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Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity

We consider the Kelvin-Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply the Carleman estimate to prove the Lipschitz stability in determining a spatial varying function in an external source term of Kelvin-Voigt model by a single measurement. Finally as a related system, we consider an isothermal compressible fluid system and apply the Carleman estimate to establish the Lipschitz stability for an inverse source problem for the compressible fluid system.

The equation (1.1) is called the Kelvin-Voigt model and is one model equation for the viscoelasticity. The viscoelasticity indicates a mixed physical property of the viscosity and the elasticity, and is frequently observed in human tissues. Thus the viscoelasticity is important for example for the medical diagnosis. In the diagnosis, the main task is to detect some anomaly of spatially varying coefficients in (1.1) which may indicate some organizational abnormalities such as tumors. Hence inverse problems of determining coefficients λ(x ′ ), µ(x ′ ), λ(x ′ ), µ(x ′ ) by available boundary data, are demanded from medical points of view. Such inverse problems of determining coefficients can be solved by the corresponding inverse source problems which can be the linearized problems of inverse coefficient problems, but we do not here discuss details for the inverse coefficient problems.
For the inverse problems related to the medical diagnosis, one of theoretical fundamentals is the Carleman estimate. The purpose of this paper is to establish a Carleman estimate for (1.1) and apply it to an inverse source problem of determining a spatially varying factor of the external source F(x). Also as is seen later, the principal part of (1.1) is a strongly coupled parabolic Lamé system ρ∂ x 0 − L λ,µ (x, D ′ ), which causes a serious difficulty for proving the Carleman estimate for (1.1). In addition to the Kelvin-Voigt model, we consider an isothermal compressible fluid dynamics system which mathematically, modulo nonlinear terms, is a first-order partial differential equation coupled with the parabolic Lamé system and apply the Carleman estimate to establish the Lipschitz stability for some inverse source problem. Now we reduce (1.1) to an integro-parabolic system. Setting Here and henceforth, for the sake of more generality, assuming in (1.6) that λ, µ depend on x = (x 0 , x ′ ) ∈ Q and x 0 ∈ (−T, T ), we set L λ, µ (x, x 0 , D ′ )u = µ(x, x 0 )∆u + ( µ(x, x 0 ) + λ(x, x 0 ))∇ ′ div u We mainly consider system (1.6) with boundary (1.7). Henceforth the coefficients ρ, λ, µ, λ, µ are assumed to satisfy (1.8) ρ, λ, µ ∈ C 2 (Q), ρ(x) > 0, µ(x) > 0, λ(x) + µ(x) > 0 for x ∈ Q More precisely, the first main purpose of this paper is to establish (1) a Carleman estimate for functions without compact supports; (2) the Lipschitz stability in an inverse source problem of determining spatially varying factor of the external force F for (1.6). A Carleman estimate is an L 2 -weighted estimate of solution u to (1.6) which holds uniformly in large parameter. Carleman estimates have been well studied for single equations (e.g., Hörmander [16], Isakov [31]). A Carleman estimate yields several important results such as the unique continuation, the energy estimate called an observability inequality and the stability in inverse problems. However for systems whose principal part is coupled, for example even for isotropic Lamé system (that is, (1.6) with λ = µ ≡ 0), the Carleman estimate is difficult to be proved for functions u whose supports are not compact in Q. In particular, for (1.6), no such Carleman estimates are not known.
In order to prove the stability global in Ω for the inverse problem with lateral data limited on some subboundary, we need a Carleman estimate for functions without compact supports. Otherwise we have to observe the extra data on the whole lateral subbboundary (0, T ) × ∂Ω.
In establishing a Carleman estimate for our system (1.6) for non-compactly supported u, we should emphasize the two main difficulties: • The principal part ρ∂ x 0 − L λ,µ (x, D ′ ) is strongly coupled.
As for Carleman estimates for functions without compact supports and applications to inverse problems for the Lamé system with L λ,µ (x, D ′ ) = 0, we refer to Bellassoued, Imanuvilov and Yamamoto [3], Bellassoued and Yamamoto [4], Imanuvilov, Isakov and Yamamoto [20], Imanuvilov and Yamamoto [26] - [29]. In this paper, we modify the arguments in those papers and establish a Carleman estimate for system (1.6) where u does not have compact support. Then we apply the Carleman estimate to an inverse source problem by modifying the method in Imanuvilov and Yamamoto [22] - [24] and Beilina, Cristofol, Li and M. Yamamoto [1] discussing scalar hyperbolic equations. As for the methodology for applying Carleman estimates to inverse problems, we refer to a pioneering paper Bukhgeim and Klibanov [6], and also Beilina and Klibanov [2], Bellassoued and Yamamoto [5], Klibanov [34,35], Klibanov and Timonov [36].
For the statement of the Carleman estimate for (1.6), we need to introduce notations and a definition.
In order to prove the Carleman estimate for the viscoelastic Lamé system, we assume the existence of a real-valued function ψ which is pseudoconvex with respect to the symbols p ρ,µ (x, ξ) and p ρ,λ+2µ (x, ξ). More precisely, we can state as follows.
We identify the differential form dω f with the vector-function: Finally we introduce norms and for any positive p we introduce the norm Now we are ready to state our first main result, a Carleman estimate as follows.
Here the constant C 7 > 0 is independent of s.
The proof of Theorem 1.1 is given in Sections 2-6.
Next we apply Carleman estimate (1.21) to an inverse source problem of determining a spatially varying factor of source term of the form F(x) := R(x)f(x ′ ). Now we assume that ρ, λ, µ are independent of Let Ω ⊂ R n be a smooth bounded domain and η ∈ (0, T ) be a fixed time moment. We consider Here R(x) is an n × n matrix function and f(x ′ ) is an R n -valued function. We further assume We consider Inverse source problem. Let functions a, b, R be given and Γ is an arbitrary fixed open subset of ∂Ω. Determine function f(x ′ ) by ∂ ν u| (0,T )× Γ and u(η, ·), ∂ x 0 u(η, ·) in Ω.
Finally, as application of Carleman estimate Theorem 1.1 in a special case of λ = µ ≡ 0, we consider an inverse source problem for an isothermal compressible fluid equations (e.g., Landau and Lifshitz [38]), which can be simplified as follows: Here ρ(x) and v(x) describe the density and the velocity field at x ∈ Q respectively. Consider the following inverse problem: Let ρ(0, ·), v(0, ·) in Ω and ∂ ν v on (−T, T ) × Γ be given. Then determine a function f(x ′ ).
Then we can choose a constant C 9 independent of (ρ k , v k ) such that 2. Proof of Theorem 1.1 Sections 2-7 are devoted to the proof of Theorem 1.1 and in Section 8 we collect necessary lemmata for the proof.
Consider the following boundary value problem Here P ρ,β (x, D) = ρ∂ 2 x 0 − β∆, q is a given vector-valued function from L 2 (Q) with n 2 −n 2 + 1 components, and a vector-valued function v 1 with (n 2 − n)/2 components is defined by where g = (g 1 , . . . , g n 2 +n 2 +1 ) is a given vector-valued function and the boundary operator B(x, D ′ ) is constructed in the following way for 1 ≤ k ≤ n, and B 2 (x ′ ) is the smooth matrix constructed in the following way: Consider an n×n matrix such that on the main diagonal we have ν n (x ′ ), the nth row is (ν 1 (x ′ ), . . . , ν n (x ′ )), the first n−1 elements of the last column are −ν 1 (x ′ ), . . . , −ν n−1 (x ′ ) and all the rest elements are zero:  If ν n (x ′ ) = 0, then the determinant of such a matrix is not equal to zero. Denote the inverse to this matrix by B 3 (x ′ ) and set w : . . , ν n w 1 − ν 1 w n , . . . , ν n w n−1 − ν n−1 w n , n j=1 ν j w j ). We are not investigating the existence of solution for problem (2.1) -(2.2), but assume the existence of function v which satisfies (2.1) -(2.2) with some g. Our current goal is to establish the Carleman estimate for solution of this problem.
Without loss of generality we can assume that where C 1 is independent of s.
The end of this section, sections 2-5, and part of section 6 are devoted to the proof of Proposition 2.1.
Let O be a domain in R n .
Definition. We say that the symbol a( y, ξ, s) ∈ W k,∞ (O × R n+1 ) belongs to the class where |β| = n j=0 β j and M( ξ, s) ≥ 1; C) For any N ∈ N the symbol a can be represented as where the functions a j have the following properties: for any λ > 1 and for all ( y, ξ, s) ∈ for any multiindex β and any and ( ξ, where the term R N satisfies the estimate . For the symbol a, we introduce the semi-norm Obviously for any k ∈ {0, 1} Obviously the pseudodifferential operators with the symbols Γ ± β belongs to the class W k,∞ cl S 1,s (B(0, δ(y * ))) for any k ∈ {0, 1} and Then there exist positive constants δ(y * ), C 11 , C 12 independent of s such that for all and s ≥ s 0 we have where ǫ(δ) → +0 as δ → +0 and Proof. It suffices to prove the statement of the lemma separately for Re w ν , Im w ν . Let v ν = Re w ν or v ν = Im w ν . We write A(y, D) = n j,k=1 a jk (y)∂ 2 y k y j . For simplicity instead of the notation a β (y, ξ) we use the notation a(y, ξ) = n k,j=1 a kj (y)ξ k ξ j , where a kj = a jk for all k, j ∈ {1, . . . , n}. We set a j (y, ξ) = ∂ y j a(y, ξ), a (j) (y, ξ) = ∂ ξ j a(y, ξ), We introduce the operators We set P (y, D, s) = L 1 (y, D, s) + L 2 (y, D, s). Then Function f ν satisfies the estimate Taking the L 2 -norm of (2.31) we have The following equality is proved in [19]: Observe that the function θ(y) is independent of w and by (1.18) Hence by definition (2.8) of the function θ there exist ℓ 0 such that for all ℓ ≥ ℓ 0 and for all Then for any y 1 = (y 1,0 , . . . , y 1,n ), y 2 = (y 2,0 , . . . , y 2,n ) ∈ B(y * , δ) ∩ supp η ℓ we have Let y * = ( y * 0 , . . . , y * n ) be some point from B(y * , δ) ∩ supp η ℓ . We claim that for any positive ǫ there exists positive δ(ǫ) independent of ℓ such that Estimate (2.38) follows from the following inequality: In order to get the last two inequalities in (2.39) we used (2.37), (1.18) and (1.19).
We introduce the form G( s, ∇v ν ) in the following way: Since a kj ( y * ) = 0 for k = j by (2.38) and (1.13) there exist a positive constant C 19 such that By (2.38) for any positive ǫ one can take a positive δ(ǫ) such that We set where in the right hand side above formula ξ and ζ are given (2.42), (2.43). Observe that where F y→ ξ is the Fourier transform given by Taking the scalar product of the function where ∂ νa w = n j=1 a nj ∂ y j w. By (2.38) for any ǫ ∈ (0, 1) there exists δ(ǫ) such that Q | s|s 2 |a( y * , ∇ϕ( y * ), ∇ϕ(y * )) − a(y, ∇ϕ, ∇ϕ)||v ν | 2 dy The inequalities (2.45) and (2.46) imply that for any positive ǫ there exists a constant C 22 such that Observe that pseudoconvexity Condition 1.1 implies that there exists a positive constant C 26 such that Therefore, from (2.49) and (2.48), for some positive constant C 27 we have the inequality Thanks to (2.34) Now we estimate the derivative of the function v ν respect to variable y n . Taking the scalar product of the function L 2 (y, D, s)v ν and v ν we have Thanks to (2.52) for any positive ǫ there exists a constant C 34 independent of s such that Using (2.33), from (2.53) and (2.51) we obtain Now we estimate some integrals on the right hand side of (2.54): Integrating by parts we have Using (2.58), (2.55) and (2.56), from (2.54) we obtain Next we estimate the difference between the boundary integrals in (2.59) and 3 (2.60) From (2.59) and (2.60), we obtain (2.28).

The short computations imply and Lemma 8.2 imply
Using Lemma 8.2 and Lemma 8.1 and (2.80) we obtain .

By Proposition 8.2 we have
.
We start with the following proposition: Proposition 6.1. Let R(x, x 0 , D ′ ) be a partial differential operator with smooth coefficients of order j and κ ≥ 0. Then there exist a constant C 10 independent of s such that (6.12) e sϕ ϕ κ for all s ≥ 1. If r 0 ∈ C 1 (Σ) and p ∈ (0, 1) there exists a constant C 11 independent of s such that (6.13) e sϕ Proof. Instead of inequality (6.12) it suffices to prove (6.14) e sϕ ϕ κ The inequality (6.12) is equivalent to (6.15) e sϕ ϕ κ where g in an arbitrary function from L 2 (0, T ). Applying the Cauchy inequality we have Here in order to obtain the last inequality, we used (1.15) and (1.16).
By (1.6) and the zero Dirichlet boundary conditions for the function u we have Next we move all the terms containing the first derivatives of the function u into the righthand side, divide both sides by µ and denote the right-hand side of obtained equality as g k . Then the first n components of the function g are defined by formula: Then by (1.1), (1.2) we have . . , g n ) on Σ.
Proof of Theorem 1.3. Let functions (ρ j , v j ), j ∈ {1, 2} satisfy the equations Then, from equations (7.16)-(7.18) we have (7.19) Let y(t) = (y 1 (t), . . . , y n (t)) be a solution to the system of ordinary differential equations The curve (t, y(t)) is the characteristic curve for the hyperbolic operator Consider the initial value problem such that the initial condition ψ 0 ∈ C 3 (Ω) satisfies Existence of such a function ψ 0 is proved in [18]. Thanks to (7.21) solution ψ ∈ C 3 (Q) to the Cauchy problem (7.22) exist, unique and satisfies the estimate Then, by (7.24) and (7.23) there exist a positive δ such that Using such a function ψ we construct the weight function ϕ by formula where λ is the large positive parameter and δ is a small positive parameter, and Condition 1.2 with β = µ and β = λ+2µ. Moreover for some positive constants C 19 , C 20 (7.28) dϕ(t, y(t)) dt < −C 19 < 0 for t ∈ [0, δ) and dϕ(t, y(t)) dt > C 20 > 0 for t ∈ (−δ, 0].

Appendix
In the appendix we formulate several lemmata which were used for the proof of Theorem 1.1. Proof of these lemmata is the same as in e.g. [21] or [30].
We estimate terms I j separately. By (2.9) we have Using the definition of the set Z 1 we have We estimate the first term in this inequality. The estimate of the second one is the same