Khaled Zennir

LOCAL EXISTENCE AND GLOBAL NONEXISTENCE OF SOLUTION FOR LOVE-EQUATIONS WITH INFINITE MEMORY

Keywords: 2010 Mathematics Subject Classification. 35L20, 35L70, 37B25, 93D15 Nonlinear Love-equation, Local Existence, Blow-up, Infinite Memory. 1

In this paper, we consider an IBVP for a nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo-Galerkin method and the weak compactness method, we prove the local existence and uniqueness of weak solution. The finite time blow up of weak solution is considered.

Introduction

Many interesting physical phenomena in which delay effects occur (e.g., population dynamics) can be modeled by partial differential equations with finite or infinite visco-elastic memory which provides a typical damping mechanism in nature. The well-posedness and stability for elasticity and visco-elasticity systems attracted lots of interests in recent years, where different types of dissipative mechanisms have been introduced to obtain diverse results. In general, the stability properties of visco-elastic system are in dependence on the form of the convolution kernel (see in this direction results in [START_REF] Beniani | Zennir Polynomial Decay of Solutions to the Cauchy Problem for a Petrovsky-Petrovsky System in R n[END_REF], [START_REF] Braik | Zennir Polynomial stability for system of 3 wave equations with infinite memories[END_REF], [START_REF] Messaoudi | Blow up and global existence in nonlinear visco-elastic wave equations[END_REF], [START_REF] Messaoudi | Blow up of solutions with positive initial energy in a nonlinear visco-elastic wave equations[END_REF], [START_REF] Messaoudi | General decay of solutions of a visco-elastic equation[END_REF], [START_REF] Messaoudi | A general stability result for a nonlinear wave equation with infinite memory[END_REF], [START_REF] Park | Stability of a von karman equation with infinite memory[END_REF], [START_REF] Kh | Global nonexistence of Solutions for a system of nonlinear visco-elastic wave equations with degenerate damping and source terms[END_REF], [START_REF] Kh | General decay of solutions for damped wave equation of Kirchhoff type with density in R n[END_REF], [START_REF] Zitouni | On the existence and decay of solution for visco-elastic wave equation with nonlinear source in weighted spaces[END_REF], . . . ). The blow up is an essential and very important phenomena to be study in the evolution PDEs, there is a different between global nonexistence which means that the local solution can't be continue to exist in time i.e. there exists a finite time blow up which is our case in this paper and the blowing up ∀t > 0, that is the solution goes to infinity for all t > 0. Blow up an effect occurs, for example, when a sea wave tumbles to the shore, when a computer breaks down as a result of electrical breakdown, when a nuclear bomb explodes and in a number of other interesting physical phenomena. (see [START_REF] Ball | Remarks on blow up and nonexistence theorems for nonlinear evolutions equations[END_REF], [START_REF] Braik | Zennir A finite-time blow-up result for a class of solutions with positive initial energy for coupled system of heat equations with memories[END_REF], [START_REF] Han | Global existence and blow-up of solutions for a system of nonlinear visco-elastic wave equations with damping and source[END_REF], [START_REF] Messaoudi | Blow up and global existence in nonlinear visco-elastic wave equations[END_REF], [START_REF] Messaoudi | Blow up of solutions with positive initial energy in a nonlinear visco-elastic wave equations[END_REF], [START_REF] Ngoc | Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions[END_REF], [START_REF] Ono | On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation[END_REF], [START_REF] Sun | Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level[END_REF], [START_REF] Yang | Blow-up of solutions for visco-elastic equations of Kirchhoff type with arbitrary positive initial energy[END_REF], [START_REF] Kh | Global nonexistence of Solutions for a system of nonlinear visco-elastic wave equations with degenerate damping and source terms[END_REF], . . . ) 1.1. Formulation of problem. Denote u = u(x, t), u = u t = ∂u ∂t (x, t), u = u tt = ∂ 2 u ∂t 2 (x, t), u x = ∂u ∂x (x, t), u xx = ∂ 2 u ∂x 2 (x, t). In this article, we consider the Love-equation in the form

u -λ 0 u x + λ 1 u x + u x x + λ t -∞ g(t -s)u xx (s)ds = F [u] -F [u]
x + f (x, t), x ∈ Ω = (0, 1), 0 < t < T, (1.1) where

F [u] = F x, t, u, u x , u , u x ∈ C 1 [0, 1] × R + × R 4 , (1.2)
and λ, λ 0 , λ 1 > 0, are constants. The given functions g, f are specified later. With F = F (x, t, y 1 , . . . , y 4 ), we put D 1 F = ∂F ∂x , D 2 F = ∂F ∂t , D i+2 F = ∂F ∂yi , with i = 10-2018 1, . . . , 4. Equation (1.1) satisfies the homogeneous Dirichlet boundary conditions: u(0, t) = u(1, t) = 0, t > 0, (1.3) and the following initial conditions u(x, -t) = u 0 (x, t), u (x, 0) = u 1 (x).

(1.4)

To deal with a wave equation with infinite history, we assume that the kernel function g satisfies the following hypothesis: (Hyp1:) g : R + → R + is a non-increasing C 1 function such that λ 0 -λ ∞ 0 g(s)ds = l > 0, g(0) > 0.

(1.5)

We need the following assumptions on source forces:

(Hyp2:) u 0 (0), u 1 ∈ H 1 0 ∩ H 2 ; (Hyp3:) f ∈ H 1 ((0, 1) × (0, T ));

(Hyp4:) F ∈ C 1 [0, 1]×[0, T ]×R 4 , such that F (0, t, 0, y 2 , 0, y 4 ) = F (1, t, 0, y 2 , 0, y 4 ) = 0 for all t ∈ [0, T ], y 2 , y 4 ∈ R.
1.2. Bibliographical notes. We start our literature review concerning visco-elastic problems with the pioneer work of Dafermos [START_REF] Dafermos | Asymptotic stability in visco-elasticity[END_REF], where the author considered a one-dimensional visco-elastic problem

ρu = cu xx - t -∞
g(t -s)u ss ds, and established various existence results and then proved, for smooth monotone decreasing relaxation functions, that the solutions go to zero as t goes to infinity.

In [START_REF] Hrusa | Global existence and asymptotic stability for a semi-linear Volterra equation with large initial data[END_REF], Hrussa considered a one-dimensional nonlinear visco-elastic equation

u = cu xx - t 0 m(t -s)(φ(u x )) x sds = f,
and proved several global existence results for large data. Here, the author also obtained a decay rate of solution. Messaoudi in [START_REF] Messaoudi | Blow up and global existence in nonlinear visco-elastic wave equations[END_REF] investigated the u=interaction between the nonlinear damping and nonlinear source in the equation

u = ∆u + t 0 g(t -s)∆uds + au |u | m = bu|u| γ ,
and showed, under suitable conditions on g, that solutions with negative energy blow up in finite time if γ > m, and continue to exist if m > γ.

Concerning problems with infinite history, we mention the work [START_REF] Giorgi | Global Attractors for a Semilinear Hyperbolic Equation in visco-elasticity[END_REF] in which considered the following semi-linear hyperbolic equation, in a bounded domain of R 3 ,

u -K(0)∆u - ∞ 0 K (s)∆u(t -s)ds + g(u) = f,
with K(0), K(∞) > 0, K ≤ 0 and gave the existence of global attractors for the problem. Next in [START_REF] Kh | Global nonexistence of Solutions for a system of nonlinear visco-elastic wave equations with degenerate damping and source terms[END_REF], proved that the solutions of a system of wave equations with visco-elastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally non-existing provided that the initial data are sufficiently large in a bounded domain; the initial energy is positive, and the strongly nonlinear functions f 1 and f 2 located in the sources satisfy an appropriate conditions. The author concentrate his studies on the role of the nonlinearities of sources. After that, in [START_REF] Liu | Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms[END_REF], the authors considered a fourth-order suspension bridge equation with nonlinear damping term and source term. The authors gave necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when p > m, they gave sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time is also established. Recently, in [START_REF] Sun | Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level[END_REF], the authors studied a three-dimensional (3D) visco-elastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history, t ≤ 0 in

u -k(0)∆u - ∞ 0 k (s)∆u(t -s)ds + |u | m-1 u = |u| p-1 u,
where the relaxation function k is monotone decreasing with k(+∞) = 1, m ≥ 1, 1 ≤ p < 6. When the source is stronger than dissipations, i.e. p > max{m, k(0)}, they obtained some finite time blow-up results with positive initial energy. In particular, they obtained the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level. In [START_REF] Hao | General Stability Result of Abstract Thermoelastic System with Infinite Memory[END_REF], the abstract thermo-elastic system is considered,

   u + Au + Bu - ∞ 0 g(s)u xx (t -s)ds -A α θ = 0 θ + kA β θθ + A α u = 0, u(-t) = u 0 (t), u (0) = u 1 , θ(0) = θ 0 , (1.6)
in which u is the displacement vector, θ is the temperature difference, and α ∈ [0, 1), β ∈ (0, 1] are constants. H is a real Hilbert space equipped with the inner product ., . and the related norm . . The operators A : D(A) → H and B : D(B) → H are self-adjoint linear positive definite operators. Under suitable conditions on the order of the coupling, the memory kernel function and the initial values, the well-posedness and the general decay rate of solution is given by semigroup theory and perturbed energy functional technique to allow a wider thermo-elastic systems. Without infinite memory term, when λ = 0 in (1.1), Triet and his collaborator in [START_REF] Triet | Existence, blow-up and exponential decay for Kirchhoff-love equations with Dirichlet conditions[END_REF] considered an IBVP for a nonlinear Kirchhoff-Love equation

u tt - ∂ ∂x B x, t, u, u 2 , u x 2 , u t 2 , u xt 2 u x + λ 1 u xt + u xtt + λu t = F x, t, u, u x , u t , u xt , u(t) 2 , u x (t) 2 , u t (t) 2 , u xt (t) 2 - ∂ ∂x G x, t, u, u x , u t , u xt , u(t) 2 , u x (t) 2 , u t (t) 2 , u xt (t) 2 + f (x, t), x ∈ Ω = (0, 1), 0 < t < T, (1.7)
u(0, t) = u(1, t) = 0, (1.8) u(x, 0) = ũ0 (x), u t (x, 0) = ũ1 (x), (1.9)
where λ > 0, λ 1 > 0 are constants and ũ0 , ũ1 ∈ H 1 0 ∩H 2 ; f , F and G are given functions. By applying the Faedo-Galerkin, the authors proved existence and uniqueness of a solution and by constructing Lyapunov functional, they proved a blow-up of the solution with a negative initial energy, and established a sufficient condition for the exponential decay of weak solutions. This last results extend their previous results in [START_REF] Triet | A Dirichlet problem for a nonlinear wave equation of Kirchhoff -Love type[END_REF] Remark 1.1. We should mention here that the presence of the term -F [u]

x can ensure only the local existence (can't be continue to be global in time), i. e., if we take -F [u]

x ≡ 0, under the same condition, (Hyp1)-(Hyp4), (3.4) and (3.10), we arrive to the blow up for all t > 0. That is the term F [u] is strong enough to prevent the existence of solutions.

This paper is organized as follows: In the second section, because of the nonlinearities, we combine a three techniques to prove the local existence of unique weak solution in Theorem 2.3. In the third section, the blow up results with negative initial energy is obtained in Theorem 3.3, under certain conditions on the sources and the function g. It is not surprising that this work is inspired from [START_REF] Ngoc | A linear recursive scheme associated with the Love's equation[END_REF], [START_REF] Ngoc | On a high-order iterative scheme for a nonlinear Love equation[END_REF], [START_REF] Triet | Existence, blow-up and exponential decay for Kirchhoff-love equations with Dirichlet conditions[END_REF], [START_REF] Kh | Global nonexistence of Solutions for a system of nonlinear visco-elastic wave equations with degenerate damping and source terms[END_REF].

Existence of a weak solution

The weak formulation. We define in the following, the weak solution to of (1.1)-

(1.4). Definition 2.1. A function u is said to be a weak solution of (1.1)-(1.4) on [0, T ] if u, u , u ∈ L ∞ (0, T ; H 1 0 ∩ H 2 )
, such that u satisfies the variational equation

1 0 u w dx + 1 0 (λ 0 u x + λ 1 u x + u x )w x dx -λ 1 0 ∞ 0 g(s)u x (t -s)dsw x dx = 1 0 f wdx + 1 0 F [u]wdx + 1 0 F [u]w x dx, (2.1)
for all test function w ∈ H 1 0 , for almost all t ∈ (0, T ). The following famous and widely used technical lemma will play an important role in the sequel.

Lemma 2.2. For any v ∈ C 1 0, T, H 1 0 we have 1 0 ∞ 0 g(s)v xx (t -s)v (t)dsdx = 1 2 d dt ∞ 0 g(s) 1 0 |v(t -s) -v(t)| 2 dxds - 1 2 d dt ∞ 0 g(s)ds 1 0 |v x (t)| 2 dx - 1 2 ∞ 0 g (s) 1 0 |v(t -s) -v(t)| 2 dxds.
Proof. It's not hard to see

1 0 ∞ 0 g(s)v xx (t -s)v (t)dsdx = - ∞ 0 g(s) 1 0 v x (t)v x (t -s)dxds = - ∞ 0 g(s) 1 0 v x (t) [v x (t -s) -v x (t)] dxds - ∞ 0 g(s) 1 0 v x (t)v x (t)dxds.
Consequently,

1 0 ∞ 0 g(s)v xx (s)v (t)dsdx = 1 2 ∞ 0 g(s) d dt 1 0 |v x (t -s) -v x (t)| 2 dxds - ∞ 0 g(s) d dt 1 2 1 0 |v x (t)| 2 dx ds, which implies, 1 0 ∞ 0 g(s)v xx (s)v (t)dsdx = 1 2 d dt ∞ 0 g(s) 1 0 |v x (t -s) -v x (t)| 2 dxds - 1 2 d dt ∞ 0 g(s)ds 1 0 |v x (t)| 2 dx - 1 2 ∞ 0 g (s) 1 0 |v x (t -s) -v x (t)| 2 dxds,
First main theorem. Various existence and uniqueness, as well as Faedo-Galerkin method, have been obtained in the lase decades for nonlinear IBVPs in Sobolev spaces (see [START_REF] Cavalcanti | Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction[END_REF], [START_REF] Han | Global existence and blow-up of solutions for a system of nonlinear visco-elastic wave equations with damping and source[END_REF], [START_REF] Mustafa | Well posedness and asymptotic behavior of a coupled system of nolinear viscoelastic equations[END_REF], [START_REF] Lasiecka | Uniform boundary stabilization of semi-linear wave equations with nonlinear boundary damping[END_REF], . . . ). Now, we consider the existence of a local solution for (1.1)-(1.4), with λ ∈ R, λ 0 , λ 1 > 0.

Theorem 2.3. Let u 0 (0), u 1 ∈ H 1 0 ∩ H 2 be given. Assume that (Hyp1)-(Hyp4) hold. Then Problem (1.1)-(1.4) has a unique local solution u and u, u , u ∈ L ∞ (0, T * ; H 1 0 ∩ H 2 ), (2.2)
for some T * > 0 small enough.

Proof of Theorem 2.3. In the first step of this proof, using linearization method for a nonlinear term to construct a linear recurrent sequence {u m }. Then, the Faedo-Galerkin method combined with the weak compactness method shows that {u m } converges to u which is exactly a unique local solution of (1.1)-(1.4).

Step 1. Let T > 0 be fixed, let M > 0, we put

K M (f ) = f 2 L 2 ((0,1)×(0,T )) + f 2 L 2 ((0,1)×(0,T )) + f x 2 L 2 ((0,1)×(0,T )) , (2.3)
F C 0 ([0,1]×[0,T ]×[-M,M ] 4 ) = sup (x,t,y1,...,y4)∈[0,1]×[0,T ]×[-M,M ] 4 |F (x, t, y 1 , . . . , y 4 )|, FM = F C 1 ([0,1]×[0,T ]×[-M,M ] 4 ) = F C 0 ([0,1]×[0,T ]×[-M,M ] 4 ) + 6 i=1 D i F C 0 ([0,1]×[0,T ]×[-M,M ] 4 ) .
For some T * ∈ (0, T ] and M > 0, we put

W (M, T * ) = v, v ∈ L ∞ (0, T * ; H 1 0 ∩ H 2 ) : v ∈ L ∞ (0, T * ; H 1 0 ), with v L ∞ (0,T * ;H 1 0 ∩H 2 ) , v L ∞ (0,T * ;H 1 0 ∩H 2 ) , v L ∞ (0,T * ;H 1 0 ) ≤ M , W 1 (M, T * ) = {v ∈ W (M, T * ) : v ∈ L ∞ (0, T * ; H 1 0 ∩ H 2 )}.
(2.4)

We can now establish the linear recurrent sequence {u m } and choosing u 0 (t) ≡ 0, suppose that

u m-1 ∈ W 1 (M, T * ), (2.5)
and associate with problem (1.1)-(1.4) the following problem.

Using Lemma 2.2, find u m ∈ W 1 (M, T * ) (m ≥ 1) which satisfies 1 0 u m w dx + 1 0 λ 0 u xm + λ 1 u xm + u xm w x dx -λ ∞ 0 g(s)ds 1 0 u xm w x dx -λ ∞ 0 g(s) 1 0 (u xm (t -s) -u xm )w x dxds = 1 0 f wdx + 1 0 F m [u]wdx + 1 0 F m [u]w x dx, ∀w ∈ H 1 0 , u m (-t) = u 0 (t), u m (0) = u 1 , t ∈ [0, T ], (2.6)
where

F m [u] = F [u m-1 ] = F x, t, u m-1 , u xm-1 , u m-1 , u xm-1 , (2.7)
Proposition 2.4. Let u 0 (0), u 1 ∈ H 1 0 ∩ H 2 be given, the first term of sequence u 0 (t) ≡ 0. Assume that (Hyp1)-(Hyp4) hold. Then there exist positive constants M , T * > 0 such that, there exists a recurrent sequence {u m } ⊂ W 1 (M, T * ) defined by (2.5)-(2.7).

Proof of Proposition 2.4. We use the standard Faedo-Galerkin method to prove our result. Consider a special orthonormal basis {w j } ∞ j=1 on H 1 0 , formed by the eigenfunctions of the operator -∂ 2 ∂x 2 . Let V k = span{w 1 , w 2 , . . . , w k } and the projections of the history and initial data on the finite-dimensional subspace V k are given by

u 0k (t) = k j=1 α (k) j (t)w j , u 1k = k j=1 β (k) j w j , (2.8)
where

α (k) j (t) = 1 0 u 0 (t)w j dx, β (k) j (t) = 1 0 u 1 w j dx. We seek k functions ϕ (k) mj (t) ∈ C 2 [0, T ], 1 ≤ j ≤ k, such that the expression in form u (k) m = k j=1 ϕ (k) mj w j , (2.9)
solves the problem

1 0 u (k) m w j dx + 1 0 u (k) xm + λ 1 u (k) xm + u (k) xm w jx dx -λ ∞ 0 g(s)ds 1 0 u (k) xm w jx dx -λ ∞ 0 g(s) 1 0 (u (k) xm (t -s) -u (k) xm (s))w jx dxds (2.10) = 1 0 f w j dx + 1 0 F m w j dx + 1 0 F m w jx dx, 1 ≤ j ≤ k, u (k) m (0)(-t) = u 0k (t), u (k) m (0) = u 1k , in which u 0k → u 0 (t) strongly in H 1 0 ∩ H 2 , u 1k =→ u 1 strongly in H 1 0 ∩ H 2 .
(2.11)

This leads to a system of ODEs for unknown functions ϕ

u (k) m u (k) m dx + 1 0 λ 0 u (k) xm + λ 1 u (k) xm + u (k) xm u (k) xm dx +λ 1 0 ∞ 0 g(s)u (k) xm (t -s)dsu (k) xm dx = 1 0 F [u (k) m ]u (k) m dx + 1 0 F [u (k) m ]u (k) xm dx + 1 0 f u (k) m dx. 10-2018
Using results in Lemma 2.2, we obtain

1 2 d dt 1 0 |u (k) m | 2 + λ 0 -λ ∞ 0 g(s)ds |u (k) xm | 2 + |u (k) xm | 2 dx dx +λ ∞ 0 g(s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxds +λ 1 1 0 |u (k) xm | 2 dx -λ ∞ 0 g (s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxds (2.12) = 1 0 F [u (k) m ]u (k) m dx + 1 0 F [u (k) m ]u (k) xm dx + 1 0 f m u (k) m dx.
Let us denote the LHS of (2.12) as

e (k) (u m ) = 1 0 |u (k) m | 2 + λ 0 -λ ∞ 0 g(s)ds |u (k) xm | 2 + |u (k) xm | 2 dx dx + λ ∞ 0 g(s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxds + 2λ 1 t 0 1 0 |u (k) xm | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxdsdτ, and
e (k) (u xm ) = 1 0 |u (k) xm | 2 + λ 0 -λ ∞ 0 g(s)ds |u (k) xxm | 2 + |u (k) xxm | 2 dx dx + λ ∞ 0 g(s) 1 0 |u (k) xxm (t -s) -u (k) xxm (t)| 2 dxds + 2λ 1 t 0 1 0 |u (k) xxm | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |u (k) xxm (t -s) -u (k) xxm (t)| 2 dxdsτ, and
e (k) (u m ) = 1 0 |u (k) m | 2 + λ 0 -λ ∞ 0 g(s)ds |u (k) xm | 2 + |u (k) xm | 2 dx dx + λ ∞ 0 g(s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxds + 2λ 1 t 0 1 0 |u (k) xm | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |u (k) xm (t -s) -u (k) xm (t)| 2 dxdsdτ.
Put

E (k) m (t) = e (k) (u m ) + e (k) (u xm ) + e (k) (u m ). (2.13)
Then

E (k) m (t) = E (k) m (0) + 2 t 0 1 0 f (s)u (k) m (s)dxds + 2 t 0 1 0 f x (s)u (k) xm (s)dxds + 2 t 0 1 0 f (s)u (k) m (s)dx ds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xxm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dxds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xm (s)dxds.
(2.14)

We need, now, to estimate

A (k) m = 1 0 |u (k) m (0)| 2 dx + 1 0 |u (k) xm (0)| 2 dx. Let w j = u (k) m
in (2.10) and integrate by parts, taking t → 0 + in the first term, to obtain

1 0 |u (k) m (0)| 2 dx + 1 0 |u (k) xm (0)| 2 dx + 1 0 λ 0 -λ ∞ 0 g(s)ds u 0kx + λ 1 u 1kx u (k) xm (0)dx +λ ∞ 0 g(s) 1 0 (u 0kx (0) -u 0kx (-s))u (k) m (0)dxds = 1 0 f (0)u (k) m (0)dx + 1 0 F m (0)u (k) m (0)dx + 1 0 F m (0)u (k) xm (0)dx.
Then

A (k) m ≤ 1 0 λ 0 -λ ∞ 0 g(s)ds u 0kx + λ 1 u 1kx + F m (0) u (k) xm (0)dx + λ ∞ 0 g(s) 1 0 (u 0kx (0) -u 0kx (-s))u (k) m (0)dxds + 1 0 f (0)u (k) m (0)dx + 1 0 F m (0)u (k) m (0)dx, ≤ 1 0 λ 0 -λ ∞ 0 g(s)ds u 0kx + λ 1 u 1kx + 2F m (0) + f (0) + λ ∞ 0 g(s)(u 0kx (0) -u 0kx (-s)) dx A (k) m 1/2 ≤ 1 0 λ 0 -λ ∞ 0 g(s)ds u 0kx + λ 1 u 1kx + 2F m (0) + f (0) 10-2018 + λ ∞ 0 g(s)(u 0kx (0) -u 0kx (-s)) dx 2 ≤ ξ, for all m, k, (2.15)
because, 1 0 |F m (0)|dx is a constant independent of m, where ξ is a constant depending only on f , u 0 , u 1 , F , λ 0 , λ, λ 1 and ∞ 0 g(s)ds. Equations (2.11), (2.13) and (2.15) imply that

E (k) m (0) = 1 0 |u 1k | 2 + λ 0 -λ ∞ 0 g(s)ds |u 0kx | 2 + |u 1kx | 2 dx dx + λ ∞ 0 g(s) 1 0 |u 0kx (-s) -u 0kx (0)| 2 dxds + 1 0 |u 1kx | 2 + λ 0 -λ ∞ 0 g(s)ds |u 0kxx | 2 + |u 1kxx | 2 dx dx + λ ∞ 0 g(s) 1 0 |u 0kxx (-s) -u 0kxx (0)| 2 dxds + A (k) m + 1 0 λ 0 -λ ∞ 0 g(s)ds |u 1kx | 2 dx + λ ∞ 0 g(s) 1 0 |u 1kx -u 1kx | 2 dxds, ≤ ξ 0 , for all m, k ∈ N, (2.16)
where ξ 0 is also a constant depending only on f , u 0 , u 1 , F , λ 0 , λ, λ 1 and ∞ 0 g(s)ds. We then now estimate the other terms of (2.14). By the Cauchy -Schwartz inequality, we obtain

E (k) m (t) ≤ ξ 0 + f 2 L 2 ((0,1)×(0,T )) + t 0 1 0 |u (k) m | 2 dsdx; + f x 2 L 2 ((0,1)×(0,T )) + t 0 1 0 |u (k) xm | 2 dxds; + f 2 L 2 ((0,1)×(0,T )) + t 0 1 0 |u (k) m | 2 dxds; + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xxm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dxds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xm (s)dxds.
(2.17) By (Hyp1), (2.13), we have

E (k) m (t) ≤ ξ 0 + f 2 H 1 ((0,1)×(0,T )) + c t 0 E (k) m (s)ds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xxm (s)dx ds + 2 t 0 1 0 F m (s)u (k) m (s)dxds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds + 2 t 0 1 0 F xm (s)u (k) xm (s)dxds.
(2.18)

We have

E (k) m (t) ≤ ξ 0 + f 2 H 1 ((0,1)×(0,T )) + c t 0 E (k) m (s)ds + T * F 2 M + t 0 1 0 |u (k) xm (s)| 2 dxds + T * F 2 M + t 0 1 0 |u (k) m | 2 dxds + 2 t 0 1 0 F xm (s)u (k) xxm (s)dx ds + T * F 2 M + t 0 1 0 |u (k) xm (s)| 2 dxds. + 2 t 0 1 0 F m (s)u (k) m (s)dxds + 2 t 0 1 0 F m (s)u (k) xm (s)dx ds (2.19)
By (Hyp1), (2.13) and remarking from (1.2) that

F xm (t) = D 1 F [u m-1 ] + D 3 F [u m-1 ]u xm-1 + D 4 F [u m-1 ]u xxm-1 +D 5 F [u m-1 ]u xm-1 + D 6 F [u m-1 ]u xxm-1 . Then E (k) m (t) ≤ ξ 0 + f 2 H 1 ((0,1)×(0,T )) + 2T * 1 + 2(1 + 4M ) 2 F 2 M + c t 0 E (k) m (s)ds.
(2.20)

We choose M > 0 sufficiently large such that

E 0 + f 2 H 1 ((0,1)×(0,T )) ≤ 1 2 M 2 , (2.21)
and then choose T * ∈ (0, T ] small enough such that

1 2 M 2 + T * 1 + 2(1 + 4M ) 2 F 2 M exp[2T * ] ≤ M 2 , (2.22)
and

k T * = 2 F 2 M T * exp[T * ] < 1, (2.23)
Then

u (k) m ∈ W (M, T * ),
|w m | 2 + λ 0 -λ ∞ 0 g(s)ds |w xm | 2 + |w xm | 2 dx +λ ∞ 0 g(s) 1 0 |w xm (t -s) -w xm (t)| 2 dxds +2λ 1 t 0 1 0 |w xm | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |w xm (t -s) -w xm (t)| 2 dxdsdτ = 2 t 0 1 0 F m+1 (s) -F m (s) w m (s)dxds (2.29) +2 t 0 1 0 F m+1 (s) -F m (s) w xm (s)dxds.
By (Hyp2)-( Hyp4), (2.3), (2.5) and (2.25) we have

1 0 |F m+1 (s) -F m (s)| 2 dx ≤ 2 FM 1 0 |w m-1 | 2 dx, then E m (t) ≤ 2 FM 1 0 |w m-1 | 2 dx + t 0 1 0 |w m | 2 dxds + 2 FM 1 0 |w m-1 | 2 dx + t 0 1 0 |w xm | 2 dxds ≤ 4 FM 1 0 |w m-1 | 2 dxT * + t 0 E m (s)ds, (2.30)
where

E m (t) = 1 0 |w m | 2 + λ 0 -λ ∞ 0 g(s)ds |w xm | 2 + |w xm | 2 dx +λ ∞ 0 g(s) 1 0 |w xm (t -s) -w xm (t)| 2 dxds +2λ 1 t 0 1 0 |w xm | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |w xm (t -s) -w xm (t)| 2 dxdsdτ.
Thanks to Gronwall's Lemma, (2.30), we get

1 0 |w m |dx ≤ k T * 1 0 |w m-1 |dx ∀m ∈ N, (2.31)
so 1 0 |u m -u m+p |dx ≤ M (1 -k T * ) -1 k m T * , ∀m, p ∈ N. (2.32)
It follows that {u m } is a Cauchy sequence in W 1 (T * ), so there exists u ∈ W 1 (T * ) such that

u m → u strongly in W 1 (T * ). ( 2

.33) 10-2018

Note that u m ∈ W 1 (M, T * ), so there exists a subsequence {u mj } of {u m } such that

u mj → u in L ∞ (0, T * ; H 1 0 ∩ H 2 ) weakly*, u mj → u in L ∞ (0, T * ; H 1 0 ∩ H 2 ) weakly*, u mj → u in L ∞ (0, T * ; H 1 0 ) weakly*, u ∈ W (M, T * ).
(2.34) By (2.3), (2.5), (2.7) and (2.34) 4 , we obtain

1 0 |F m (t) -F [u](t)|dx ≤ 2(1 + 2M ) FM 1 0 |u m-1 -u|dx, (2.35)
Then (2.33) and (2.35) imply

F m → F [u] strongly in L ∞ (0, T * ; L 2 ), (2.36)
Let us passing to limit in (2.6), (2.7) as m = m j → ∞, by (2.33), (2.34) and (2.36), there exists u ∈ W (M, T * ) satisfying

1 0 u w dx + 1 0 (λ 0 u x + λ 1 u x + u x )w x dx -λ 1 0 ∞ 0 g(s)u x (t -s)dsw x dx = 1 0 f wdx + 1 0 F [u]wdx + 1 0 F [u]w x dx, (2.37)
for all test function w ∈ H 1 0 , for almost all t ∈ (0, T ) and satisfying the initial conditions. Uniqueness. Let u 1 , u 2 be two weak solutions of (1.1)-(1.4), such that

u 1 , u 2 ∈ W 1 (M, T * ).
(2.38)

Then v = u 1 -u 2 satisfies 1 0 v w dx + 1 0 (λ 0 v x + λ 1 v x + v x )w x dx -λ 1 0 ∞ 0 g(s)v x (t -s)dsw x dx = 1 0 F [u 1 ] -F [u 2 ] wdx + 1 0 F [u 1 ] -F [u 2 ] w x dx, (2.39)
for all test function w ∈ H 1 0 , for almost all t ∈ [0, T ]. Taking v = w in (2.39) 1 and integrating with respect to t, for

e(t) = 1 0 |w | 2 + λ 0 -λ ∞ 0 g(s)ds |w x | 2 + |w x | 2 dx +λ ∞ 0 g(s) 1 0 |w x (t -s) -w x (t)| 2 dxds +2λ 1 t 0 1 0 |w x | 2 dxds -λ t 0 ∞ 0 g (s) 1 0 |w x (t -s) -w x (t)| 2 dxdsdτ,
we obtain

e(t) = 1 0 F [u 1 ] -F [u 2 ] v dx + 1 0 F [u 1 ] -F [u 2 ] v x dx.
(2.40)

On the other hand, by (Hyp2)-(Hyp4), we deduce from (2.3), that

1 0 |F [u -1] -F [u 2 ]|dx ≤ 2c(1 + 2M ) FM e 1/2 (s). (2.41)
Then

E(t) ≤ [4c(1 + 2M ) FM t 0 e(s)ds.
Thanks again to Gronwall's Lemma, we have E ≡ 0, i.e., u 1 ≡ u 2 .

Theorem 2.3 is completely proved.

Blow up

We further prove that if (3.10) hold, then the blow up of any weak solution (3.1) for a finite time occurs when the initial energy is negative. Here, we consider (1.1)- (1.4)

with f = 0, f 1 = F (t, x, u, u x ) ∈ C 1 (R 2 ; R), f 2 = F (t, x, u, u x ) ∈ C 1 (R 2 ; R) as follows      u -λ 0 u x + λ 1 u x + u x x + λ t -∞ g(t -s)u xx (s)ds = f 1 (u, u x ) -f 2 (u, u x ) x , x ∈ (0, 1), 0 < t < T * , (3.1)
with the boundary conditions

u(0, t) = u(1, t) = 0, t > 0, (3.2)
and the following initial conditions:

u(x, -t) = u 0 (x, t), u (x, 0) = u 1 (x), (3.3)
We have proved in the previous section, the existence of local weak solution of (1.1)-(1.4) in the Theorem 2.3. Furthermore, let us assume that there exist F ∈ C 2 (R 2 ; R) and the constants p, q > 2;

d 1 , d 2 > 0, such that ∂F ∂u (u, v) = f 1 (u, v), ∂F ∂v (u, v) = f 2 (u, v), uf 1 (u, v) + vf 2 (u, v) ≥ d 1 F(u, v), ∀(u, v) ∈ R 2 , F(u, v) ≥ d 2 (|u| p + |v| q ), ∀(u, v) ∈ R 2 ; (3.4)
We introduce the energy functional E(t) associated with system (3.1)-(3.3)

E(t) = 1 2 1 0 |u | 2 dx + 1 2 1 0 λ 0 -λ ∞ 0 g(s)ds |u x | 2 dx + 1 2 1 0 |u x | 2 dx + 1 2 λ 1 0 ∞ g(s)|u x (t) -u x (t -s)| 2 dsdx - 1 0 F(u, u x )dx. (3.5)
It is note hard to see this Lemma (Using Lemma 2.2).

Lemma 3.1. Suppose that (Hyp1) holds. Let u be solution of system (3.1)- (3.3).

Then the energy functional (3.5) is a non-increasing function, i.e., for all t ≥ 0,

d dt E(t) = -λ 1 1 0 |u x | 2 dx + 1 2 λ 1 0 ∞ 0 g (s)|u x (t) -u x (t -s)| 2 dsdx. (3.6)
For reader, we state this Lemma with its proof.

Lemma 3.2. Let ν > 0 be a real positive number and let L(t) be a solution of the ordinary differential inequality

dL(t) dt ≥ ξL 1+ν (t), (3.7)
defined in [0, ∞). If L(0) > 0, then the solution does not exist for t ≥ L(0) -ν ξ -ν ν -1 .

Proof. The direct integration of (3.7) gives

L -ν (0) -L -ν (t) ≥ ξνt
Thus, we get the following estimate:

L ν (t) ≥ L -ν (0) -ξνt -1 . (3.8)
It is clear that the right-hand side of (3.8) is unbounded for

ξνt = L -ν (0).
Lemma 3.2 is proved.

Our goal is to prove that when the initial energy is negative, the solution of system (3.1) blows up in finite time under the (3.4), (3.10). The regularity obtained by (2.2) implies that problem (1.1)-(1.4) admits a unique strong solution

u ∈ C 1 ([0, T * ]; H 1 0 ∩ H 2 ), u ∈ L ∞ (0, T * ; H 1 0 ∩ H 2 ). (3.9)
Second main theorem. Our result here reads as follows.

Theorem 3.3. Assume that (3.4) hold. Assume further that E(0) < 0 for any u 0 (0), u 1 ∈ H 1 0 ∩ H 2 hold. There exist a number r, 2 < r < min{p, q}, r < d

1 d 2 such that ∞ 0 g(s)ds < λ 0 (1 + r/2) λ 1/2r + 1 + r/2 . ( 3

.10)

Then, the unique weak solution u of (3.1)-(3.3) blows up in finite time.

Proof. Let

H(t) = -E(t), ∀t ∈ [0, T * ). ( 3

.11)

By multiplying the equation in (3.1) by -u , integrating over [0, 1] and using Lemma 2.2, we obtain

- d dt H(t) = λ 1 1 0 |u x | 2 dx - 1 2 λ 1 0 ∞ 0 g (s)|u x (t) -u x (t -s)| 2 dsdx ≥ 0, ∀t ∈ [0, T * ). (3.12)
Consequently, E(0) < 0, we have

H(0) = - 1 2 1 0 |u 1 | 2 dx - 1 2 1 0 λ 0 -λ ∞ 0 g(s)ds |u 0x (0)| 2 dx - 1 2 1 0 |u 1x | 2 dx -λ 1 0 ∞ 0 g(s)|u 0x (0) -u 0x (-s)| 2 dsdx + 1 0 F(u 0 (x, 0), u 0x (x, 0))dx, (3.13) inequality (3.12) implies 0 < H(0) ≤ H(t), ∀t ∈ [0, T * ). ( 3.14)
Using (Hyp1), to get

H(t) - 1 0 F(u, u x )dx = - 1 2 1 0 |u | 2 dx - 1 2 1 0 λ 0 -λ ∞ 0 g(s)ds |u x | 2 dx -λ 1 0 ∞ 0 g(s)|u x (t) -u x (t -s)| 2 dsdx - 1 2 1 0 |u x | 2 dx ≤ 0, ∀t ∈ [0, T * ). (3.15) One implies 0 < H(0) ≤ H(t) ≤ 1 0 F(u, u x )dx. (3.16)
Then, we define a functionals and introduce

M (t) = 1 2 1 0 u 2 dx, N (t) = λ 1 1 2 1 0 |u x | 2 dx + 1 0 u x u x dx, (3.17
L(t) = H 1-σ (t) + εM (t) + εN (t), (3.18)
for ε small enough and

0 < σ < 1, 2/(1 -2σ) ≤ min{p, q}. (3.19)
We now show that L(t) satisfies the differential inequality in Lemma 3.2. By taking the derivative of (3.18) and using (3.1) and Lemma 2.2, we obtain

L (t) = (1 -σ)H -σ (t)H (t) + ε 1 0 |u | 2 dx + ε 1 0 |u x | 2 dx -ε λ 0 -λ ∞ 0 g(s)ds 1 0 |u x | 2 dx + ελ ∞ 0 g(s) 1 0 [u x (t -s) -u x (s)]u x dxds + ε 1 0 f 1 (u, u x )udx + ε 1 0 f 2 (u, u x )u x dx. (3.20)
By the Cauchy-Schwarz and Young inequalities, we find

∞ 0 g(s) 1 0 [u x (t -s) -u x (s)]u x dxds ≤ ∞ 0 g(s) 1 0 |u x (t -s) -u x (s)| 2 dx 1 0 |u x | 2 dxds ≤ γ ∞ 0 g(s) 1 0 |u x (t -s) -u x (s)| 2 dxds + ∞ 0 g(s)ds 4γ 1 0 |u x | 2 dx, γ > 0.
Therefore,

L (t) ≥ (1 -σ)H -σ (t)H (t) + ε 1 0 |u | 2 dx + ε 1 0 |u x | 2 dx -ε(λ 0 -λ ∞ 0 g(s)ds -λ ∞ 0 g(s)ds 4γ ) 1 0 |u x | 2 dx + ελγ ∞ 0 g(s) 1 0 |u x (t -s) -u x (s)| 2 dxds + ε 1 0 f 1 (u, u x )udx + ε 1 0 f 2 (u, u x )u x dx. (3.21)
By (3.4), we obtain

1 0 f 1 (u, u x )udx + 1 0 f 2 (u, u x )u x dx ≥ d 1 1 0 F(u, u x )dx, 1 0 F(u, u x )dx ≥ d 2 1 0 |u x (t)| q dx + 1 0 |u(t)| p dx . (3.22)
Then, it follows since (3.19) and due to the inequality At this point,we can find positive constants ν > 0 such that L (t) ≥ εν H(t) + Similarly, we obtain Moreover, we note that L 1/(1-σ) (t) ≤ c H 1-σ (t) + This yields L (t) ≥ a 0 L 1/(1-σ)(t) , ∀t ≥ 0.

(3.30)

Finally, as a result of simple integration of (3.30), completes the proof of Theorem 3.3.

  mi . Based on standard existence theory for ODE, System (2.10) admits a unique solution ϕ(k) mj , 1 ≤ j ≤ k on interval [0, T ], by(2.5) and the argument in[START_REF] Coddington | Theory of ordinary differential equations[END_REF], the proof is completed. A priori estimates. The next estimates prove that there exist positive constants M , T * > 0 such that u (k) m ∈ W (M, T * ), for all m and k. We partially estimate the terms of the associated energy. Multiplying the equation in (1.1) by u

  )

( 1 - 1 0|u | 2 dx + ε 1 0 1 0+ εd 1 d 2 1 0 1 0 1 0a 3 =) 1 0) 1 0 1 0 0 g(s) 1 0 1 0 1 0

 111111131110111 σ)H -σ (t)H (t) > 0, ∀t ∈ [0, T * ), thatL (t) ≥ ε |u x | 2 dx -ε λ 0 -λ |u x (t -s) -u x (s)| 2 dxds |u x (t)| q dx + |u(t)| p dx . (3.23)Adding and substituting rE(t), 2 < r < min{p, q}; by using the definition of H(t),|u x (t -s) -u x (s)| 2 dxds + ε(d 1 d 2 -r) d 1 d 2 -r > 0. Then, estimate (3.24) becomes L (t) ≥ ε(1 + r 2 |u | 2 dx + ε(1 + r 2 |u x | 2 dx + εpH(t) + ελa 2 |u x | 2 dx + ελa 1 ∞ |u x (t) -u x (t -s)|2 dxds + εa 3 |u x (t)| q dx + |u(t)| p dx . (3.25)

1 0 1 0|u x | 2 dx + 1 0 1 s = 1 . 1 1 0|u| p dx + 1 0|u | 2

 111111112 |u x (t) -u x (t -s)| 2 dxds + |u x | 2 dx .(3.26) Thus, we can choose ε > 0 small enough such thatL(t) ≥ L(0) > 0, ∀t ∈ [0, T * ). (3.27) Further, by (3.19), Holder and Young inequalities, we obtain We take s = 2(1 -σ) to get τ |u| p dx + H(t) , ∀t ∈ [0, T * ). dx + H(t) .

1 0u x u x dx 1 / 1 0 1 0

 1111 (1-σ) ≤ c |u x | q dx + |u x | 2 dx + H(t) .

|u| p dx + 1 0|u | 2 dx + 1 0|u x | 2 dx + 1 0|u x | q dx + 1 0

 1111 |u x | 2 dx .

  Consider (2.28) with w = w m , and then integrating in t, we obtain by results in Lemma 2.2

	1
	0

for all m and k.

(2.24)

Pass to the limit. By (2.22), there exists a subsequence of {u

(2. [START_REF] Ono | On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation[END_REF] Passing to limit in (2.10), (2.11), it is clear to see that u m is satisfying (2.6), (2.7) in L 2 (0, T * ). Furthermore, (2.6) 1 and (2.25) 4 imply that

This completes the proof of Proposition 2.4.

Step 2. Let the Banach space

with respect to the norm

(2.27)

We will show the convergence of {u m } to the solution of our problem in the next Lemma.

Lemma 2.5. Let (Hyp1)-(Hyp4) hold. Then Proof. We use the result obtained in Proposition 2.4 and the compact embedding theorems.

Existence. We proved that {u m } is a Cauchy sequence in W 1 (T * ). In order to do this, let w m = u m+1 -u m . Then w m satisfies E-mail address: khaledzennir4@yahoo.vom


Pierre-David L Étourneau

Guillaume Bal

RESOLUTION AND DE-ALIASING IN ULTRASOUND ELASTOGRAPHY

Keywords: Imaging, elastography, inverse problems

We present an imaging technique for the recovery of the displacement field of an ensemble of random point scatterers, as is commonly needed in the medical imaging method called elastography. We show that by using a small number of well-localized sources and a small acquisition aperture, it is possible to recover the displacement through the construction of an imaging functional based on the ratio of the correlations of the signals received. We perform a resolution analysis of the method and show that displacements are well-reconstructed only when their spatial variations are sufficiently small.

Introduction

Elastography is a medical imaging modality aiming to reconstruct the elastic properties of tissues; see for instance [START_REF] Ammari | A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements[END_REF][START_REF] Gennisson | Transient elastography in anisotropic medium: Application to the measurement of slow and fast shear wave speeds in muscles[END_REF][START_REF] Jiang | Approximate steady state models for magnetic resonance elastography[END_REF][START_REF] Kruse | Tissue characterization using magnetic resonance elastography: preliminary results[END_REF][START_REF] Mclaughlin | Calculating tissue shear modulus and pressure by 2D log-elastographic methods[END_REF][START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF][START_REF] Ophir | Elastography: A quantitative method for imaging the elasticity of biological tissues[END_REF][START_REF] Ophir | Elastography: Imaging the elastic properties of soft tissues with ultrasound[END_REF][START_REF] Sandrin | Shear Modulus Imaging with 2-D Transient Elastography[END_REF]. In ultrasound elastography, this is done by probing elastic displacements by sound waves. As elastic (shear) waves propagate slowly (on the order of meters per second) through tissues, they generate displacements of randomly spaced scatterers. The displacement of these scatterers is then imaged using ultrasound echoes which travel at a speed much higher than the shear waves. This paper concerns the modeling of ultrasonic wave propagation through the random environment and the reconstruction of the displacements induced by shear waves from available measurements.

The displacement of a scatterer hit by a plane wave results in a phase shift in the scattered signal that is proportional to that displacement. This is the basic ingredient for all imaging algorithms in ultrasound elastography and more generally ultrafast ultrasound imaging, see [START_REF] Kasai | Real-time twodimensional blood flow imaging using an autocorrelation technique[END_REF][START_REF] Bonnefous | Time domain formulation of pulse-doppler ultrasound and blood velocity estimation by cross correlation[END_REF][START_REF] Gianmarco F Pinton | Rapid tracking of small displacements with ultrasound[END_REF][START_REF] Provost | 3D ultrafast ultrasound imaging in vivo[END_REF] and their references. A careful analysis of ultrasonic wave propagation in the random environment and a modeling of the influence of the elastic displacements on the measured ultrasonic waves was performed in [START_REF] Bal | Local asymptotic analysis of transient elastography reconstructions[END_REF]. In that paper, it is shown that three-dimensional displacements may be reconstructed when the displacements are sufficiently small and with sufficiently slow spatial variations. In the present paper, we revisit this problem. In a slightly different geometry for the acquisition of data, we show that the smallness constraint on the displacements may be removed by an appropriate treatment of aliasing effects. We also provide theoretical and numerical evidence that the reconstruction of spatially highly-fluctuating displacements is not feasible with the available data.

The rest of the paper is structured as follows. Section 2 introduces our modeling of ultrasound propagation in random media. Our reconstructions are based on the ratio of measurements (see (2.1) below) at successive frames of displacement. It is such ratios that provide information about the displacements that is asymptotically independent of the unknown properties of the random medium, at least provided that the local variations of the displacements be not too large. Section 2 also presents our main assumptions on the probing source. In order to perform explicit calculations and for concreteness, we assume that the source is a localized in phase space; see (2.2) below. Similar conclusions on the reconstruction of displacements may be drawn for other source configurations (see [START_REF] Bal | Local asymptotic analysis of transient elastography reconstructions[END_REF] for a spatially but not angularly localized source term).

The imaging functional and the main theoretical results of the paper are given in Section 3. As in [START_REF] Bal | Local asymptotic analysis of transient elastography reconstructions[END_REF], we observe that an appropriate ratio of available measurements provides information about the spatial tissue displacements in the form of a phase shift; see (3.3). This is a consequence of Fourier analysis, and has been used experimentally in the past (e.g. [START_REF] Gianmarco F Pinton | Rapid tracking of small displacements with ultrasound[END_REF], displacement imaging; [START_REF] Kasai | Real-time twodimensional blood flow imaging using an autocorrelation technique[END_REF][START_REF] Bonnefous | Time domain formulation of pulse-doppler ultrasound and blood velocity estimation by cross correlation[END_REF], blood flow tracking) using a source and receiver in the same location. Physical and measurement sources of error in those specific setups were also studied in [START_REF] Gill | Measurement of blood flow by d: accuracy and sources of error[END_REF][START_REF] Bilgen | Error analysis in acoustic elastography. i. displacement estimation[END_REF].

The reconstruction of a displacement from one such ratio requires that the local variations of the displacement be sufficiently small to avoid aliasing effects (a phenomenon known as "decorrelation" in correlation-based elastography, [START_REF] Gianmarco F Pinton | Rapid tracking of small displacements with ultrasound[END_REF][START_REF] Bilgen | Error analysis in acoustic elastography. i. displacement estimation[END_REF]). Using a collection of such ratios at different frequencies to obtain better localization, we propose a method that overcomes the aliasing effect and allows us to reconstruct arbitrarily large displacements. These main results are stated in Theorem 1. A salient feature of the ratios (3.3) are that they are asymptotically statistically stable. This means that as the number of random scatterers increases to infinity, the ratios become independent of the (unknown) realization of the random medium. In practice, measurements still involve a certain amount of randomness. A conservative estimate of the size of this random contribution is provided in Theorem 1.

The results of Theorem 1 hold only when the spatial displacements have sufficiently small spatial variations. More precisely, we assume that on the support of the probing wave-packet, the elastic displacement is relatively constant. This assumption is presented more concretely in Hypothesis 3.2 below. Its importance was also noticed before through the study of linear displacement models in echo mode [START_REF] Bilgen | Error analysis in acoustic elastography. i. displacement estimation[END_REF], leading to large variance of the displacement estimators when it is violated. The salient feature is as follows: if λ is the central wavelength of the probing wave-packet, then the variations of the displacement have to be small at the scale √ λ. This means that the resolution one expects from ultrasound elastography is not of order λ, as is the case in standard ultrasound reconstructions, but much less favorable. For instance for the applications we have in mind (biological tissues), the central wavelength is generally on the order of a fraction of a millimeter (∼ 10 -4 m). Therefore, under the setup considered in this work, one can only expect a resolution on the order of centimeters (∼ 10 -2 m). As one can see from the proof of Theorem 1, the available measurements depend not only on the displacement but also on unknown properties of the random medium when Hypothesis 1 is violated. This resolution limitation is confirmed in the numerical simulations presented in section 4, where additional numerical reconstructions and displays of the aliasing effect are presented. Section 5 offers some concluding remarks.

Modeling

We consider the following setting: an acoustic (sound wave) source s(x,t) located in the vicinity of a point x 0 ∈ R 3 emits a pulse which interacts with a cluster of point scatterers located at positions {x n } N n=1 ∈ C ⊂ R 3 . The resulting field u(y,t) is recorded at points {y} M m=1 in the far-field outside the cluster C. Then, the location of each scatterer is perturbed by a vector-valued function φ(x) assumed to be defined in all of R 3 , and equal to 0 outside C (in this case, the new position of the scatterer originally located at x n becomes x n + φ(x n )).The same sound wave source is fired and the resulting signal u φ (y,t) is recorded at the same locations {y} M m=1 . Our objective is the reconstruction of φ(x). To do so, we study the ratio of the time-dependent signals, i.e., R(t,y;x 0 , ξ) = u φ (t,y;x 0 , ξ)

u(t,y;x 0 , ξ) . (2.1)
Physically, in elastography, the displacement φ(x) is generated by propagating elastic shear waves. These latter travel at a speed much below that of acoustic (pressure) waves (see [START_REF] Ammari | A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements[END_REF][START_REF] Gennisson | Transient elastography in anisotropic medium: Application to the measurement of slow and fast shear wave speeds in muscles[END_REF][START_REF] Jiang | Approximate steady state models for magnetic resonance elastography[END_REF][START_REF] Kruse | Tissue characterization using magnetic resonance elastography: preliminary results[END_REF][START_REF] Mclaughlin | Calculating tissue shear modulus and pressure by 2D log-elastographic methods[END_REF][START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF][START_REF] Ophir | Elastography: A quantitative method for imaging the elasticity of biological tissues[END_REF][START_REF] Ophir | Elastography: Imaging the elastic properties of soft tissues with ultrasound[END_REF][START_REF] Sandrin | Shear Modulus Imaging with 2-D Transient Elastography[END_REF] and references therein). This phenomenon is known in the community as the separation of scale between the speed of shear and pressure waves. As a consequence, the generated displacement φ(x) appears static to the traveling acoustic waves.

In addition and for all that follows, we shall further make use of the following hypotheses: the background speed for acoustic waves is homogeneous with speed c. The scatterers are point like and their locations are generated uniformly at random in a bounded region C and have scattering coefficients {σ n } which are i.i.d. random variable with mean E[σ n ] = 0 and variance E[σ 2 n ] = σ 2 < ∞. Furthermore, the Born approximation (single scattering) is valid.

The latter are fairly generic hypotheses for the problem at hand and are widely used in the context of elastography []. In particular, and as mentioned above, we have in mind applications concerning biological tissues. In this context, the point scatterers generally take the form of red blood cells of cell nuclei [] which have relatively small scattering cross-section WRITE SMTG and are found at a density such that it is reasonable to use the single-scattering hypothesis FINISH CITE. WATER BACKGROUND? Finally, we make use of one more crucial hypothesis regarding the displacement field φ(•) which is specific to the problem we are treating. It is introduced in Hypothesis 3.2 below. In short, it states that if β is the typical (rescaled) wavelength of the probing source then the local fluctuations of the displacement in a region of size β 2 must be small. As noticed in [START_REF] Bal | Local asymptotic analysis of transient elastography reconstructions[END_REF], this scale is related to the Heisenberg uncertainty principle. As will become clear in the derivation of Theorem 1, the probing signal s is required to be focused both spatially and angularly (in frequency) in order to allow for a reconstruction of φ(x). Such a focusing can only be performed at best with a focusing of order β 2 for a signal of wavelength of order β by the uncertainty principle.

We will show that, under such hypotheses, if sufficiently many sources and receivers are used, it is possible to recover the quantity φ(x) with a small error and a resolution that depends on the bandwidth of the sources and the aperture (this statement is made precise in Theorem 1 below).

For all that follows, we shall use the following notation: |x| represents the norm of the vector x and x = x |x| represents its direction (unit vector).

The original (forward) problem of wave propagation in a inhomogeneous medium ask for a function u(x,t) such that,

∂ 2 ∂t 2 u(t,x) -c(x) 2 ∆u(t,x) = f (x,t) in R n × (0,∞) u(0,x) = 0, u t (0,x) = 0 on R n × {t = 0}
where ∆ is the usual Laplacian. It can be assumed without loss of generality (w.l.o.g.) that c(x) takes the form (fluctuations about the mean),

c(x) = c 2 1 + S(x) c 2
where c is some fixed positive constant and S(x) takes the form (under our point scatterer hypothesis),

S(x) = N n=1 σ n δ(x -x n )
δ(•) being a Dirac delta function, and the {x n } N n=1 are i.i.d random variables distributed uniformly in a compact region C, i.e., x n ∼ U(C). Substitution implies that u(t,x) must solve the problem,

∂ 2 ∂t 2 u(t,x) -c 2 ∆u(t,x) = S(x)∆u(t,x) in R n × (0,∞), u(0,x) = 0, u t (0,x) = 0 on R n × {t = 0}.
Now introduce the Green's function,

G(t,x) = 1 4π|x| δ t - |x| c
which is the fundamental solution to the problem,

∂ 2 ∂t 2 u(t,x) -c 2 ∆u(t,x) = δ(x)δ(t) in R n × (0,∞), u(0,x) = 0, u t (0,x) = 0 on R n × {t = 0}.
Solving the re-formulated problem through the Green's function leads to the Lipmann-Schwinger equation,

u(t,x) = u 0 (t,x) + G(t -τ,x -χ)S(χ)∆u(τ,χ)dτ dχ where u 0 (t,x) = G(t -τ,x -χ)f (χ,τ )dτ dχ
represents the source term. Then, under the hypothesis of Born single scattering, the solution u(t,x) can be replaced by the source term in the integral to give,

u(t,x) = u 0 (t,x) + G(t -τ,x -χ)S(χ)∆u 0 (τ,χ)dτ dχ.
It is always possible to subtract the known ballistic term (u 0 (t,x)) from the received signal. Therefore, we shall hereafter focus our attention to the second term in the previous equation.

We will further assume that the (ballistic) source is such that,

u 0 (t,x; ξ, ) = γ ,ν ((x -x 0 ) -ct ξ)
where 0 < < 1 is the scale of the displacement, x 0 is a fixed location and γ ,ν (•) is of the form,

γ ,ν ((x -x 0 ) -ct ξ) = 1 √ 2πν e 2πi (x-x 0 )• ξ-ct β e |(x-x0)-ct ξ| 2 2ν 2 (2.2)
for some fixed 0 < β,ν. In other words, we assume that the source is such that its initial profile (a modulated Gaussian) is translated along the direction ξ at speed c. As shown in the following lemma, such hypothesis is asymptotically true in a homogeneous medium, Lemma 2.1. Let,

η ,ν (x,t) = e 2πi(x•χ-c|χ|t) e 2π 2 ν 2 χ- ξ β 2 dχ γ ,ν (x -ct ξ) = 1 √ 2πν e 2πi x• ξ-ct β e -|x-ct ξ| 2 2ν 2
where 0 < < 1 and 0 < β,ν.

Then, if ν = β/2 √ log( ) δ for some δ > 0, η ,ν (x,t) = γ ,ν (x -ct ξ) + O δ 2 and if ν = β-ρ 2
for some 0 < ρ < β then,

lim →0 η ,ν (x,t) -γ ,ν (x -ct ξ) ∞ = 0. Proof. First let, R := inf r > 0 : χ>r e -2π 2 ν 2 |χ| 2 dχ <
and note that,

R = O log( ) ν Then, η (x,t) = |χ- ξ 1+β |<R e 2πi(x•χ-c|χ|t) e -2π 2 ν 2 χ- ξ β 2 dχ + O( ) = |ζ|<R e 2πi x•ζ+x• ξ β -c ζ+ ξ β t e -2π 2 ν 2 |ζ| 2 dζ + O( ) after the change of variable ζ = χ - ξ β . Assume that: ν = β/2 √ log( ) δ
for some small δ > 0. Then,

2β |ζ| 2 = O β δ 2 β ζ • ξ = O β/2 δ for |ζ| < R , which implies that, ζ + ξ β = 1 β 2β |ζ| 2 + 2 β ζ • ξ + 1 1/2 = 1 β + ζ • ξ + O δ 2
from the Taylor series for the square root. Then, using the Taylor series for the complex exponential gives,

η (x,t) = e 2πi x• ξ-ct β |ζ|<R e 2πi(x-ct ξ)•ζ e -2π 2 ν 2 |ζ| 2 dχ + O δ 2 = 1 √ 2πν e 2πi x• ξ-ct β e -|x-ct ξ| 2ν 2 + O δ 2
since the Fourier transform of a Gaussian is also a Gaussian. Finally, if we take δ = ρ 2 log( ) for some 0 < ρ < β, one gets,

O δ 2 = O ( ρ log( )) → 0 0 and ν = β-ρ 2
→ 0 0. Note that for each fixed set of parameters, the expression η (x,t) is a solution to the wave equation in a homogeneous medium, being a superposition of plane waves. The previous proposition therefore states that if we pick the support of the wavelet (region where Gaussian is O(1)) "large enough" relative to the frequency β , the original waveform translates through space as a whole with only small error. We shall hereafter assume that ν has been picked appropriately so that this hypothesis holds, i.e.,

ν = O( β/2 ) (2.3)

Imaging technique and resolution analysis

In this section, we introduce our imaging technique for the displacement making use of dispersionless sources as described in the previous section. After removing the source term and substituting the explicit form for the source, it is seen that the received signals for the random and displaced random fields respectively take the form,

u(t,x;x 0 , ξ) = N n=1 σ n 4π|y -x n | γ ,ν x n -x 0 + c t - |y -x n | c ξ (3.1) u φ (t,x;x 0 , ξ) = N n=1 σ n 4π|y -x n | γ ,ν x n -φ(x n ) -x 0 + c t - |y -x n -φ(x n )| c ξ (3.2)
where we have also assumed to simplify notation (and by abuse of notation) that,

∆u 0 (τ,χ) = γ ,ν ((x -x 0 ) -ct ξ).
Indeed both u 0 and ∆u 0 are localized Gaussian beams that only differ by a constant that does not modify our subsequent analyses. We further rely on the following hypothesis regarding the displacement φ(•): Definition 3.1. Given a fixed time t, receiver location y, fixed location x 0 and unit vector ξ, we define the following vectors,

x(y,t;x 0 , ξ) = x 0 + (ct -|y|) ξ α(y,x,t;x 0 , ξ) = x -x 0 -c t - |y -x| c
ξ and the set,

S = S (x * ) := x ∈ R d : |x -x * | < .
Hypothesis 3.2. Given ,β,ν,δ > 0, the displacement field φ(x) satisfies the following inequalities,

||φ(x)|| ∞ < ν 1+δ log(ν -1 )
and,

[φ(x) -φ(x )] I S ν √ 2 log(ν) (|x -x |) ∞ < β+δ for all x,x ∈ R 3 and β,δ > 0.
The hypothesis requires that the displacement be slightly smaller than the size of the support of the wavelet sent through the medium, i.e., O(ν). It also requires that the variation of the displacement around its mean over the support of the wavelet be uniformly smaller than the carrier frequency O( β ). Since by the previous section ν = O( β/2 ) β , this entails a separation of scale; the local variations of the displacement should be significantly smaller than the amplitude of the displacement. This is the regime for which the theory presented here applies.

Remark Note that if φ(x) ∈ C 1 (R 3 ), the above inequality is satisfied if ∇φ(x )I S ν √ 2 log(ν) (|x -x |) ∞ < β+δ for all x,x ∈ R 3 .
Our main result is the following,

Theorem 3.3. Let y be such that ||φ(x)||∞ |y| , Diam(C) |y| = O( κ ) and assume Hypothesis 3.2 is satisfied. Then, R(t,y;x 0 , ξ) = u φ (t,y;x 0 , ξ) u(t,y;x 0 , ξ) = e 2πi φ(x(y,t))• γ β + E N (t,y;x 0 , ξ) (3.3)
where u(t,y) and u φ (t,y) are the signals recorded at time t and location y, before and after displacement, from a localized source emitted at x 0 and traveling in the ξ direction, and where |γ|γ = ξ -ŷ. The quantity E N (t,y;x 0 , ξ) is a stochastic error term such that, lim

N →∞ P |E N (t,y;x 0 , ξ)| > ζ ≤ O(1 -e -1 Γ 1 +Γ 2 ν δ/2 ζ ) + O ν δ log(ν -1 ) + O 2δ ν .
Finally, if such quantity is available for vectors γ such that |γ| ∈ [γ -

β µ , γ + β µ ] then, I(χ) = γ+ β µ γ- β µ R(t,y;x 0 , ξ)e -2πi χ 1+β |γ| d|γ| = sinc χ -φ(x(y,t)) • γ µ + O   γ+ β µ γ- β µ |E N (t,y;x 0 , ξ)|d|γ|   .
The theorem says the following: under the separation of scale hypothesis, the ratio of the signals before and after displacement takes the form of a phase shift corresponding to the magnitude of the displacement in the direction γ. When the number of scatterers on the support of a wavelet is large (but small enough so that the Born approximation holds), this ratio is statistically stable and can in theory be used to recover φ(x(y,t))• γ β up to a 2π factor. Indeed, the periodicity of the complex exponential leads to ambiguity/aliasing if |φ(x(y,t)) • γ| > β . It is possible to remedy this problem through the use of a small sampling aperture. This is the content of the second part of the theorem which says that if measurements are available on a specific sampling arc spanning O β µ radians, it is possible to resolve this ambiguity by computing the functional I(χ) which then possess a single maximum located at φ(x(y,t)) • γ. Upon computing this quantity for d linearly independent unit vectors γj (d being the ambient dimension) and undo the projections, it is possible to recover the full displacement up to a small error. A diagram is presented in Figure 3.1 to summarize the various geometric quantities introduced earlier. We now proceed to the analysis and start with a few preparatory lemmas which are proved in appendix.. Lemma 3.1. Let y be such that Diam(C) |y| = O( κ ) for some κ 1 and 0 < < 1. Then,

(y -x n ) = ŷ + O( κ ) Lemma 3.2. If ||φ(x)||∞ |y| = O( κ ) for some κ 1 then, α(y,x n + φ(x n ),t) = x n -x 0 + (|y -x n | -ct) ξ + φ(x n ) -(φ(x n ) • y) ξ + O( κ )
and,

• I S ν √ 2 log(ν -1 ) α(y,x n ,t;x 0 , ξ) + O(ν) = e 2πi φ(x(y,t))•( ξ-y) β 1 + O φ(x n ) -φ(x(y,t;x 0 , ξ)) • ( ξ -y) β e -|xn-x 0 +|y-xn| ξ-ct ξ| 2 2ν 2 • I S ν √ 2 log(ν -1 ) α(y,x n ,t;x 0 , ξ) + O(ν) = e 2πi φ(x(y,t;x 0 , ξ))•( ξ-y) β e -|xn -x 0 +|y-xn | ξ-ct ξ| 2 2ν 2 I S ν √ 2 log(ν -1 ) α(y,x n ,t;x 0 , ξ) + O( δ ) + O(ν)
by Hypothesis 3.2. Finally, using results from Lemma 3.2 and 3.3 and substituting leads to,

u φ (t,x) = σ n 4π|y -x n | e 2πi φ(x(y,t)•( ξ-y) β e 2πi ((xn -x 0 )• ξ-ct+|y-xn |) β e -|xn -x 0 +|y-xn | ξ-ct ξ| 2 2ν 2 +O ν δ log(ν -1 ) + O δ .
Summing over the scatterers indices n provides the desired result.

We also have the following corollary which is a simple consequence of the previous proposition, Corollary 3.5. The ratio of the signals received before and after displacement takes the form, R(t,y;x 0 , ξ) = u φ (t,y;x 0 , ξ)

u(t,y;x 0 , ξ) = e 2πi φ(x(y,t))• γ β + N n=1 σn 4π|y-xn| e n u(t,y;x 0 , ξ)
where γ = ξ -ŷ. We now treat the stochastic error term, E N (t,y;x 0 , ξ) = N n=1 σn 4π|y-xn| e n u(t,y;x 0 , ξ) .

From Proposition 3.4, it is seen that the numerator, although stochastic, is bounded by a small quantity almost surely. On the other hand, it appears possible for the denominator to be so small that the error term "blows up". We show below that when the number of scatterers in the support of a wavelet is large, this occurs only with small probability, Lemma 3.4. Denote,

X n = σ n 4π|y -x n | e 2πi ((xn -x 0 )• ξ-ct+|y-xn |) β e -|xn -x 0 +|y-xn| ξ-ct ξ| 2 2ν 2 Y n = σ n 4π|y -x n | e n
where e n is as in Proposition 3.4. Then, if the scattering coefficients {σ n } are i.i.d. with mean zero and variance 1,

E[X n ] = 0, E[Y n ] = 0 E[X n Ȳm ] = 0, E[X n Xm ] = 0, E[Y n Ȳm ] = 0
for n = m. Lemma 3.5. Let X n be as in Lemma 3.4. Then, if the scattering coefficients {σ n } are i.i.d. with mean zero and variance 1,

Γ 1 = E[X n Xn ] = O(1) > 0 Γ 2 = E[X n X n ] = O( β/2 )
for all time and receiver location. Proposition 3.6. The error term,

E N (t,y;x 0 , ξ) = N n=1 σn 4π|y-xn| e n u φ (t,y;x 0 , ξ) is such that, lim N →∞ P |E N (t,y;x 0 , ξ)| > β ≤ O(1 -e -1 Γ 1 +Γ 2 ν δ/2 β ) + O ν δ log(ν -1 ) + O 2δ ν .
Proof. The error term can be written as,

E N (t,y;x 0 , ξ) = 1 N N n=1 N m=1 X n Ȳm 1 N N n=1 N m=1 X n Xm
where we divided both the numerator and denominator by 1

N N n=1 X n . First con- sider, |µ N | = E 1 N N n=1 N m=1 X n Ȳm ≤ E[|X 1 ||Y 1 |] = O ν δ log(ν -1 ) + O( δ )
by Lemma 3.4 and Proposition 3.4. Also note that |X n | is uniformly bounded almost surely. Then consider some constant α > 0 and,

P 1 N N n=1 N m=1 X n Ȳm -µ N > α ≤ 1 N 2 α 2 N n=1 N m=1 N p=1 N q=1 E X n Ȳm Xp Y q -µ 2 N (3.4)
by Chebyshev inequality. By the properties described in Lemma ??, there are at most O(N 2 ) nonzero terms in the sum, and they are bounded by a quantity of order O ν 2δ log(ν -1 ) + O( 2δ ) (again because |Y m | is of such order and |X n | is bounded almost surely) . Thus, using the previous result about µ N , one gets that,

P 1 N N n=1 N m=1 X n Ȳm > α ≤ O ν 2δ log(ν -1 ) α 2 + O 2δ α 2
Then, let β > 0 and consider

P E N (t,y;x 0 , ξ) > β = P 1 N N n=1 N m=1 X n Ȳm 1 N N n=1 N m=1 X n Xm > β ∩ 1 N N n=1 N m=1 X n Ȳm ≤ α + P 1 N N n=1 N m=1 X n Ȳm 1 N N n=1 N m=1 X n Xm > β ∩ 1 N N n=1 N m=1 X n Ȳm > α ≤ P 1 N N n=1 N m=1 X n Xm < α β + O ν 2δ log(ν -1 ) α 2 + O 2δ α 2
by Eq.(3.4). Note that,

1 N N n=1 N m=1 X n Xm = 1 √ N N n=1 X n 2 Therefore, P 1 N N n=1 N m=1 X n Xm < α β = P 1 √ N N n=1 X n ∈ z ∈ C : |z| < α β
but since the {X n } are i.i.d. complex random variable such that,

E[X n ] = 0 E[X n Xn ] = Γ 1 = O(1) > 0 E[X n X n ] = Γ 2 = O( β/2 )
(by Lemma 3.5), one gets from the central limit theorem for complex i.i.d. random variables that, lim

N →∞ P 1 √ N N n=1 X n ∈ z ∈ C : |z| < α β = P X ∈ z ∈ C : |z| < α β
where,

X ∼ N (0,Γ 1 ,Γ 2 )
that is, the probabilty tends to that of a complex Gaussian random variable with covariance matrix: Γ 1 I and relation matrix: Γ 2 I, where I is the identity. Its probability density is proportional to,

f X (z) = e - Γ 1 Γ 2 1 -Γ 2 2 |z| 2 + Γ 2 Γ 2 1 -Γ 2 2 (z) 2 ≤ e -1 Γ 1 +Γ 2 |z| 2 since |Γ 2 | < |Γ 1 | for small enough. Thus, P X ∈ z ∈ C : |z| < α β ∼≤ |z|< √ α β e -1 Γ 1 +Γ 2 |z| 2 = π(Γ 1 + Γ 2 )(1 -e -1 Γ 1 +Γ 2 α β ).
Therefore, lim

N →∞ P 1 N N n=1 N m=1 X n Y m 1 N N n=1 N m=1 X n X m > β ≤ O(1 -e -α Γ 1 +Γ 2 α β ) + O ν 2δ log(ν -1 ) α 2 + O 2δ α 2
for all α > 0. Assuming α = ν δ/2 one gets, lim

N →∞ P 1 N N n=1 N m=1 X n Y m 1 N N n=1 N m=1 X n X m > β ≤ O(1 -e -1 Γ 1 +Γ 2 ν δ/2 β ) + O ν δ log(ν -1 ) + O 2δ ν as claimed.
In particular, since ν = O( β/2 ) by Eq.(2.3), the upper bound for the probability goes to zero as goes to zero. Thus, for a displacement satisfying Hypothesis 3.2, if the number of scatterers in a region of radius O( β/2 ) is large, one can expect the error term to be small. e 2πi φ(x(y,t;x 0 , ξ))•γ

β |γ| + E N (t,y;x 0 , ξ) e -2πi χ-γ β |γ| d|γ| = sinc χ -φ(x(y,t;x 0 , ξ)) • γ µ + O   γ+ β µ γ- β µ |E N (t,y;x 0 , ξ)|d|γ|   ,
which concludes our proof.

Discussion and numerical results

In this section, we treat of the numerical aspects of the scheme. First, we discuss the behavior of the stochastic error term followed by the discretization of the imaging functional obtained in Eq. (3.3). Then, we provide a few examples of the reconstruction scheme and demonstrate that it behaves according to the theory. In all that follows, and as described earlier, the medium is constituted by a homogeneous background with speed of sound c = 1 perturbed by point scatterers (spatial Poisson process) with parameter λ large enough so that the average number of scatterers N lying in a ball of radius O(ν) (support of a wave packet) is on the order of O( 103 ). The scattering cross-section σ of each scatterer is a random variable with mean 0 and finite variance. Also recall that is a small parameter representing the size of the displacement (scale of the problem). We chose ν = β-0.1 2 following Eq.(2.3) and β = 1, unless otherwise stated. This is summarized in Table 4.2.

Stochastic error term

From Proposition 3.3, it is seen that the ratio R(t,y;x 0 , ξ) consists in the sum a complex exponential plus a stochastic term which goes to zero in probability as the number of scatterers in the support of a wave packet goes to infinity. A typical example of the value of the ratio as a function of | γ| is shown in Figure 4.1, and a typical example for the functional I(χ) is shown in Figure 4.2. This corroborates the result: both the real and imaginary parts take the form of slightly perturbed sinusoids which, upon integration, leads to a slightly perturbed sinc(•) function.

We would also like to add that the bound on the probability in Proposition 3.3 is suboptimal. This can be seen from a few numerical examples provided in Figure 4.3. For these cases, we chose γ = x, β ∈ {1,0.75}, δ ∈ {1,1.25}, ν = 10

β-0.1 2
and plotted the empirical estimate of the probability,

P |E N (t,y;x 0 , ξ)| > 1
as a function of ∈ [10 -4 ,1], taking N = 5000 and after performing 2000 independent trials to compute an empirical estimate of the probability. The results are shown on a logarithmic scale. It is seen that the empirical probability does indeed decay although at a rate which appears faster than our estimate.

Discrete measurements

In practice, measurements are generally available at discrete points on a spherical sampling manifold in the far field. In this sense, it is only possible to compute a discrete version of the integral in the functional,

I(χ) = γ+ β µ γ- β µ e 2πi φ(x(y,t;x 0 , ξ))•γ β e -2πi χ-γ β |γ| d|γ|
introduced in Theorem 3.3. In this section, we treat of the effects of such discretization.

For this purpose, assume that γ is fixed and that the ratio R(t,y;x 0 , ξ) can be measured at points (y m , ξm ) which are such that, where m = -M,-M + 1,...,M -1. In this case, the discretization takes the form,

I(χ) ≈ 1 2M M -1 m=-M e 2πi (φ(x(y,t;x 0 , ξ))•γ-χ) β |γ|m
which is seen to be a trapezoid rule applied to the integrand

f (x) = e 2πi (φ(x(y,t;xo , ξ))•γ-χ) β x
, i.e., a complex exponentials with a maximum frequency of,

f max = ||φ(x)|| ∞ β = O -β/2 .
From classical Fourier analysis, it is known that at least,

2M ∼ 2 β µ 1 β/2 = 2 β/2
µ discretization points should be used in order to avoid aliasing. Therefore, for a fixed displacement size and parameter µ, a finite number of measurements on the order of O β/2 µ is sufficient.

The parameter µ corresponds to the width of the sinc(•) appearing in the functional, which in turns is related to the accuracy of the approximation; the larger µ, the less accurate the approximation and vice-versa. Figure 4.4 shows the example of a typical functional computed with two different values of µ (∈ {0.1,0.5}). It is seen that the width of the main peak of the sinc(•) function is five times smaller in the case of µ = 0.1 (left, Figure 4.4) than for µ = 0.5 (right, Figure 4.4). In the presence of measurement noise, the numerical value corresponding to the approximation has an (unknown) absolute error bounded by µ.

Finally, if less that 2M points are used, one expect aliasing in which case the functional will exhibit multiple peaks as shown in Figure 4.5. Under this situation, it is not possible to uniquely determine an approximation for φ(x) • γ. where the integral has been discretized using a uniform quadrature with too few points (x-axis in units of ). The functional exhibits multiple local maxima of roughly the same amplitude. It is not possible to determine an appropriate approximation to the displacement in this case.

Reconstructions

The numerical reconstruction experiments were carried out in 2D as follows : first, N point scatterers {x n } are disposed uniformly at random in a region of size

C = -1 α , 1 α × [-4ν,4ν],
where α > 0 is a parameter defined below. For a fixed scale , the number N is chosen such that the average number of scatterers on the support of a wavelet of the form (2.2) is of order O( 103 ). For a fixed vector γ and location x ∈ C, wave packets with direction ξ, initial location x 0 were chosen in such a way that x = x 0 + c t ξ for some t > 0. A simulation is then launched with a localized source (Eq.(2.2)), and the (single-)scattered field u(y,t;x 0 , ξ) is recorded at y in the far region and at time t. The receiver location y is chosen in such a way that, ξŷ

| ξ -ŷ| = γ
as described in the previous section. This is repeated for various vectors ξ chosen in such a way that,

| ξ -ŷ| = 2 ξ • γ ∈ [γ - β µ , γ + β µ ]
for some (pre-determined) γ, µ. This experiment is repeated with the original scatterers displaced by a quantity φ(x) so that {x n } → {x n + φ(x n )} providing us with the scattered field u φ (y,t;x 0 , ξ) as well.

Once this information has been acquired, the ratio R(t,y;x 0 , ξ) is taken and the functional,

I(χ) = γ+ β µ γ- β µ R(t,y;x 0 , ξ)e -2πi χ β |γ| d|γ|
is computed. We finally identify the location of its maximum which provides our estimate for φ(x) • γ at x.

We set out to study the behavior of the reconstruction as a function of the size and variations of the displacement. We chose γ ∈ {ê x , êy } and considered various points xm of the form, and different values of ∈ {1e -1 ,5e -2 ,1e -2 }. The displacements were chosen to be of the form

ν 1+δ φ i (αx)
for φ i (x) among three sets of functions shown in Table 4.1. This scaling corresponds to that of Hypothesis 3.2. Finally, we set β = 1 and δ = 0.5. All these values are summarized in Table 4.2. Let us conclude this section by some remarks on the spatial resolution of the method. As described in Proposition 1, we expect a good reconstruction of φ when The above limitations in the spatial resolution are not only a specificity of the method. Close inspection of the formula for the measurement (3.2) and the derivation of Proposition 3.4 shows that the error term e n there mixes the displacement φ and the statistical properties of the random cloud of points parameterized by (x n ,σ n ). The reconstruction of φ independent of the realization of the unknown medium depends on the aforementioned separation of scales that allow us to prove as in Proposition 3.4 that the term involving e n is negligible. The above numerical simulations show that reconstructions are no longer stable, and we claim no longer feasible, when e n fails to be small, i.e., when Hypothesis (3.2) fails to hold.

Conclusion

In conclusion, we have presented an imaging technique for the displacement field of an ensemble of randomly distributed points scatterers in a homogeneous medium. The technique relies on the hypothesis that the Born approximation is valid and that a separation of scale exists between the size of the displacement and its local fluctuations. We demonstrated that when such hypothesis is satisfied, the ratio of the signal obtained from localized sources before and after displacement takes the form of a sum between a complex exponential and a random term which goes to zero in probability as the number of scatterers in the support goes to infinity. The latter complex exponential has frequency equal to the magnitude of the displacement at a particular (known) location and in a particular (known) direction. We argued that this information can be used to reconstruct an approximation of the displacement field, and that the quality of the approximation can be increased by using a small discrete aperture. Finally, we provided numerical reconstruction results demonstrating that the method works in the prescribed regime.

|y -x n -φ(x n )| 2 = |y -x n | 2 -2φ(x n ) • (y -x n ) + |φ(x n )| 2 = |y -x n | 2 1 -2 φ(x n ) |y -x n | • y -x n + |φ(x n )| 2 |y -x n | 2 .
Then, taking the square root and using its Taylor expansion gives, where we used the Taylor expansion for the exponential. following Eq.(2.3).

Fig. 3 . 1 .

 31 Fig. 3.1. Experimental setup: a localized source travels from x 0 in the direction ξ. It encounters scatterers at the point x (taken to be the origin) which produces a response recorded at a receiver located at y in the far field. The ratio of the responses before (u) and after (u φ ) displacement contains information about the projection of the displacement at the point x along the direction γ = ξ-ŷ | ξ-ŷ| .

e 2 = 2 + 2 Proposition 3 . 4 . 2 =

 222342 2πi α(y,xn +φ(xn ),t;x 0 , ξ)• ξ β = e 2πiφ(xn)•( ξ-ŷ) e 2πi (xn-x 0 )• ξ+|y-xn |-ct β + O( κ-β ). Lemma 3.3. If Hypothesis 3.2 is satisfied and ||φ(x)||∞ |y| = O( κ ) for some κ 1 then, for fixed values of y, t, x 0 and ξ, e -|α(y,xn +φ(xn ),t;x 0 , ξ)| 2 2ν e -|xn -x 0 +(|y-xn |-ct) ξ| 2 2ν O ν δ log(ν -1 ) + O κ-β ν Let y be such that ||φ(x)||∞ |y| , Diam(C) |y| = O( κ ) and assume Hypothesis 3.2 is satisfied. Then, u φ (t,x) = e 2πi φ(x(y,t)•( ξx) β N n=1 σ n 4π|y -x n | e 2πi ((xn -x 0 )• ξ-ct+|y-xn |) β e -|xn -x 0 +|y-xn | ξ-ct ξ| -x n | e n where e n is an error term of order O ν δ log(ν -1 ) + O( δ ) almost surely. Proof. Consider, e 2πi φ(xn)•( ξy) β e -|xn-x 0 +|y-xn| ξ-ct ξ| 2 2ν e 2πi φ(xn)•( ξy) β e -|xn-x 0 +|y-xn| ξ-ct ξ| ν -1 ) α(y,x n ,t;x 0 , ξ) + O(ν) = e 2πi φ(x(y,t;x 0 , ξ))•( ξy) β e 2πi φ(xn )-φ(x(y,t))•( ξx) β e -|xn -x 0 +|y-xn | ξ-ct ξ| 2 2ν 2

  We are now ready to prove Theorem 3.3. The first part of the theorem follows from Corollary 3.5 and Proposition 3.6. We now show the second part. It is assumed that the sampling manifold is such that one has access to measurements corresponding to |γ| ∈ [γ -β µ , γ + β µ ] for some γ ∈ [0,2] and µ > 0. Then, γ2 γ1 R(t,y;x 0 , ξ)e -2πi χ β |γ| d|γ| = e 2πi γ

Fig. 4 . 1 .

 41 Fig. 4.1. Typical example for the value of the ratio R(t,y;x 0 , ξ) as a function of | γ| where γ = ξ -ŷ. Left: real part, Right: imaginary part. The ratio corresponds to a complex exponential plus random noise. The dominant frequency encode information about the local displacement.

Fig. 4 . 2 .

 42 Fig. 4.2. Typical example for the value of the imaging functional I(χ). The imaging functional corresponds to a sinc(•) function of width µ plus some random noise. The location of the maximum corresponds to the value of the local displacement projected along a direction γ = ξ-ŷ | ξ-ŷ| .

Fig. 4 . 3 .

 43 Fig. 4.3. Probability that the modulus of the error term is larger than 1 as a function of (logarithmic scale). Left: β = 1, δ = 1 and ν = 10 β-ρ 2 empirical (blue) vs theoretical bound (black);

Fig. 4 . 4 .

 44 Fig. 4.4. Example of the imaging functional I(χ) computed for two different values of the aperture parameter µ (x-axis in units of ). Left: µ = 0.1, Right: µ = 0.5. A larger µ implies less resolution, i.e., a larger width for the sinc(•) function.

Fig. 4 . 5 .

 45 Fig. 4.5. Example of the imaging functional I(χ)where the integral has been discretized using a uniform quadrature with too few points (x-axis in units of ). The functional exhibits multiple local maxima of roughly the same amplitude. It is not possible to determine an appropriate approximation to the displacement in this case.

  m = -M,...,M and M = 200. We also picked γ = 1, µ = 0.05, α = β+δ ν 1+δ , ν = β-0.1 2

  are sufficiently small, i.e., when is small under this setup. The results are shown in Figure4.6, 4.7 and 4.8. The first feature one notices is that very good reconstruction is indeed observed for the smallest (left figure). However, as fluctuations increase, the reconstruction becomes poorer. This is particularly evident on Figure4.6 on the middle figures where the displacement in regions of small local variations (extrema of the sine curve) are well-reconstructed whereas those of larger local variations are not.

Fig. 4 . 6 .

 46 Fig. 4.6. Reconstruction (blue) and true value (black) of the x-displacement (top) and ydisplacement (bottom) for φ 1 (x) for = 1e -2 ,5e -2 ,1e -1 (left to right). The reconstructions fails where the local variations are large but succeeds otherwise.

Fig. 4 . 7 .

 47 Fig. 4.7. Reconstruction (blue) and true value (black) of the x-displacement (top) and ydisplacement (bottom) for φ 2 (x) for = 1e -2 ,5e -2 ,1e -1 (left to right). The reconstructions fails where the local variations are large but succeeds otherwise.

Fig. 4 . 8 .

 48 Fig. 4.8. Reconstruction (blue) and true value (black) of the x-displacement (top) and ydisplacement (bottom) for φ 3 (x) for = 1e -2 ,5e -2 ,1e -1 (left to right). The reconstructions fails where the local variations are large but succeeds otherwise.

2 -n |y| + |x n | 2 |y| 2 1/ 2 = 1 -ŷ • x n |y| + 1 2 |x n | 2 |y| 2 + O ŷ • x n |y| 2 = 1 +-x n |y| 1 |y-xn| |y| 4 =

 2221222114 2y • x n + |x n | 2 1/2 = 1 -2ŷ • x O ( κ )through the Taylor expansion for the square root and by assumption. Then,(y -x n ) = y -x n |y -x n | = y (ŷ + O( κ )) 1 1 + O( κ ) = ŷ + O( κ )by the geometric series.Proof of Lemma 3.2 Consider first,

2 = 2 + 2 + 2 ν 2 + O κ-β ν 2 through Lemma 3 . 2 .- 1 ) 2 2ν 2 + 2 +O-β ν 2 + 2 +

 22222232122222 |y -x n -φ(x n )| = |y -x n | 1 -φ(x n ) |y -x n | • y -x n + |φ(x n )| 2 |y -x n | |y -x n | -φ(x n ) • y + O ( κ ).Then write,e 2πi α(y,xn+φ(xn),t)• ξ β = e 2πi (xn+φ(xn)-x 0 -ct ξ+|y-xn | ξ-(φ(xn)• y) ξ)• ξ β +O( κ-β ) = e 2πiφ(xn)•( ξ-ŷ) e 2πi (xn -x 0 )• ξ+|y-xn |-ct β + O( κ-β )through the Taylor series for the exponential and after re-arranging.Proof of Lemma 3.3 First write,|α(y,x n + φ(x n ),t;x 0 , ξ)| 2 2ν 2 = |x n -x 0 + |y -x n | ξ -ct ξ| 2 2ν φ(x n ) -(φ(x n ) • y) ξ, • (x n -x 0 + |y -x n | ξ -ct ξ) ν φ(x n ) -(φ(x n ) • y) ξThen consider, e -|α(y,xn ,t;x 0 , ξ)|α(y,x n ,t;x 0 , ξ)| e -|α(y,xn ,t)| O(ν).Further note that,x n + (|y -x n | -ct) ξ --φ(x n ) + (φ(x n ) • ŷ) ξ ≤∼ ν log(ν)and therefore,x n + (|y -x n | -ct) ξ = O ν log(ν)by Hypothesis 3.2. This further implies that,φ(x n ) -(φ(x n ) • y) ξ • (x n -x 0 + |y -x n | ξ -ct ξ) ν 2 = O ν δ log(ν -1 ) .Thus, e -|α(y,xn ,t;x 0 , ξ),x n ,t;x 0 , ξ) e -|xn -x 0 +(|y-xn |-ct) ξ| 2 2ν ν δ log(ν -1 ) + O κO(ν) = e -|xn -x 0 +(|y-xn |-ct) ξ| 2 2ν O ν δ log(ν -1 ) + O κ-β ν 2

Proof of Lemma 3. 4 2 = 0 E 2 • E 1 4π|y 2 = 0 E 2 = E 1 4π|y -x 1 | 2 = 2 = 2 = 2 =

 42021202112222 Simply note that,E[X n ] = E[σ n ]E 1 4π|y -x n | e 2πi (xn • ξ-ct+|y-xn |) β e -|xn+|y-xn | ξ-ct ξ| 2 2ν [Y n ] = E[σ n ]E 1 4π|y -x n | e n = 0by independence and the fact that the σ n 's have mean zero. Similarly,E[X n Ȳm ] = E[σ n ]E[σ m ]E 1 4π|y -x n | e 2πi (xn • ξ-ct+|y-xn |) β e -|xn +|y-xn | ξ-ct ξ| n Xm ] = E[σ n ]E[σ m ]E 1 4π|y -x n | e 2πi (xn • ξ-ct+|y-xn |) β e -|xn +|y-xn | ξ-ct ξ| 2 2ν -x m | e -2πi (xm • ξ-ct+|y-xm|) β e -|xm+|y-xm| ξ-ct ξ| 2 2ν [Y n Ȳm ] = E[σ n ]E[σ m ]E ēn 4π|y -x n | E ēm 4π|y -x m | = 0This concludes the proof of the lemma.Proof of Lemma 3.5 First, note that,Γ 1 = E[X n Xn ] > 0 = E[σ 2 n ]E 1 4π|y -x n | 2 e -|xn +|y-xn | ξ-ct ξ| 2 ν 2 e -|x 1 +|y-x 1 | ξ-ct ξ| 2 ν E[X n X n ] = E[σ 2 n ]E 1 4π|y -x n | 2 e 2πi (xn • ξ-ct+|y-xn|) β e -|xn +|y-xn | ξ-ct ξ| 2 ν E 1 4π|y -x 1 | 2 e 2πi (x 1 • ξ-ct+|y-x 1 |) β e -|x 1 +|y-x 1 | ξ-ct ξ| 2 ν O β ν = O β/2 .

Table 4 .

 4 1. Displacement functions φ i (•) used throughout numerical simulations.

	φ i	x-component	y-component
	φ 1	cos(2π x)	1.5 sin(π/2x)
	φ 2	e -(x-y) 2 0.02	I |y|<1/2 (y) dy -0.5 e -(x-y) 2 0.02	I |y|<1/2 (y) dy
	φ 3	2I |x|<1/2 x + 1 2	-1.75I |x|<3/4 x + 3 4

Table 4 . 2 .

 42 Values of parameters for numerical experiments.

The work of GB was partially funded by AFOSR Grant NSSEFF-FA9550-10-1-0194 and NSF grant DMS-1408867.