(0,T)×,
𝒖=𝒇on(0,T)×Ω,
𝒖=+𝝎×𝒚on(0,T)×𝒮,
m+𝒮𝝈(𝒖,p)𝒏𝑑𝜸=in(0,T),
𝑰0𝝎+𝒮𝒚×𝝈(𝒖,p)𝒏𝑑𝜸=in(0,T),
𝒖(0,)=in,
(0)=,𝝎(0)=.

Here and 𝝎 represents respectively the linear and angular velocity of the rigid body. Let us note that in this simplified fluid-rigid body system, the structure domain 𝒮 is fixed. We assume that the density ρ𝒮 of the rigid body is a positive constant. In particular, the mass m and the inertia tensor I0 are defined as follows:

m=ρ𝒮μ3(𝒮),𝑰0=ρ𝒮𝒮|𝒙|2𝑰3-(𝒙𝒙)d𝒙,

where μ3 denotes the Lebesgue measure in 3 and where 𝑰3 is the 3×3 identity matrix.

In dimension N=2, the above system is slightly modified: ω, I0, equation (1.11) and equation (1.13) write respectively

𝒖=+ω𝒚,I0ω+𝒮𝒚𝝈(𝒖,p)𝒏𝑑𝜸=0,

where

[x1x2]=[-x2x1].

Finally, I0 is defined by

I0=ρ𝒮𝒮|𝒙|2𝑑𝒙.

System (1.1)–(1.5) is a linear simplification of the classical Navier–Stokes system

𝒖t+𝒖𝒖-𝐝𝐢𝐯𝝈(𝒖,p)=in(0,T)×,
div𝒖=0in(0,T)×,
𝒖=𝒇on(0,T)×Ω,
𝒖=on(0,T)×𝒮,
𝒖(0,)=in,

and system (1.8)–(1.15) is a linear simplification of the “full” fluid-rigid body system that can be written in dimension N=2 as

𝒖t+𝒖𝒖-𝐝𝐢𝐯𝝈(𝒖,p)=,t(0,T),𝒙(t),
div𝒖=0,t(0,T),𝒙(t),
𝒖=𝒇,t(0,T),𝒙Ω,
𝒖=+ω(𝒙-𝒉),t(0,T),𝒙𝒮(t),
m+𝒮(t)𝝈(𝒖,p)𝒏𝑑𝜸=,t(0,T),
I0ω+𝒮(t)(𝒙-𝒉)𝝈(𝒖,p)𝒏𝑑𝜸=0,t(0,T),
𝒉=,t(0,T),
θ=ω,t(0,T),
𝒖(0,)=,𝒙(0),
(0)=,ω(0)=0,
𝒉(0)=𝒉0,θ(0)=θ0,
𝒮(t)=𝒉(t)+𝑹θ(t)𝒮0.

Here 𝒉(t)2 and θ(t) are respectively the center of mass and the orientation of 𝒮(t). In particular, the center of mass of 𝒮0 is located at . We have denoted by 𝑹θ the matrix of rotation of angle θ. Contrary to system (1.8)–(1.15), here the solid is moving (equation (1.33)). Let us emphasize that system (1.8)–(1.15) is important to study system (1.22)–(1.33): for instance, this linear system is used in [36, 37] to prove the existence of strong solutions for system (1.22)–(1.33) with the aid of a fixed point argument. Let us also note that equations (1.26), (1.27) for the rigid body are the Newton laws.

As for the previous systems, the idea is to take some particular choice of 𝒇 and to measure the corresponding Cauchy force given by (1.7) in order to obtain geometrical information on 𝒮(t). However, here there is an important difference: applying 𝒇 at the boundary of Ω makes the rigid body moves through the (unknown) trajectory (𝒉(t),θ(t)). Moreover, with such a boundary condition, it could possible that the rigid body touches Ω and it is not clear what happens after this contact (see [34]).

We also consider a simplification of system (1.22)–(1.33) obtained by assuming that the Reynolds number is very small. In that case, neglecting the inertia forces, the 3D version of system (1.22)–(1.33) can be approximated by

-𝐝𝐢𝐯𝝈(𝒖,p)=,t(0,T),𝒙(t),
div𝒖=0,t(0,T),𝒙(t),
𝒖=𝒇,t(0,T),𝒙Ω,
𝒖=+𝝎×(𝒙-𝒉),t(0,T),𝒙𝒮(t),
𝒮(t)𝝈(𝒖,p)𝒏𝑑𝜸=t(0,T),
𝒮(t)(𝒙-𝒉)×𝝈(𝒖,p)𝒏𝑑𝜸=t(0,T),
𝒉=,t(0,T),
𝑸=A(𝝎)𝑸,t(0,T),
𝒮(t)=𝒉(t)+𝑸(t)𝒮0,t(0,T).

The map A is defined as follows:

A(𝒓)=[0-r3r2r30-r1-r2r10],𝒓3.

This map is related to the vector product by the formula

A(𝒓)𝒙=𝒓×𝒙.

Let us remark that the above system is not linear since 𝒮(t) is not given. This system is studied in [7] where the identifiability of the rigid body is obtained through the measurement of the Cauchy forces on the boundary. Like system (1.22)–(1.33), the solid moves through the action of 𝒇 on this system.

These geometrical inverse problems for fluid systems were already considered in [1] where the authors tackle the problem of recovering the shape and location of a fixed obstacle in a viscous incompressible fluid modeled by the Navier–Stokes system. They show the identifiability of the fixed obstacle: if 𝒇 not identically equal to , then the mapping that associates to 𝒮 the measurement given by (1.7) is one-to-one. They also prove a stability result: if two measurements are close, it implies that the two corresponding obstacles are close. Extensions of this result in the case of a fixed obstacle are obtained in [9] and in [10]. In [2], the authors consider a similar problem in the 2D case and for a fluid modeled by the Stokes system. They develop an integral method in order to recover the structure. The identifiability result of [1] is extended in [7] to the case of a moving rigid body, but only in the case of the stationary Stokes system. In the case of a potential fluid (thus inviscid), one can use, in 2D, complex analysis ([5, 6]) to detect a moving rigid body of particular shape (ball, ellipse) if the fluid fills the exterior of the structure domain.

Numerical aspects are considered in [3]: the authors use shape optimization techniques to detect a fixed obstacle in a viscous incompressible fluid. They prove in particular that the shape Hessian is compact and thus that the problem is ill-posed.

Here we are interested in obtaining geometrical information on the obstacle such as the distance from a fixed point to the obstacle or its convex hull. The problem of finding the distance from a fixed point was considered in [16], in the case of a fixed obstacle in a stationary Stokes fluid. In that study, they use a method based on complex geometrical solutions that was introduced in [35] and that has been applied in several inverse problems ([12, 30, 31, 8, 29, 15], etc.). In order to recover the convex hull of the obstacle, Ikehata introduced the enclosure method and used it in [22, 21, 23], etc. The above references were devoted to works on stationary problems. The case of the heat equation was considered in [13] with the use of complex geometrical solutions and [24, 27, 28] for the enclosure method.

In this work, we consider both methods to deal with nonstationary fluid or fluid-structure systems. More precisely, we use the approach in [27] in order to deal with the nonstationary Stokes system. A first step consists in considering the Laplace transform of the system in order to transform it into a stationary Stokes-type system. Then we show that if (𝒗^α,q^α) is a family of solutions the same (stationary) system but on the whole domain Ω (see (2.1)–(2.2)), then a quantity (see (2.7)) based on the measurement given by (1.7) behaves in similar way as the H1 norm of 𝒗^α on 𝒮 as α goes to (Theorem 2.1). The idea is then to construct solutions 𝒗^α so that the H1 norm on 𝒮 gives geometrical information on the domain. One of the difficulties in this construction comes from the fact that here the test functions are divergence free. In particular, in the case of the distance of 𝒮 to a point 𝒙0, we need to impose 𝒙0ch(Ω) and N=3. These hypothesis are not considered in the case of the heat equation (see [27]).

The above method can not be adapted to the case of nonlinear systems such as (1.17)–(1.21) and (1.22)–(1.33). As a consequence, for these nonlinear systems we use complex geometrical solutions constructed in [16]. This allows us to recover only some partial information, and more precisely, at the contrary to the linear case, we lose one of the inequalities. Nevertheless, these two approaches give some first results in the case of nonstationary fluid systems.

The plan of the paper is the following: in Section 2, we state our main results, for the linear systems and for the nonlinear systems. We recall some preliminaries in Section 3, that allow us to prove our first main result in Section 4: the relation between the measurement and the H1 norm of 𝒗^α on 𝒮, as explained above. Then in Section 5, we construct 𝒗^α in order to recover the convex hull of 𝒮 and in Section 6, we construct 𝒗^α in order to recover the distance from a fixed point to 𝒮. Section 7 is devoted to inverse problems for the nonlinear systems: we use there complex geometrical solutions.

2 Main results

Let us first describe the method used to recover geometric information on the obstacle 𝒮 in the case of the linear systems (1.1)–(1.5) and (1.8)–(1.15).

First we consider a family (𝒗^α,q^α)C2(Ω¯)×C1(Ω¯) of solutions of a Stokes system

α𝒗^α-𝐝𝐢𝐯𝝈(𝒗^α,q^α)=inΩ~,
div𝒗^α=0inΩ~,

for some domain Ω~Ω and for α>0.

We then consider 𝒇α defined by

𝒇α(t,𝒙):=χα(t)𝒗^α(𝒙),

with χαC([0,T]) such that χα(0)=0 and χα(t)>0 in (0,T] and such that

0Te-αtχα(t)𝑑t=1.

For instance, in what follows, we take

χα(t)=α2t1-(1+αT)e-αT,t[0,T].

In particular,

𝒇^α(𝒙):=0Te-αt𝒇α(t,𝒙)𝑑t=𝒗^α(𝒙),𝒙Ω.

We can remark that since 𝒇α is given by (2.3), then it satisfies the condition

Ω𝒇α𝒏𝑑𝜸=0on(0,T).

The above equation allows us to consider the solution (𝒖α,pα) of the Stokes system (1.1)–(1.5), with the boundary condition

𝒖α=𝒇αon(0,T)×Ω.

Let us set

Eα:=Ω0Te-αt(𝒗^α𝝈(𝒖α,pα)𝒏-𝒖α𝝈(𝒗^α,q^α)𝒏)𝑑t𝑑𝜸.

We are now in a position to state our first main result.

Assume (𝐯^α,q^α) satisfies (2.1)–(2.2) and (𝐮α,pα) is the solution of system (1.1)–(1.5) with (2.3) and (2.6). Then Eα defined by (2.7) satisfies

(𝒮α|𝒗^α|2+2|𝑫(𝒗^α)|2d𝒙)-Cα2e-αT𝒗^αH1(Ω)2Eα
C(α+1)(𝒮|𝒗^α|2+2|𝑫(𝒗^α)|2d𝒙)+Cα2e-αT𝒗^αH1(Ω)2.

The above result and the two corollaries below correspond the closure method associated with the evolutionary Stokes system. A general framework for this method in the case of heat type equations is developed in [25]. The first extension of this method to a system of partial differential equation was developed in [26].

The first corollary of Theorem 2.1 corresponds to the reconstruction of the support function h𝒮 of 𝒮. Let us recall that for any subset G of 3, the support function hG of G is defined by

hG(𝜿)=sup𝒙G𝜿𝒙,𝜿𝕊2,

where 𝕊2 is the unit sphere of 3. This function is classically used in the theory of convex sets (see, for instance, [4, p. 26]). In particular, if G is convex,

G={𝒙3:𝒙𝜿hG(𝜿) for all 𝜿𝕊2}.
(Recovering the support function)

Assume S is of class C2. There exists a family of solutions (𝐯^α,q^α) of (2.1)–(2.2) such that the solution (𝐮α,pα) of (1.1)–(1.5) with (2.6) and (2.3) verifies

limα+12αlog(Eα)=h𝒮(𝜿).

The second corollary of Theorem 2.1 allows us to obtain the distance d(𝒙0,𝒮) of 𝒮 to a point 𝒙0ch(Ω) (the convex hull of Ω).

(Recovering the distance to a point)

Assume N=3, S is of class C2 and 𝐱0ch(Ω). There exists a family of solutions (𝐯^α,q^α) of (2.1)–(2.2) such that the solution (𝐮α,pα) of (1.1)–(1.5) with (2.6) and (2.3) verifies

limα+12αlog(Eα)=-d(𝒙0,𝒮).

In contrast to [27, 28], in the above result, we have to assume that 𝒙0ch(Ω). This restriction comes from the fact that we need in our construction that the family (𝒗^α,q^α) satisfies the condition div𝒗^α=0. In [27, 28], the authors also manage to reconstruct the smallest sphere centered at a point and enclosing the obstacle. Here, we cannot extend their construction since we need the free divergence condition for 𝒗^α.

We have similar results for the linear system (1.8)–(1.15):

Assume S is of class C2.

  1. (1)There exists a family of solutions (𝒗^α,q^α) satisfying (2.1)–(2.2) such that the solution (𝒖α,pα,α,𝝎α) of (1.8)–(1.15) with (2.3), (2.6) verifies
    limα+12αlog(Eα)=h𝒮(𝜿).
  2. (2)Assume 𝒙0ch(Ω) and N=3. There exists a family of solutions (𝒗^α,q^α) satisfying (2.1)–(2.2) such that the solution (𝒖α,pα,α,𝝎α) of (1.8)–(1.15) with (2.3), (2.6) verifies
    limα+12αlog(Eα)=-d(𝒙0,𝒮).

The proof of the previous theorem is completely similar to the proof of Theorem 2.1, with the same families constructed in Corollary 2.2 and Corollary 2.3. Therefore, we omit its proof.

In the case of the nonlinear system (1.34)–(1.42), we use a family of solutions (𝒗α,qα)C2(Ω¯)×C1(Ω¯) of

-𝐝𝐢𝐯𝝈(𝒗α,qα)=inΩ~,
div𝒗α=0inΩ~,

for some domain Ω~Ω. Here α>0 is a parameter in the construction of these solutions that eventually goes to . We then consider 𝒇α defined by

𝒇α(𝒙):=𝒗α(𝒙).

As in the linear case, we then consider the solution (𝒖α,pα) of systems (1.34)–(1.42) (respectively (1.22)–(1.33), and (1.17)–(1.21)), with the boundary condition

𝒖α=𝒇αon(0,T)×Ω.

We set

Fα:=Ω(𝒗α𝝈(𝒖α,pα)𝒏-𝒖α𝝈(𝒗α,qα)𝒏)𝑑𝜸.

As explained in the previous section, one difficulty for stating result for this system is that the rigid body can touch Ω. We thus assume that for all regular 𝒇,

d(𝒮(t),Ω)>0for all t[0,T].

Such an hypothesis is satisfied for instance in the case where 𝒮0 and Ω are balls (see [17, 18, 19]).

We fix 𝒙0ch(Ω) (the convex hull of Ω) and d>0. Then, we have the following results.

Assume S is of class C2, d>0 and 𝐱0ch(Ω). Assume also that (2.15) holds. Then, there exists a family of solutions (𝐯α,qα) satisfying (2.10)–(2.11) such that the solution (𝐮α,pα,α,𝛚α) of (1.34)–(1.42) with (2.13) verifies:

  1. (1)If d<d(𝒙0,𝒮(t)), then FαCAα for some constants C>0 and A(0,1).
  2. (2)If d>d(𝒙0,𝒮(t)), then FαCBα for some constants C>0 and B>1 and for α>1.

The above result is based on the construction of spherical geometrical optics solutions. In the case of Stokes-type system, such a construction has been done in [16]. Let us point out that in their method use the Hahn–Banach theorem. In the case of the Calderon problem, another construction that is not using the Hahn–Banach theorem is done in [20].

For systems (1.22)–(1.33), and (1.17)–(1.21), we slightly modify the boundary condition by using (𝒗α,qα) depending on time and satisfying (2.10)–(2.11) for all t and we consider the following measurement:

Kα:=0TΩ(𝒗α𝝈(𝒖α,pα)𝒏-𝒖α𝝈(𝒗α,qα)𝒏)𝑑t𝑑𝜸-0TΩ(𝒇α𝒏)|𝒇α|22𝑑𝜸.

In the case of system (1.22)–(1.33), we need to assume again (2.15) to prevent possible contacts. Again this condition is satisfied for instance in the case where 𝒮0 and Ω are balls (see [17, 18, 19]). It is probably true for other geometries but up to now this has not been proven.

For both systems (1.17)–(1.21) and (1.22)–(1.33), we also impose that N=2 since we are working with regular solutions and for N=3 the existence of global (in time) regular solutions is an open problem. In particular, in the case N=3, one should need to show that the times Tα of existence of the family of solutions (𝒗α,qα) can be chosen uniformly with respect to α.

Suppose N=2. Assume S is of class C2, d>0 and 𝐱0ch(Ω). There exists a family of solutions (𝐯α,qα) satisfying (2.10)–(2.11) such that:

  1. (1)The solution (𝒖α,pα) of (1.17)–(1.21) with (2.13) verifies the following implication: if (Kα)α>α0 is bounded, then d<d(𝒙0,𝒮).
  2. (2)The solution (𝒖α,pα,α,ωα) of (1.22)–(1.33) with (2.13) verifies the following implication: if (Kα)α>α0 is bounded, then d<d(𝒙0,𝒮(t)) for all t[0,T].

As explained in the introduction, the above result is only partial since with the other case (as in Theorem 2.6) is not present here. As it appear in the proof, it would imply to prove an estimate on the solutions (𝒖α,pα,α,ωα) for system (1.22)–(1.33).

For simplicity, we suppress in the proofs below the explicit dependence on α in the notation. For example, we write 𝒗^ instead of 𝒗^α.

3 Preliminaries

Assume 𝐯^H1(Ω) such that div𝐯^=0 in Ω. Consider a pair (𝐰,π)H1(F)×L2(F) such that div𝛔(𝐰,π)L2(F). Then there exists a constant C=C(Ω,S) such that

|Ω𝒗^𝝈(𝒘,π)𝒏d𝜸|C𝒗^H1(Ω)(𝑫(𝒘)L2()+𝐝𝐢𝐯𝝈(𝒘,π)L2()),
|𝒮𝒗^𝝈(𝒘,π)𝒏d𝜸|C𝒗^H1(𝒮)(𝑫(𝒘)L2()+𝐝𝐢𝐯𝝈(𝒘,π)L2()).
Proof.

We use [14, p. 176, relations (III.3.31) and (III.3.32)]: there exists 𝑽^H1() such that

div𝑽^=0in ,𝑽^=𝒗^on Ω,𝑽^=on 𝒮,

with

𝑽^H1()C𝒗^H1/2(Ω)C𝒗^H1(Ω).

We then use integration by parts

Ω𝒗^𝝈(𝒘,π)𝒏𝑑𝜸=𝑽^𝝈(𝒘,π)𝒏𝑑𝜸=𝐝𝐢𝐯𝝈(𝒘,π)𝑽^𝑑𝒙+2𝑫(𝒘):𝑫(𝑽^)d𝒙,

and (3.1) follows from (3.3).

The proof of (3.2) is similar, we consider (instead of 𝑽^) a function 𝑾^H1() such that

div𝑾^=0in ,𝑾^=on Ω,𝑾^=𝒗^on 𝒮,

with

𝑾^H1()C𝒗^H1/2(𝒮)C𝒗^H1(𝒮).

The proof of the lemma is complete. ∎

Assume 𝐟=χ𝐯^, with χH1(0,T), 𝐯^H3/2(Ω) satisfying

χ(0)=0,Ω𝒗^𝒏𝑑𝜸=0.

Then:

  1. (1)There exists a unique solution (𝒖,p) of system (1.1)–(1.5)
    𝒖L2(0,T;H2())C([0,T];H1())H1(0,T;L2()),
    pL2(0,T;H1()/);
  2. (2)There exists a unique solution (𝒖,p,,𝝎) of system (1.8)–(1.15) satisfying (3.4), (3.5) and ,𝝎H1(0,T).

The above result is quite classical for system (1.1)–(1.5) and is similar for system (1.8)–(1.15). We only give here some ideas of the proof. Note that the particular form of 𝒇 is not needed to obtain the result and the result remains true for more general boundary conditions.

Proof.

Using, for instance, [33], there exists 𝑽^H2() such that

div𝑽^=0in ,𝑽^=𝒗^on Ω,𝑽^=on 𝒮.

Using this lifting, we consider the change of variables

𝑼=𝒖-χV^

and the equations for (𝑼,p) can be written as

𝑼t-𝐝𝐢𝐯𝝈(𝑼,p)=𝑭in(0,T)×,
div𝑼=0in(0,T)×,
𝑼=on(0,T)×,
𝑼(0,)=in,

with

𝑭=χΔ𝑽^-χ𝑽^L2(0,T;L2()).

To end the proof, one can write (3.6)–(3.9) with the Stokes operator 𝑨=-𝑷0Δ as

𝑼+𝑨𝑼=𝑷0F,

where 𝑷0:L2()0 is the Leray projection on

0:={𝒘L2():div𝒘=0,𝒘𝒏=0on}.

Using that 𝑨 is self-adjoint and positive, we obtain the result.

For system (1.8)–(1.15), we can proceed with the same proof. Using the lifting V^, we are reduced to solve

𝑼t-𝐝𝐢𝐯𝝈(𝑼,p)=𝑭in(0,T)×,
div𝑼=0in(0,T)×,
𝑼=on(0,T)×Ω,
𝑼=+𝝎×𝒚on(0,T)×𝒮,
m+𝒮𝝈(𝑼,p)𝒏𝑑𝜸=𝑭in(0,T),
I0𝝎+𝒮𝒚×𝝈(𝑼,p)𝒏𝑑𝜸=𝝎𝑭in(0,T),
𝑼(0,)=in,
(0)=,𝝎(0)=,

with 𝑭 given by (3.10) and

𝑭=-2χ𝒮𝑫(V^)𝒏𝑑𝜸,𝝎𝑭=-2χ𝒮𝒚×𝑫(V^)𝒏𝑑𝜸.

We recall that the operator 𝑫 is defined in (1.6). It is classical that 𝑫𝒘= on 𝒮 if and only if there exist 𝒘,𝝎𝒘3 such that 𝒘(𝒚)=𝒘+𝝎𝒘×𝒚 for 𝒚𝒮 (see, for instance, in [32, p. 51]).

If we extend 𝑼 and 𝑭 in 𝒮 by

𝑼(t,𝒚)=(t)+𝝎(t)×𝒚,𝑭(t,𝒚)=𝑭(t)+𝝎𝑭(t)×𝒚,

then the above system can be written as

𝑼+𝑨𝑼=𝑷𝑭,

with

:={𝒘L2(Ω):div𝒘=0,𝒘𝒏=0onΩ,𝑫𝒘=in𝒮},
𝑫(𝑨):={𝒘H1(Ω):𝒘=onΩ,𝒘|H2},
𝓐𝒘:={-Δ𝒘in,(𝒮𝑫(𝒘)𝒏𝑑𝜸)+(𝒮𝒚×𝑫(𝒘)𝒏𝑑𝜸)×𝒙,𝒙𝒮,
𝑨:=𝑷𝓐

and 𝑷:L2(Ω) is the orthogonal projection. We have 𝑷𝑭L2(0,T;) and it is proved in [37] that 𝑨 is self-adjoint and positive in , and this allows us to prove the result by using classical result on parabolic systems. ∎

Assume that (𝐮,p) is the solution of (1.1)–(1.5) with 𝐟 defined by (2.3). Then for α large enough, there exists a constant C (independent of α) such that

𝒖L(0,T;L2())+𝑫(𝒖)L2(0,T;L2())Cα2𝒗^H1(Ω).
Proof.

Let us multiply (1.1) by 𝒖 and integrate by parts:

ddt(|𝒖|22)𝑑𝒙+2|𝑫(𝒖)|2𝑑𝒙=χ(t)Ω𝝈(𝒖,p)𝒏𝒗^𝑑𝜸.

In the above relation, we have used (2.3). Then we use the same function 𝑽^ used in the proof of Lemma 3.1 and integrate by parts:

Ω𝝈(𝒖,p)𝒏𝒗^𝑑𝜸=𝝈(𝒖,p)𝒏𝑽^𝑑𝜸
=𝐝𝐢𝐯𝝈(𝒖,p)𝑽^+2𝑫(𝒖):𝑫(𝑽^)d𝒙
=ddt𝒖𝑽^𝑑𝒙+2𝑫(𝒖):𝑫(𝑽^)d𝒙.

Combining (3.12) and (3.15) and integrating on (0,t), we obtain

12|𝒖(t)|2𝑑𝒙+0t2|𝑫(𝒖)|2𝑑𝒙𝑑s=χ(t)𝒖(t)𝑽^𝑑𝒙-0tχ𝒖𝑽^𝑑𝒙𝑑s+0t2χ𝑫(𝒖):𝑫(𝑽^)d𝒙ds.

Using Gronwall’s lemma, we deduce the existence of constant depending only on T such that

sup(0,T)|𝒖|2𝑑𝒙+0T2|𝑫(𝒖)|2𝑑𝒙𝑑sC𝒗^H1(Ω)2χH1(0,T)2.

With the choice (2.4), taking α large enough, we conclude that there exists C=C(T,Ω,𝒮)>0 such that (3.11) holds. ∎

We can obtain in a similar way the following lemma.

Assume that (𝐮,p,,𝛚) is the solution of (1.8)–(1.15) with 𝐟 defined by (2.3). Then for α large enough, there exists a constant C (independent of α) such that

𝒖L(0,T;L2(Ω))+𝑫(𝒖)L2(0,T;L2(Ω))Cα2𝒗^H1(Ω).

In the above result, we have extended 𝒖 in 𝒮 by setting

𝒖(t,𝒙):=(t)+𝝎(t)×𝒙,𝒙𝒮.

Finally, we end this section by recalling existence results for systems (1.17)–(1.21) and (1.22)–(1.33), for N=2.

Assume N=2 and assume 𝐟=χ𝐯^, with χH1(0,T), 𝐯^H3/2(Ω) satisfying

χ(0)=0,Ω𝒗^𝒏𝑑𝜸=0.

Then the following hold:

  1. (1)There exists a unique solution (𝒖,p) of system (1.17)–(1.21) with (3.4), (3.5).
  2. (2)Assume (2.15). There exists a unique solution (𝒖,p,,ω) of system (1.22)–(1.33) satisfying 𝒉,θH2(0,T) and
    𝒖L2(0,T;H2((𝒉,θ)))C([0,T];H1((𝒉,θ)))H1(0,T;L2((𝒉,θ))),
    pL2(0,T;H1((𝒉,θ))/).

The first result is classical and the second result was proved in [36]. It is possible to prove the first result by using a fixed point approach: one can consider the mapping

𝑭-(𝒖)𝒖,

where (𝒖,p) is the solution of

𝒖t-𝐝𝐢𝐯𝝈(𝒖,p)=𝑭in(0,T)×,
div𝒖=0in(0,T)×,
𝒖=on(0,T)×𝒮,
𝒖=𝒇on(0,T)×Ω,
𝒖(0,)=in.

Using the Banach fixed point theorem and the above mapping, we can obtain the local in time existence of system (1.17)–(1.21). Then, we derive H1 estimate (that is possible since N=2) to deduce the global in time existence.

For system (1.22)–(1.33), the approach is similar but with several additional difficulties. First since we are working with a moving domain, it is convenient to consider a change of variables 𝑿(t,):(0)(t) (construct from 𝒉,θ) and transform 𝒖 in 𝒖~:=𝐂𝐨𝐟(𝑿)(𝒖𝑿) (where 𝐂𝐨𝐟(𝑿) is the transpose of the cofactor matrix of 𝑿) and p in p~:=(det𝑿)(p𝑿). In the above proposition, (3.16)–(3.17) means that

𝒖~L2(0,T;H2((0)))C([0,T];H1((0)))H1(0,T;L2((0))),p~L2(0,T;H1((0))/).

Then we can consider a fixed point as above but with using (1.8)–(1.15) instead of (1.1)–(1.5) and where in the application (3.18) we have to add nonlinear terms coming from the change of variables (see [36] for more details).

4 Proof of Theorem 2.1

Let us define for all α>0,

𝒖^(𝒙):=0Te-αt𝒖(t,𝒙)𝑑t,p^(𝒙):=0Te-αtp(t,𝒙)𝑑t.

Then, we deduce from (1.1)–(1.5) that

α𝒖^-𝐝𝐢𝐯𝝈(𝒖^,p^)=-e-αT𝒖(T)in,
div𝒖^=0in,
𝒖^=𝒇^onΩ,
𝒖^=on𝒮,

with 𝒇^ defined by (2.5).

We consider the solution (𝒘^,r^) of the problem

α𝒘^-𝐝𝐢𝐯𝝈(𝒘^,r^)=in,
div𝒘^=0in,
𝒘^=𝒇^onΩ,
𝒘^=on𝒮.

The couple (𝒖^-𝒘^,p^-r^) satisfies the system

α(𝒖^-𝒘^)-𝐝𝐢𝐯𝝈((𝒖^-𝒘^),(p^-r^))=-e-αT𝒖(T)in,
div(𝒖^-𝒘^)=0in,
(𝒖^-𝒘^)=on.

Taking the inner product of (4.1) with 𝒖^-𝒘^ and integrating by parts, we obtain

α𝒖^-𝒘^L2()2+4𝑫(𝒖^-𝒘^)L2()21αe-2αT𝒖(T)L2()2.

Since (𝒗^,q^) satisfies (2.1)–(2.2), the couple (𝒗^-𝒘^,q^-r^) is solution of the system

α(v^-𝒘^)-𝐝𝐢𝐯𝝈((𝒗^-𝒘^),(q^-r^))=in,
div(𝒗^-𝒘^)=0in,
(𝒗^-𝒘^)=𝒗^on𝒮,
(𝒗^-𝒘^)=onΩ.

Taking the inner product of (4.3) with (𝒗^-𝒘^) and integrating by parts, it follows

0=α|𝒗^-𝒘^|2+2|𝑫(𝒗^-𝒘^)|2d𝒙-𝒮𝝈((𝒗^-𝒘^),(q^-r^))n𝒗^𝑑𝜸.

Taking the inner product of (4.3) with 𝒗^, taking the inner product of (2.1) with (𝒗^-𝒘^) and integrating by parts yield

0=-𝝈((𝒗^-𝒘^),(q^-r^))𝒏𝒗^𝑑𝜸+𝝈(𝒗^,q^)𝒏(𝒗^-𝒘^)𝑑𝜸.

The above relation implies

0=-Ω𝝈((𝒗^-𝒘^),(q^-r^))𝒏𝒗^𝑑𝜸-𝒮𝝈((𝒗^-𝒘^),(q^-r^))𝒏𝒗^𝑑𝜸+𝒮𝝈(𝒗^,q^)𝒏𝒗^𝑑𝜸.

Taking the inner product of (2.1) with 𝒗^ and integrating by parts on 𝒮, we obtain

𝒮α|𝒗^|2+2|𝑫(𝒗^)|2d𝒙+𝒮𝝈(𝒗^,q^)𝒏𝒗^𝑑𝜸=0.

Combining (4.4), (4.5) and (4.6), we deduce

-Ω𝝈((𝒗^-𝒘^),(q^-r^))𝒏𝒗^𝑑𝜸=α|𝒗^-𝒘^|2+2|𝑫(𝒗^-𝒘^)|2d𝒙+𝒮α|𝒗^|2+2|𝑫(𝒗^)|2d𝒙.

We are now in a position to deal with Eα defined by (2.7). First we rewrite it as

Eα=Ω(𝒗^𝝈(𝒖^,p^)𝒏-𝒖^𝝈(𝒗^,q^)𝒏)𝑑𝜸=Ω𝒇^(𝝈(𝒖^,p^)𝒏-𝝈(𝒗^,q^)𝒏)𝑑𝜸.

We can split Eα into two parts:

Eα=Ω𝒇^(𝝈(𝒘^,r^)𝒏-𝝈(𝒗^,q^)𝒏)𝑑𝜸+Ω𝒇^(𝝈(𝒖^,p^)𝒏-𝝈(𝒘^,r^)𝒏)𝑑𝜸.

The second term on the right-hand side of the above relation can be estimated by using (3.1):

|Ω𝒇^(𝝈(𝒖^,p^)𝒏-𝝈(𝒘^,r^)𝒏)d𝜸|C𝒗^H1(Ω)(𝑫(𝒖^-𝒘^)L2()+𝐝𝐢𝐯𝝈(𝒖^-𝒘^,p^-r^)L2())

and combining the above estimate with (4.1), we obtain

|Ω𝒇^(𝝈(𝒖^,p^)𝒏-𝝈(𝒘^,r^)𝒏)d𝜸|C𝒗^H1(Ω)(𝑫(𝒖^-𝒘^)L2()+α𝒖^-𝒘^L2()+e-αT𝒖(T)L2()).

Gathering the above inequality, (4.2) and Lemma 3.3, we finally deduce that, for α1,

|Ω𝒇^(𝝈(𝒖^,p^)𝒏-𝝈(𝒘^,r^)𝒏)d𝜸|Cα2e-αT𝒗^H1(Ω)2.

To estimate the first term on the right-hand side of (4.8), we use (4.7), (4.4) and (3.2)

α|𝒗^-𝒘^|2+2|𝑫(𝒗^-𝒘^)|2d𝒙=𝒮𝝈((𝒗^-𝒘^),(q^-r^))n𝒗^𝑑𝜸
C𝒗^H1(S)(𝑫(𝒗^-𝒘^)L2()+𝐝𝐢𝐯𝝈((𝒗^-𝒘^),(q^-r^))L2()).

Therefore, using (4.3) we deduce that, for α1,

α|𝒗^-𝒘^|2+2|𝑫(𝒗^-𝒘^)|2d𝒙C(α+1)(𝒮|𝒗^|2+2|𝑫(𝒗^)|2d𝒙).

We conclude from (4.8), (4.9), (4.7) and (4.10) the relation (2.8).

5 Proof of Corollary 2.2

The aim of this section is to prove Corollary 2.2, and in particular to construct a family (𝒗^,q^) depending on α>0 allowing to recover the support function hS defined by (2.9).

The proof is similar to the one in [22] or in [25], but we include here the proof for completeness.

We set

𝒗^(𝒙):=eα𝜿𝒙,q^(𝒙)=0,𝒙3,

with

,𝜿𝕊2,𝜿=0.

We can check that

Δ𝒗^(𝒙)=α𝒗^(𝒙),div𝒗^=0,

so that (𝒗^,q^) is a solution of (2.1), (2.2).

In order to estimate Eα, we first recall the following proposition (see [22, Proposition 3.2]).

Assume G is an open subset of R3. If G is of class C2, then for any 𝛋S2, there exist constants M=M𝛋>0, ε=ε𝛋>0 and p=p𝛋[0,1] such that

μ2({𝒙G:𝒙𝜿=hG(𝜿)-r})Mrpfor all r(0,ε),

where μ2 denotes the Lebesgue measure of R2.

As can be seen in the remaining part of the proof, we only need relation (5.2), and thus the corollary is valid for “regular sets” in this sense (see [22] for more details about this notion). Let us introduce the following notation:

G𝜿(δ):={𝒙G:hG(𝜿)-δ<𝒙𝜿hG(𝜿)}.

Now we are in a position to prove Corollary 2.2. First, it is straightforward from the definition of the support function (recalled in (2.9)) that

𝒮e2α𝜿𝒙𝑑𝒙μ3(𝒮)e2αh𝒮(𝜿),

where μ3 is the Lebesgue measure in 3. Second,

𝒮e2α(𝜿𝒙-h𝒮(𝜿))𝑑𝒙𝒮𝜿(δ)e2α(𝜿𝒙-h𝒮(𝜿))𝑑𝒙=0δ{𝒙𝒮:𝜿𝒙-h𝒮(𝜿)=-r}e2α(𝜿𝒙-h𝒮(𝜿))𝑑𝒙𝑑r=0δμ2({𝒙𝒮:𝜿𝒙-h𝒮(𝜿)=-r})e-2αr𝑑rM0δrpe-2αr𝑑rMe-2αδδp+1p+1.

Then, if we take δ=α-1/2, we obtain

𝒮e2α𝜿𝒙𝑑𝒙C2(𝒮,𝜿)e2αh𝒮(𝜿)1α(p+1)/2.

Setting β=p+12[0,1], we deduce

C21αβe2αh𝒮(𝜿)𝒮e2α𝜿𝒙𝑑𝒙C1e2αh𝒮(𝜿).

Using (5.1), we can check that

𝒮|𝒗^|2𝑑𝒙=𝒮e2α𝜿𝒙𝑑𝒙and𝒮|𝑫(𝒗^)|2𝑑𝒙=α|𝜿+𝜿2|2𝒮e2α𝜿𝒙𝑑𝒙.

We can also see that

𝒗^H1(Ω)2C(1+α)e2αhΩ(𝜿),

where C=C(Ω) is a positive constant. Therefore, from (5.3), (5.4) and (5.5), (2.8) we obtain

Cα(1-β)e2αh𝒮(𝜿)-Cα2e-αT(α+1)e2αhΩ(𝜿)EαC(α+1)2e2αh𝒮(𝜿)+Cα2e-αT(α+1)e2αhΩ(𝜿).

Since

α(β+1)(α+1)e-αTe2αhΩ(𝜿)e-2αh𝒮(𝜿)0

and

α2(α+1)-1e-αTe2αhΩ(𝜿)e-2αh𝒮(𝜿)0

as α+, (5.6) implies

logC2α+(1-β)log(α)2α+h𝒮(𝜿)+o(1)12αlog(Eα)logC2α+log((α+1)α)2α+h𝒮(𝜿)+o(1)

for α+. This allows us to conclude the proof of Corollary 2.2.

6 Proof of Corollary 2.3

In this section, we prove Corollary 2.3. In order to do this, we construct a family (𝒗^,q^) depending on α>0 allowing to recover the distance d(𝒙0,𝒮) of 𝒮 to a point 𝒙0ch(Ω).

In order to construct (𝒗^,q^), we use spherical coordinates for a frame centered in 𝒙0 and such that the e3 direction is parallel to a plane separating 𝒙0 and Ω. More precisely, every point of the space is defined by its spherical coordinates (r,θ,φ)+×[0,π]×[0,2π] through the formula

x1=rsinθcosφ,x2=rsinθsinφ,x3=rcosθ.

Since 𝒙0ch(Ω), we can assume that Ω is contained in a region of the form {(r,θ,φ):r>0,θ1<θ<θ2}, where 0<θ1<θ2<π.

With the customary abuse of notation, the same symbol is used for the function of 𝒙=(x1,x2,x3) and of (r,θ,φ). In the orthonormal basis (𝒆r,𝒆θ,𝒆φ) associated to the spherical coordinates, we take

𝒗^(r,θ,φ):=e-αrrsinθ𝒆φ,q^(r,θ,φ)=0,r>0,θ1<θ<θ2.

In what follows, we write

g(r,θ):=e-αrrsinθ.

We are going now to use several classical formulas of operators in spherical coordinates (see, for instance, [11, pp. 285–287]). First, for the divergence, we have

div𝒗^=1rsinθgφ=0.

We also have the Laplacian operator in spherical coordinates:

Δ𝒗^=(Δ𝒗^)r𝒆r+(Δ𝒗^)θ𝒆θ+(Δ𝒗^)φ𝒆φ,

with

(Δ𝒗^)r=-2r2sinθgφ=0,(Δ𝒗^)θ=-2cosθr2sin2θgφ=0,
(Δ𝒗^)φ=2gr2+1r22gθ2+2rgr+cosθr2sinθgθ-1r2sin2θg.

Some calculation gives

gr=-(α+1r)g,
2gr2=(α+2αr+2r2)g,
gθ=-cosθsinθg,
2gθ2=(-1+2sin2θ)g.

Inserting (6.5)–(6.8) in (6.4) yields

(Δ𝒗^)φ=(α+2αr+2r2-1r2+2r2sin2θ-2αr-2r2-cos2θr2sin2θ-1r2sin2θ)g=αg.

The above relation, (6.2), (6.4) and (6.3) imply

Δ𝒗^=α𝒗^,

so that (𝒗^,q^) defined by (6.1) is a solution of (2.1), (2.2).

We can thus use this family and apply Theorem 2.1 to prove Corollary 2.3. More precisely, this corollary will be proved if we can estimate the integrals of 𝒗^, 𝑫(𝒗^) and 𝒗^. We use again classical formula for differential operators in spherical coordinates (see, for instance, [11, pp. 285–287]): setting

Mij=𝑴𝒆i𝒆ji,j{r,θ,φ},

we have

(𝒗^)rr=(𝒗^)θθ=(𝒗^)rθ=(𝒗^)θr=0,
(𝒗^)φφ=1rsinθgφ=0,
(𝒗^)θφ=-gcosθrsinθ,(𝒗^)φθ=1rgθ,
(𝒗^)φr=gr,(𝒗^)rφ=-gr,

and

𝑫(𝒗^)rr=𝑫(𝒗^)θθ=𝑫(𝒗^)rθ=𝑫(𝒗^)θr=0,
𝑫(𝒗^)φφ=1rsinθgφ=0,
𝑫(𝒗^)θφ=𝑫(𝒗^)φθ=12(1rgθ-gcosθrsinθ),
𝑫(𝒗^)rφ=𝑫(𝒗^)φr=12(gr-gr).

Using (6.5) and (6.7), we deduce

𝑫(𝒗^)rφ=-(α2+1r)gand𝑫(𝒗^)θφ=-cosθrsinθg.

The above relation implies

I:=𝒮α|𝒗^|2+2|𝑫(𝒗^)|2d𝒙=𝒮(2α+4αr+41r2sin2θ)|g|2r2sinθdrdθdφ.

Using the hypothesis on 𝒙0 and Ω, we can assume that

𝒮{(r,θ,φ):0<r1r<r2, 0<θ1<θ<θ2<π}.

We can take r1 such that

r1=min𝒮r=min𝒙𝒮|𝒙-𝒙0|=d(𝒙0,𝒮).

From (6.14), we can assume that

sinθ>s*>0in𝒮.

In what follows, α is taken large enough (for instance, α>1). Using (6.14), (6.15) and (6.16), we can estimate I defined by (6.13) as

Iμ3(𝒮)(2α+4αr1+41r12(s*)2)e-2αr1r12(s*)2C1(𝒮)(α+1)e-2αd(𝒙0,𝒮).

The lower bound on the integral is obtained from the following result that is proved, for instance, in [27, Proposition 3.2].

Assume S is of class C2. There exists 𝛄R such that

lim infαα𝜸e2αd(𝒙0,𝒮)𝒮e-2α|𝒙-𝒙0|𝑑𝒙>0.

Using the above proposition and (6.14), we deduce that

IC(𝒮)(2α+4αr2+41r22)1r22α-𝜸e-2αd(𝒙0,𝒮)C2(𝒮)α1-𝜸e-2αd(𝒙0,𝒮).

On the other hand, using (6.9)–(6.12), (6.5), (6.6), we can check (as in (6.13))

𝒗^H1(Ω)2C(Ω)Ω(1+α+1r2)|g|2r2sinθdrdθdφC(Ω)(α+1)e-2αd(𝒙0,Ω).

Therefore, by the same kind of reasoning as in the end of Section 5, we conclude the proof of Corollary 2.3.

7 Spherical geometrical optics solutions

In this section, we prove Theorem 2.6 and Theorem 2.8 by using the spherical geometrical optics solutions. Let us first recall the following result proved in [16]:

For all 𝐱0ch(Ω) (the convex hull of Ω) and d>0, there exists a family (𝐯α,qα)C2(Ω¯)×C1(Ω¯) such that

-𝐝𝐢𝐯𝝈(𝒗α,qα)=𝑖𝑛Ω~,
div𝒗α=0𝑖𝑛Ω~,

for some domain Ω~Ω and for α>0 and such that for α>α0,

cα2(dd(𝒙0,𝒮))2α𝒮|𝒗α|2𝑑𝒙Cα2(dd(𝒙0,𝒮))2α

and

cα4(dd(𝒙0,𝒮))2α𝒮|𝑫(𝒗α)|2𝑑𝒙Cα4(dd(𝒙0,𝒮))2α.

Here c and C are constants that may depend on 𝒮.

7.1 Proof of Theorem 2.6

For simplicity, we suppress in the proofs below the explicit dependence on α in the notation. For example, we write 𝒗 instead of 𝒗α.

Multiplying (1.34) by 𝒖, integrating by part and using (1.35)–(1.39), we obtain

Ω𝝈(𝒖,p)𝒏𝒇𝑑𝜸=(t)2|𝑫(𝒖)|2𝑑𝒙.

Multiplying (7.1) by 𝒗, integrating by part and using (2.12) we deduce

Ω𝝈(𝒗,q)𝒏𝒇𝑑𝜸=Ω2|𝑫(𝒗)|2𝑑𝒙.

Multiplying (7.1) by a smooth divergence free map w and integrating on 𝒮(t), we obtain

𝒮(t)𝝈(𝒗,q)𝒏w𝑑𝜸+2𝒮(t)𝑫(𝒗):𝑫(w)d𝒙=0.

Consequently, taking particular choices of w, we have

𝒮(t)𝝈(𝒗,q)𝒏𝑑𝜸=𝒮(t)𝒙×𝝈(𝒗,q)𝒏𝑑𝜸=0.

Then multiplying (7.1) by 𝒖, integrating on (t), integrating by parts and using (7.5) implies

Ω𝝈(𝒗,q)𝒏𝒇𝑑𝜸=(t)2𝑫(𝒗):𝑫(𝒖)d𝒙.

Combining (7.3), (7.6) and (7.4),

Ω𝝈(𝒖-𝒗,p-q)𝒏𝒇𝑑𝜸=2𝒮(t)|𝑫(𝒗)|2𝑑𝒙+2(t)|𝑫(𝒗-𝒖)|2𝑑𝒙.

On the other hand, combining (1.34)–(1.39), (7.1)–(7.2) and (7.5), we deduce

-𝐝𝐢𝐯𝝈(𝒖-𝒗,p-q)=in(t),t(0,T),
div(𝒖-𝒗)=0in(t),t(0,T),
(𝒖-𝒗)=onΩ,t(0,T),
(𝒖-𝒗)=+𝝎×(𝒙-𝒉)-𝒗on𝒮(t),t(0,T),
𝒮(t)𝝈(𝒖-𝒗,p-q)𝒏𝑑𝜸=,t(0,T),
𝒮(t)(𝒙-𝒉)×𝝈(𝒖-𝒗,p-q)𝒏𝑑𝜸=,t(0,T).

Therefore, multiplying (7.7) by 𝒖-𝒗, using (7.8)–(7.12), and applying Lemma 3.1 and the Korn inequality, we deduce

(𝒖-𝒗)(t)H1((t))C𝒗H1(𝒮(t)).

Consequently, we obtain

2𝒮(t)|𝑫(𝒗)|2𝑑𝒙Ω𝝈(𝒖-𝒗,p-q)𝒏𝒇𝑑𝜸C(𝒮(t)2|𝑫(𝒗)|2+|𝒗|2d𝒙).

Using Theorem 7.1, we obtain

cα4(dd(𝒙0,𝒮(t)))2αΩ𝝈(𝒖-𝒗,p-q)𝒏𝒇𝑑𝜸C(α2+α4)(dd(𝒙0,𝒮(t)))2α.

If d<d(𝒙0,𝒮(t)), then the above estimate yields

FαC(dd(𝒙0,𝒮(t)))α.

If d>d(𝒙0,𝒮(t)), then we deduce

Fαc(dd(𝒙0,𝒮(t)))2α.

We conclude the proof of Theorem 2.6.

7.2 Proof of Theorem 2.8

We only prove the result for system (1.22)–(1.33). A similar and simpler proof can be done for the Navier–Stokes system (1.17)–(1.21).

We modify the function (𝒗α,qα) of Theorem 7.1 by multiplying it by a function χC([0,T]) such that χ(0)=0, χ>0 in (0,T]. This modification allows us to have regular solutions for system (1.22)–(1.33) or for the Navier–Stokes system (1.17)–(1.21) if N=2 (see Proposition 3.5).

First, the Reynolds formula implies

ddt(t)|𝒖|22𝑑𝒙=(t)𝒖t𝒖𝑑𝒙+𝒮(t)𝒖𝒏|𝒖|22𝑑𝜸.

On the other hand, an integration by parts gives

(t)(𝒖)𝒖𝒖𝑑𝒙=(t)𝒖𝒏|𝒖|22𝑑𝜸=𝒮(t)𝒖𝒏|𝒖|22𝑑𝜸+Ω𝒇𝒏|𝒇|22𝑑𝜸.

Multiplying (1.22) by 𝒖, using (7.13)–(7.14) and integrating by parts yields

0=ddt(t)|𝒖|22𝑑𝒙+(t)2|𝑫(𝒖)|2𝑑𝒙+Ω(𝒇𝒏)|𝒇|22𝑑𝜸+m𝒉′′𝒉+I0𝝎𝝎-Ω𝝈(𝒖,p)𝒏𝒇𝑑𝜸.

Let us extend 𝒖 in 𝒮(t) by

𝒖(t,𝒙)=(t)+ω(t)(𝒙-𝒉(t))in 𝒮(t).

We also define a global density function ρ as

ρ(t,𝒙):={1if 𝒙(t),ρ𝒮if 𝒙𝒮(t).

Using (1.16), we can prove that

ddt𝒮(t)ρ𝒮|𝒖|22𝑑𝒙=m𝒉′′(t)𝒉(t)+I0ω(t)ω(t).

Combining the above equation with (7.15) and using the notation (7.16)–(7.17), we deduce

Ω𝝈(𝒖,p)𝒏𝒇𝑑𝜸=ddtΩρ|𝒖|22𝑑𝒙+(t)2|𝑫(𝒖)|2𝑑𝒙+Ω(𝒇𝒏)|𝒇|22𝑑𝜸.

Multiplying (7.1) by 𝒗, integrating by part and using (2.12), it follows

Ω𝝈(𝒗,q)𝒏𝒇𝑑𝜸=Ω2|𝑫(𝒗)|2𝑑𝒙.

Using (7.5) and using (2.12) and multiplying (7.1) by 𝒖, we obtain

Ω𝝈(𝒗,q)𝒏𝒇𝑑𝜸=(t)2𝑫(𝒗):𝑫(𝒖)d𝒙.

By combining (7.18), (7.19) and (7.20), we deduce

Ω𝝈(𝒖-𝒗,p-q)𝒏𝒇𝑑𝜸=𝒮(t)2|𝑫(𝒗)|2𝑑𝒙+(t)2|𝑫(𝒗-𝒖)|2𝑑𝒙+ddtΩρ|𝒖|22𝑑𝒙+Ω(𝒇𝒏)|𝒇|22𝑑𝜸.

We deduce that

0TΩ𝝈(𝒖-𝒗,p-q)𝒏𝒇𝑑𝜸𝑑t-0TΩ(𝒇𝒏)|𝒇|22𝑑𝜸𝑑t0T𝒮(t)2|𝑫(𝒗)|2𝑑𝒙𝑑t.

As a consequence, if the observation Kα defined by (2.16) remains bounded as α, then it implies that

0T𝒮(t)|𝑫(𝒗)|2𝑑𝒙𝑑t

is also bounded as α. From Theorem 7.1, this yields that for almost all t[0,T], d<d(𝒙0,𝒮(t)). Since 𝒉 and 𝑸 are continuous, it implies that

𝒮(t)B(𝒙0,d)=for all t[0,T].

This ends the proof of Theorem 2.8.

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    Raymond J.-P., Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 6, 921–951.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Alvarez C., Conca C., Friz L., Kavian O. and Ortega J. H., Identification of immersed obstacles via boundary measurements, Inverse Problems 21 (2005), no. 5, 1531–1552.

  • [2]

    Alves C. J. S., Kress R. and Silvestre A. L., Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations, J. Inverse Ill-Posed Probl. 15 (2007), no. 5, 461–481.

  • [3]

    Badra M., Caubet F. and Dambrine M., Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci. 21 (2011), no. 10, 2069–2101.

  • [4]

    Bonnesen T. and Fenchel W., Theory of Convex Bodies, BCS Associates, Moscow, 1987.

  • [5]

    Conca C., Cumsille P., Ortega J. and Rosier L., On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems 24 (2008), no. 4, Article ID 045001.

  • [6]

    Conca C., Malik M. and Munnier A., Detection of a moving rigid solid in a perfect fluid, Inverse Problems 26 (2010), no. 9, Article ID 095010.

  • [7]

    Conca C., Schwindt E. L. and Takahashi T., On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems 28 (2012), no. 1, Article ID 015005.

  • [8]

    Dos Santos Ferreira D., Kenig C. E., Sjöstrand J. and Uhlmann G., Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys. 271 (2007), no. 2, 467–488.

  • [9]

    Doubova A., Fernández-Cara E., González-Burgos M. and Ortega J. H., A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 6, 1213–1238.

  • [10]

    Doubova A., Fernández-Cara E. and Ortega J. H., On the identification of a single body immersed in a Navier–Stokes fluid, European J. Appl. Math. 18 (2007), no. 1, 57–80.

  • [11]

    Duvaut G., Mécanique des Milieux Continus, Dunod, Paris, 1998.

  • [12]

    Eskin G. and Ralston J., On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems 18 (2002), no. 3, 907–921.

  • [13]

    Gaitan P., Isozaki H., Poisson O., Siltanen S. and Tamminen J., Probing for inclusions in heat conductive bodies, Inverse Probl. Imaging 6 (2012), no. 3, 423–446.

  • [14]

    Galdi G. P., An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd ed., Springer Monogr. Math., Springer, New York, 2011.

  • [15]

    Heck H., Li X. and Wang J.-N., Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J. 56 (2007), no. 5, 2489–2510.

  • [16]

    Heck H., Uhlmann G. and Wang J.-N., Reconstruction of obstacles immersed in an incompressible fluid, Inverse Probl. Imaging 1 (2007), no. 1, 63–76.

  • [17]

    Hesla T. I., Collisions of smooth bodies in viscous fluids: A mathematical investigation, Ph.D. thesis, University of Minnesota, 2005.

  • [18]

    Hillairet M., Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1345–1371.

  • [19]

    Hillairet M. and Takahashi T., Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal. 40 (2009), no. 6, 2451–2477.

  • [20]

    Ikehata M., How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl. 7 (1999), no. 3, 255–271.

  • [21]

    Ikehata M., Reconstruction of a source domain from the Cauchy data, Inverse Problems 15 (1999), no. 2, 637–645.

  • [22]

    Ikehata M., Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 8 (2000), no. 4, 367–378.

  • [23]

    Ikehata M., Inverse scattering problems and the enclosure method, Inverse Problems 20 (2004), no. 2, 533–551.

  • [24]

    Ikehata M., An inverse source problem for the heat equation and the enclosure method, Inverse Problems 23 (2007), no. 1, 183–202.

  • [25]

    Ikehata M., The framework of the enclosure method with dynamical data and its applications, Inverse Problems 27 (2011), no. 6, Article ID 065005.

  • [26]

    Ikehata M. and Itou H., On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data, Inverse Problems 28 (2012), no. 12, Article ID 125003.

  • [27]

    Ikehata M. and Kawashita M., The enclosure method for the heat equation, Inverse Problems 25 (2009), no. 7, Article ID 075005.

  • [28]

    Ikehata M. and Kawashita M., On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems 26 (2010), no. 9, Article ID 095004.

  • [29]

    Kenig C. E., Sjöstrand J. and Uhlmann G., The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591.

  • [30]

    Nakamura G. and Uhlmann G., Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math. 118 (1994), no. 3, 457–474.

  • [31]

    Nakamura G. and Uhlmann G., Erratum: “Global uniqueness for an inverse boundary value problem arising in elasticity” [Invent. Math. 118 (1994), no. 3, 457–474], Invent. Math. 152 (2003), no. 1, 205–207.

  • [32]

    Raviart P.-A. and Thomas J.-M., Introduction à l’analyse numérique des équations aux dérivées partielles, Coll. Math. Appl. Maîtrise, Masson, Paris, 1983.

  • [33]

    Raymond J.-P., Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 6, 921–951.

  • [34]

    San Martín J. A., Starovoitov V. and Tucsnak M., Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002), no. 2, 113–147.

  • [35]

    Sylvester J. and Uhlmann G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169.

  • [36]

    Takahashi T., Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003), no. 12, 1499–1532.

  • [37]

    Takahashi T. and Tucsnak M., Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech. 6 (2004), no. 1, 53–77.

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