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.
This paper is devoted to reconstructing an unknown structure included in a bounded cavity () filled by a viscous incompressible fluid. More precisely, we aim to obtain some geometrical information on by measurement on the boundary 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 is a compact connected subset of Ω with nonempty interior and that is connected.
In the first part of the article, the fluid equations that we consider are the nonstationary Stokes system
In the above system, are the velocity and the pressure of the fluid. Moreover, we have denoted by the Cauchy stress tensor, which is defined by the Stokes law as
where is the identity matrix of , with denoting the space of real square matrices of order N, and where is the strain tensor defined by
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 in (1.4) and to measure the corresponding Cauchy force
in order to deduce information on the obstacle . Here and in all what follows, denotes the unit outer normal to the fluid domain.
We also consider in this paper the following linear fluid–rigid body system:
Here and 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 is fixed. We assume that the density of the rigid body is a positive constant. In particular, the mass m and the inertia tensor are defined as follows:
where denotes the Lebesgue measure in and where is the identity matrix.
Finally, is defined by
Here and are respectively the center of mass and the orientation of . In particular, the center of mass of is located at . We have denoted by 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 and to measure the corresponding Cauchy force given by (1.7) in order to obtain geometrical information on . However, here there is an important difference: applying at the boundary of makes the rigid body moves through the (unknown) trajectory . Moreover, with such a boundary condition, it could possible that the rigid body touches and it is not clear what happens after this contact (see ).
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
The map is defined as follows:
This map is related to the vector product by the formula
Let us remark that the above system is not linear since is not given. This system is studied in  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 on this system.
These geometrical inverse problems for fluid systems were already considered in  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 not identically equal to , then the mapping that associates to 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  and in . In , 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  is extended in  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 : 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 , 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  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  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  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 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 norm of on as α goes to (Theorem 2.1). The idea is then to construct solutions so that the norm on 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 to a point , we need to impose and . These hypothesis are not considered in the case of the heat equation (see ).
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 . 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 norm of on , as explained above. Then in Section 5, we construct in order to recover the convex hull of and in Section 6, we construct in order to recover the distance from a fixed point to . Section 7 is devoted to inverse problems for the nonlinear systems: we use there complex geometrical solutions.
2 Main results
First we consider a family of solutions of a Stokes system
for some domain and for .
We then consider defined by
with such that and in and such that
For instance, in what follows, we take
We can remark that since is given by (2.3), then it satisfies the condition
Let us set
We are now in a position to state our first main result.
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 . The first extension of this method to a system of partial differential equation was developed in .
The first corollary of Theorem 2.1 corresponds to the reconstruction of the support function of . Let us recall that for any subset G of , the support function of G is defined by
where is the unit sphere of . This function is classically used in the theory of convex sets (see, for instance, [4, p. 26]). In particular, if G is convex,
Corollary 2.2 (Recovering the support function)
The second corollary of Theorem 2.1 allows us to obtain the distance of to a point (the convex hull of Ω).
Corollary 2.3 (Recovering the distance to a point)
In contrast to [27, 28], in the above result, we have to assume that . This restriction comes from the fact that we need in our construction that the family satisfies the condition . 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 .
Assume is of class .
for some domain . Here is a parameter in the construction of these solutions that eventually goes to . We then consider defined by
As explained in the previous section, one difficulty for stating result for this system is that the rigid body can touch . We thus assume that for all regular ,
We fix (the convex hull of Ω) and . Then, we have the following results.
If , then for some constants and .
If , then for some constants and and for .
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 . 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 .
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 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 since we are working with regular solutions and for the existence of global (in time) regular solutions is an open problem. In particular, in the case , one should need to show that the times of existence of the family of solutions can be chosen uniformly with respect to α.
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 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 instead of .
Assume such that in Ω. Consider a pair such that . Then there exists a constant such that
We use [14, p. 176, relations (III.3.31) and (III.3.32)]: there exists such that
We then use integration by parts
The proof of (3.2) is similar, we consider (instead of ) a function such that
The proof of the lemma is complete. ∎
Assume , with , satisfying
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 is not needed to obtain the result and the result remains true for more general boundary conditions.
Using, for instance, , there exists such that
Using this lifting, we consider the change of variables
and the equations for can be written as
where is the Leray projection on
Using that is self-adjoint and positive, we obtain the result.
with given by (3.10) and
If we extend and in by
then the above system can be written as
and is the orthogonal projection. We have and it is proved in  that is self-adjoint and positive in , and this allows us to prove the result by using classical result on parabolic systems. ∎
Let us multiply (1.1) by and integrate by parts:
Using Gronwall’s lemma, we deduce the existence of constant depending only on T such that
We can obtain in a similar way the following lemma.
In the above result, we have extended in by setting
Assume and assume , with , satisfying
Then the following hold:
The first result is classical and the second result was proved in . It is possible to prove the first result by using a fixed point approach: one can consider the mapping
where is the solution of
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 estimate (that is possible since ) 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 (construct from ) and transform in (where is the transpose of the cofactor matrix of ) and p in . In the above proposition, (3.16)–(3.17) means that
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  for more details).
4 Proof of Theorem 2.1
Let us define for all ,
with defined by (2.5).
We consider the solution of the problem
The couple satisfies the system
Taking the inner product of (4.1) with and integrating by parts, we obtain
Taking the inner product of (4.3) with and integrating by parts, it follows
The above relation implies
Taking the inner product of (2.1) with and integrating by parts on , we obtain
We are now in a position to deal with defined by (2.7). First we rewrite it as
We can split into two parts:
The second term on the right-hand side of the above relation can be estimated by using (3.1):
and combining the above estimate with (4.1), we obtain
Therefore, using (4.3) we deduce that, for ,
5 Proof of Corollary 2.2
We can check that
In order to estimate , we first recall the following proposition (see [22, Proposition 3.2]).
Assume G is an open subset of . If is of class , then for any , there exist constants , and such that
where denotes the Lebesgue measure of .
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  for more details about this notion). Let us introduce the following notation:
where is the Lebesgue measure in . Second,
Then, if we take , we obtain
Setting , we deduce
Using (5.1), we can check that
We can also see that
as , (5.6) implies
for . 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 depending on allowing to recover the distance of to a point .
In order to construct , we use spherical coordinates for a frame centered in and such that the direction is parallel to a plane separating and Ω. More precisely, every point of the space is defined by its spherical coordinates through the formula
Since , we can assume that Ω is contained in a region of the form , where .
With the customary abuse of notation, the same symbol is used for the function of and of . In the orthonormal basis associated to the spherical coordinates, we take
In what follows, we write
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
We also have the Laplacian operator in spherical coordinates:
Some calculation gives
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 , and . We use again classical formula for differential operators in spherical coordinates (see, for instance, [11, pp. 285–287]): setting
The above relation implies
Using the hypothesis on and Ω, we can assume that
We can take such that
From (6.14), we can assume that
The lower bound on the integral is obtained from the following result that is proved, for instance, in [27, Proposition 3.2].
Assume is of class . There exists such that
Using the above proposition and (6.14), we deduce that
7 Spherical geometrical optics solutions
For all (the convex hull of Ω) and , there exists a family such that
for some domain and for and such that for ,
Here c and C are constants that may depend on .
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 instead of .
Multiplying (7.1) by a smooth divergence free map w and integrating on , we obtain
Consequently, taking particular choices of w, we have
Consequently, we obtain
Using Theorem 7.1, we obtain
If , then the above estimate yields
If , then we deduce
We conclude the proof of Theorem 2.6.
7.2 Proof of Theorem 2.8
We modify the function of Theorem 7.1 by multiplying it by a function such that , in . 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 (see Proposition 3.5).
First, the Reynolds formula implies
On the other hand, an integration by parts gives
Let us extend in by
We also define a global density function ρ as
Using (1.16), we can prove that
We deduce that
As a consequence, if the observation defined by (2.16) remains bounded as , then it implies that
is also bounded as . From Theorem 7.1, this yields that for almost all , . Since and are continuous, it implies that
This ends the proof of Theorem 2.8.
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