A global uniqueness result for acoustic tomography of moving fluid

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A global uniqueness result for acoustic tomography of moving fluid

Alexey Agaltsov

To cite this version:

Alexey Agaltsov. A global uniqueness result for acoustic tomography of moving fluid. 2015. �hal-

01122389�

A global uniqueness result for acoustic tomography of moving uid

A. D. Agaltsov

We consider a model time-harmonic wave equation of acoustic to- mography of moving uid in an open bounded domain in dimen- sion d ≥ 2 . We give global uniqueness theorems for related inverse boundary value problem at xed frequency.

Keywords: inverse boundary value problems, time-harmonic wave equation, acoustic tomography

Subjects: partial dierential equations, mathematical physics AMS classication: 35R30 (Inverse problems), 35Q35 (PDEs in connection with uid mechanics)

1 Introduction

Consider the operator

L

= −∆ − 2iA(x) · ∇ + V (x), (1) where ∆ is the standard Laplacian, x ∈ D , A ∈ W

(D, R

) , V ∈ L

(D, R ) , D is an open bounded domain in R

( d ≥ 2 ). In the present article we study an inverse boundary value problem for the equation L

ψ = 0 in D .

As in [AN], [RW], [RE] we consider the equation L

ψ = 0 as a model equation for a time-harmonic ( e

) pressure ψ in moving uid. In this setting

A(x) = ω

c

(x) v(x), V (x) = − ω

c

(x) ,

where v is the uid velocity vector, c is the sound speed, ω is the frequency.

Suppose that 0 is not a Dirichlet eigenvalue for operator L

in D . Then the Dirichlet problem

( L

ψ = 0 in D,

ψ|

= f, (2)

1Centre de Mathematiques Appliquees, Ecole Polytechnique Route de Saclay

91128 PALAISEAU Cedex, France

2Lomonosov Moscow State University, GSP-1, Leninskie Gory

119991 MOSCOW, Russia email: [email protected]

is uniquely solvable for ψ ∈ H

(D) for any f ∈ H

(∂D) . The Dirichlet-to- Neumann map Λ

sends f ∈ H

(∂D) to Λ

f ∈ H

(∂D) dened by the formula

Λ

f =

+ i(A · ν)f, (3) where ν is the unit exterior normal to ∂D and

∈ H

(∂D) can be dened, in particular, by the following formula:

h

, ui = Z

D

∇ψ(x)∇ u(x) ˜ − 2i˜ u(x)A(x)∇ψ(x) + ˜ u(x)V (x)ψ(x)

dx, (4) for u ∈ H

(∂D) and arbitrary u ˜ ∈ H

(D) with u| ˜

= u . Note that since ψ satises L

ψ = 0 , the right hand side of the above formula doesn't depend on the choice of u ˜ .

The inverse boundary value problem for equation L

ψ = 0 in D consists in nding A , V from Λ

. In the case when coecients A , V can be complex- valued there is an obstruction to the unique solvability of this problem caused by the gauge invariance of the map Λ

with respect to the gauge transformations

( A → A + ∇ϕ,

V → V − i∆ϕ + (∇ϕ)

+ 2A∇ϕ,

where ϕ ∈ W

(D, C ) , ϕ|

= 0 , see, e.g., [KU] ( d ≥ 3 ), [GT] ( d = 2 ).

However, in the case of real-valued coecients A , V there is no gauge non- uniqueness as it was observed, for example, in [AN].

In addition, in general case, under some regularity assumptions on ∂D , A and V , the Dirichlet-to-Neumann map Λ

uniquely determines the two-form dA and the function q in D and the tangential component of A on ∂D , where

dA = X

1≤k<l≤d

∂A

∂x

− ∂A

∂x

dx

∧ dx

, q = V + i∇ · A − A · A,

(5)

and A = (A

, . . . , A

) .

In particular, it was shown in [KU] that in dimension d ≥ 3 the map Λ

uniquely determines dA and q in D if A ∈ L

(D, C

) and V ∈ L

(D, C ) . And in dimension d = 2 it was shown in [GT] that if D is a smooth Riemann surface with boundary (in particular, if D is a planar domain with ∂D ∈ C

) then the map Λ

uniquely determines dA and q provided that A ∈ W

(D, R

) , V ∈ W

(D, C ) , p > 2 .

In addition, concerning the identiability of tangential components of A on

the boundary, it was proved in [BS] that if ∂D ∈ C

( d ≥ 3 ) or ∂D ∈ C

,

α ∈ (0, 1) ( d = 2 ) and if A ∈ C(D, C

) , V ∈ L

(D, C ) then Λ

uniquely

determines A − ν(A · ν) on ∂D , where ν is the unit exterior normal eld to ∂D .

In the present article we combine the aforementioned results in order to

obtain the following global uniqueness results in the case when coecients A ,

V are real-valued.

Theorem 1. Let D be a bounded simply connected domain with path connected boundary in R

( d ≥ 3 ) with ∂D ∈ C

. Let A

, A

∈ W

(D, R

) and V

, V

∈ L

(D, R ) . If Λ

= Λ

, then A

= A

, V

= V

.

Theorem 2. Let D be a bounded simply connected domain in R

with ∂D ∈ C

. Let A

, A

∈ W

(D, R

) and V

, V

∈ W

(D, R ) with p > 2 . If Λ

= Λ

then A

= A

and V

= V

.

Theorems 1 and 2 are proved in Section 3. In Section 2 we present formulas and equations for nding A , V from dA , q , A − ν(A · ν)|

.

2 Formulas and equations for nding A , V

In this section we suppose that D is a bounded contractible domain with path connected C

boundary in R

( d ≥ 2 ). By contractibility we mean that there exists F ∈ C

(D ×[0, 1], D) such that F

≡ x e , F

= id

, where F

(x) = F (x, t) , e x is some xed point in D and id

is the identity mapping on D . We also suppose that A ∈ W

(D, R

) , V ∈ L

(D, R ) . Given dA , q as in (5) in D and A − ν(A · ν ) on ∂D , we can nd A , V in the following way:

1. Dene A e = ( A e

, . . . , A e

) ∈ W

(D, R

) by the formula

A e

= X

i<j

Z

0

∂F

∂t

∂F

∂x

− ∂F

∂t

∂F

∂x

∂A

∂x

◦ F

− ∂A

∂x

◦ F

dt, (6) where k = 1 , . . . , d ; A = (A

, . . . , A

) , F

= (F

, . . . , F

) and ◦ denotes the composition of maps, i.e.

◦ F

(y) =

i

(F

(y)) , y ∈ D . 2. Fix x

∈ ∂D . Dene ϕ

∈ C

(∂D) by the formula

ϕ

(x) =

d

X

k=1

Z

x⁰

A

(y) − A e

(y)

dy

, x ∈ ∂D, (7) where A

= A − ν (A · ν ) , A e

= A e − ν( A e · ν ) , A

= (A

, . . . , A

) , A e

= ( A e

, . . . , A e

) , ν is the unit exterior normal eld to ∂D and integration is over an arbitrary C

curve on ∂D linking x

to x .

3. Find the unique generalized solution ϕ ∈ W

(D, R ) to ( ∆ϕ = Im q − ∇ · A e in D,

ϕ|

= ϕ

. (8)

4. Coecients A , V are given by the following formulas:

( A = A e + ∇ϕ,

V = q − i∆ϕ − i∇ · A e + A e · A e + 2 A e · ∇ϕ + (∇ϕ)

.

This algorithm will be justied in Section 4.

3 Proofs of Theorems 1, 2

We will prove Theorems 1 and 2 simultaneously. Let D , A

, A

, V

, V

satisfy the conditions of Theorem 1 (resp. Theorem 2) and suppose that Λ

= Λ

.

Using Theorem 1.1 of [BS] we obtain that A

− ν(A

· ν )

|

= A

− ν (A

· ν )

|

, (9) where ν is the unit exterior normal eld to ∂D . Using Theorem 1.1 of [KU]

(resp. Theorem 1.1 of [GT]) we get

dA

= dA

in D, (10)

q

= q

in D, (11)

where

dA

= X

1≤k<l≤d

∂A

∂x

− ∂A

∂x

dx

∧ dx

, q

= V

+ i∇ · A

− A

· A

, where A

= (A

, . . . , A

) , j = 1 , 2 .

Since the domain D is simply connected it follows from (10) that there exists ϕ ∈ W

(D, R ) such that

A

− A

= ∇ϕ in D. (12) In dimension d = 2 it follows from simple connectedness of D and from smoothness of ∂D that ∂D is path connected. Formulas (9), (12) and path connectedness of ∂D imply that ϕ is constant on ∂D .

Using (11), (12) we obtain that

V

− V

= −i∆ϕ − (∇ϕ)

+ 2A

∇ϕ in D.

Taking the imaginary part of this equation we obtain the equation ∆ϕ = 0 in D . Since ϕ is constant on ∂D and ϕ ∈ W

(D) it follows that ϕ is constant in D . Hence A

= A

and V

= V

. Theorems 1 and 2 are proved.

4 Justication of the algorithm of Section 2

It follows from formula (6) that d A e = dA . More precisely, if we denote by F

dA the pullback of the form dA by the map F and by ι

we denote the interior product with the vector eld

on {(x, t) ∈ D × [0, 1]} , then

d

X

k=1

A e

dx

= Z

0

(ι

F

dA) dt,

and the equality d A e = dA follows from the Cartan magic formula L

= d ◦ ι

+ ι

◦ d , where L

is the Lie derivative along

, d is the exterior derivative on {(x, t) ∈ D × [0, 1]} and ◦ denotes the composition of maps.

Hence we can dene ϕ ∈ W

(D, R ) by the formula

ϕ(x) = Z

˜ x

d

X

k=1

(A

− A e

)dx

, x ∈ D,

where x ˜ ∈ D is some xed point and integration is over an arbitrary C

curve in D linking x ˜ to x . Then ∇ϕ = A − A e in D and this implies that ϕ|

diers by constant from ϕ

dened in (7). We also obtain from (5) the equation

V = q − i∆ϕ − i∇ · A e + A e · A e + 2 A e · ∇ϕ + (∇ϕ)

in D.

Taking into account that V is real-valued and separating the imaginary part in the latter equation we obtain (8). Since Im q and ∇ · A e belong to L

(D, R ) , the problem (8) is uniquely solvable for ϕ ∈ W

(D, R ) . Thus, the algorithm of Section 2 is justied.

5 Aknowledgements

This work was fullled within the framework of research carried out under the supervision of Prof. R. G. Novikov.

6 References

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[BS] R. M. Brown, M. Salo, Identiability at the boundary for rst-order terms, Appl. Anal. 85 (67), 2006, 735749

[GT] C. Guillarmou, L. Tzou, Identication of a connection from Cauchy data on a Riemann surface with boundary, Geom. Funct. Anal. 21 (2), 2011, 393418

[KU] K. Krupchyk, G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrodinger operator with a bounded magnetic potential, Comm. Math. Phys. 327 (3), 2014, 9931009

[RW] D. Roussef, K. B. Winters, Two-dimensional vector ow inversion by dirac- tion tomography, Inverse Problems 10, 1994, 687697

[RE] M. N. Rychagov, H. Ermert, Reconstruction of uid motion in acoustic

diraction tomography, J. Acoust. Soc. Am. 99 (5), 1996, 30293035