Some inverse problems for the Burgers equation and related systems

Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Contents lists available at ScienceDirect Communications in Nonlinear Science and Numerical Simulation journal homepage: www.elsevier.com/locate/cnsns Some inverse problems for the Burgers equation and related systems J. Apraiz a,∗, A. Doubova b, E. Fernández-Cara b, M. Yamamoto c aUniversidad del País Vasco, Facultad de Ciencia y Tecnología, Dpto. Matemáticas, Barrio Sarriena s/n 48940 Leioa (Bizkaia), Spain bUniversidad de Sevilla, Dpto. EDAN e IMUS, Campus Reina Mercedes, 41012 Sevilla, Spain cThe University of Tokyo, Japan article info Article history: Received 18 June 2021 Accepted 9 November 2021 Available online 23 November 2021 MSC: 35R30 35Q53 35G50 65M32 Keywords: Inverse problems Uniqueness Burgers equation Nonlinear systems Numerical reconstruction abstract In this article we deal with one-dimensional inverse problems concerning the Burgers equation and some related nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of the spatial interval from some appropriate boundary observations of the solution. Depending on the properties of the initial and boundary data, we prove uniqueness and non-uniqueness results. In addition, we also solve some of these inverse problems numerically and compute approximations of the interval sizes. ©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction This paper deals with some inverse problems for nonlinear time-dependent PDEs in one spatial dimension. The analysis and solution of inverse problems of many kinds has recently increased a lot because of their relevance in many applications: elastography and medical imaging, seismology, potential theory, ion transport problems or chromatography, finances, etc.; see for instance [3,9,15]. The variety of inverse problems is huge in comparison with their direct analogs and many inverse problems coming from very classical and basic direct problems wait for theoretical and numerical research. Let us mention the monographs by Bellassoued and Yamamoto [2], Isakov [13], Romanov [16] and Hasanov and Romanov [10], where many theoretical and numerical aspects of inverse problems for partial differential equations are depicted. In this paper, we consider problems related to the identification of the size of the spatial interval where a timedependent governing nonlinear equation must be satisfied. We will focus on the Burgers equation and some variants, satisfied for (x,t)∈(0, ℓ)×(0,T). We will assume that the equation is complemented with boundary and initial conditions corresponding to known data, respectively for x∈ {0, ℓ}and t=0. Then, we will try to determine the width ℓof the spatial interval from some extra information, for instance given at x=0. The main goals will be to establish or discard uniqueness ∗Corresponding author. E-mail addresses: [email protected] (J. Apraiz), [email protected] (A. Doubova), [email protected] (E. Fernández-Cara), [email protected] (M. Yamamoto). https://doi.org/10.1016/j.cnsns.2021.106113 1007-5704/©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons. org/licenses/by-nc-nd/4.0/). J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 and to compute approximations of the solutions to the inverse problems. Related questions have been analyzed recently for the linear heat and wave equations in [1]. The plan is the following. In Section 2, we consider the viscous Burgers equation under several different circumstances. Sections 3and 4respectively deal with the Burgers equation coupled to a heat equation and the variable density Burgers system. Finally, we present the results of some numerical experiments in Section 5. Throughout this paper, ∥·∥and (·,·) will stand for the usual L2norm and scalar product, respectively. In the particular case of the space L2(0, ℓ), we will sometimes write (·,·)ℓin order to make the length ℓexplicit. The symbol Cwill denote a generic positive constant. 2. Some positive and negative results for the viscous Burgers equation Let us consider the following system for the Burgers equation: ⎧ ⎨ ⎩ ut−uxx +uux=0,0<x< ℓ, 0<t<T, u(0,t)=η(t),u(ℓ, t)=0,0<t<T, u(x,0) =u0(x),0<x< ℓ. (1) The unknown u=u(x,t) can be interpreted (for example) as the velocity of the particles of a homogeneous viscous fluid in a tube where the flow is allowed only lengthwise. It can also be viewed as the car traffic density in a road in a simplified model, see for instance [14]. The main inverse problem for (1) is the following: IP-1: Fix u0=u0(x)and η=η(t)in (1) in appropriate spaces and assume that β:= ux|x=0is known. Then, find ℓ. We are first interested in proving uniqueness. More precisely, the following question is in order: Uniqueness for IP-1: Let uℓand uLbe the solutions to (1) respectively associated to the spatial intervals (0, ℓ)and (0,L). Assume that the corresponding observations uℓ x(0,·)and uL x(0,·)coincide, that is, uℓ x(0,t)=uL x(0,t)in (0,T).(2) Then, do we have ℓ=L? In the sequel, we will provide some positive and negative answers to this question, depending on the kind of imposed boundary or initial data. 2.1. The simplest cases: zero initial and/or boundary data 2.1.1. Case I: η≡ 0and u0≡0 If u0≡0, we get uniqueness: Theorem 2.1. Assume that 0< ℓ ≤L, η∈L∞(0,T)satisfies η≡ 0and u0≡0. Let uℓand uLbe the solutions to (1) respectively corresponding to ℓand L and let us assume that, for some M >0, |uℓ x(x,t)| ≤ M in (0, ℓ)×(0,T)and |uL x(x,t)| ≤ M in (0,L)×(0,T) (3) and (2) holds. Then, ℓ=L. Proof. The proof is standard. It can be achieved by contradiction, assuming that ℓ < L. Indeed, note that uℓ∈L∞((0, ℓ)× (0,T)) and uL∈L∞((0,L)×(0,T)). If we set v:= uℓ−uL, one has vt−vxx +vuℓ x+uLvx=0 in (0, ℓ)×(0,T) and also v(0,t)=0 and vx(0,t)=0 in (0,T). Consequently, from the unique continuation property of the heat equation (see [17]), we have v=0 in (0, ℓ)×(0,T). This yields uL(x,t)=0 in (ℓ, L)×(0,T) and then (again from unique continuation) uL≡0, which is an absurd. □ 2.1.2. Case II: η≡0and u0≡ 0 Let us show that, as in the case of the linear heat equation (see [1]), non-uniqueness holds in general. More precisely, a counter-example to uniqueness can be found. We will follow three steps: 1- Using the Cole–Hopf transformation (named after J.D. Cole and E. Hopf’s works [6,11], respectively), we will rewrite (1) as a system for the heat equation. 2- Then, we will prove a result similar to [1, Proposition 2.1] and we will deduce non-uniqueness for the inverse problem corresponding to the heat equation with Neumann boundary conditions. 3- Finally, coming back to the original variables, we will be able to conclude. 2 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 The Cole–Hopf transformation is given by ϕ(x,t)=Me−1 2∫x 0u(ξ,t)dξ or, equivalently, u(x,t)= −2ϕx(x,t) ϕ(x,t), ϕ(0,t)≡M,(4) where Mis a positive constant. Using (4), the Burgers system (1) can be rewritten in the form ⎧ ⎨ ⎩ ϕt−ϕxx =0,0<x< ℓ, 0<t<T, ϕx(0,t)=0, ϕx(ℓ, t)=0,0<t<T, ϕ(x,0) =ϕ0(x),0<x< ℓ, (5) where we have introduced ϕ0(x):= Me−1 2∫x 0u0(ξ)dξ. Let us denote by λnand ˜ϕn(resp. µnand ˜ ψn) the eigenvalues and eigenfunctions of the Neumann Laplacian in (0, ℓ) (resp. (0,L)). Then, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λn:= n2π2 ℓ2,n∈N∪{0}, ˜ϕn(x):= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ √2 ℓcos (nπx ℓ),n∈N, 1 √ℓ,n=0,0<x< ℓ, and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ µn:= n2π2 L2,n∈N∪{0}, ˜ ψn(x):= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ √2 Lcos (nπx L),n∈N, 1 √L,n=0,0<x<L. The solutions to (5) corresponding to ℓand Lcan be defined for all t>0. They are respectively given by ϕℓ(x,t)=∞ ∑ n=0 (ϕ0,˜ϕn)ℓe−λnt˜ϕn(x),0<x< ℓ, t>0 (6) and ϕL(x,t)=∞ ∑ n=0 (ϕ0,˜ ψn)Le−µnt˜ ψn(x),0<x<L,t>0.(7) Recall that these scalar products are respectively given by (f,g)ℓ:= ∫ℓ 0 f(x)g(x)dx and (f,g)L:= ∫L 0 f(x)g(x)dx. For any set K, let us denote by #Kthe cardinal of K. Then, the following holds: Proposition 2.2. If L/ℓ ∈Q, then there exist initial data ϕ0verifying #{n:(ϕ0,˜ϕn)ℓ= 0} = #{n:(ϕ0,˜ ψn)L= 0} = 1,(8) such that ϕℓ xx(0,t)=ϕL xx(0,t)for all t >0. Thus, we can have non-uniqueness with initial data ϕ0satisfying (8) even if |L−ℓ| is arbitrarily small. Proof. Let m0,n0∈Nbe given such that n0<m0and ℓ=n0L/m0, that is, m0/L=n0/ℓ. Let us choose k1,n1∈Nsuch that n1=k1m0/n0. Note that λk1=k2 1π2 ℓ2=n2 1π2 L2=µn1 and set ϕ0(x):= cos (k1πx ℓ)+a=cos (n1πx L)+a,x∈R,(9) 3 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 where ais a real constant. The functions in (6) and (7) corresponding to this ϕ0are the following: ϕℓ(x,t)=a+e−k2 1π2 ℓ2tcos (k1π ℓx)(10) and ϕL(x,t)=a+e−n2 1π2 L2tcos (n1π Lx).(11) Consequently, ϕℓ x(0,t)=ϕL x(0,t)=0.□ From (4),(10) and (11), we get uℓ(x,t)=2k1π ℓ e−k2 1π2 ℓ2tsin (k1π ℓx) e−k2 1π2 ℓ2tcos (k1π ℓx)+a and uL(x,t)=2n1π L e−n2 1π2 L2tsin (n1π Lx) e−n2 1π2 L2tcos (n1π Lx)+a . If ais sufficiently large, these functions are well defined, solve the Burgers systems respectively in (0, ℓ)×(0,T) and (0,L)×(0,T) for u0(x)=2k1π ℓ sin (k1π ℓx) cos (k1π ℓx)+a=2n1π L sin (n1π Lx) cos (n1π Lx)+a and, moreover, satisfy (3). This ends the proof of non-uniqueness in this case. □ 2.2. Results where η(t)≡ 0and u0(x)≡ 0 In order to prove uniqueness when both ηand u0are nonzero (and ηis sufficiently large), we need an auxiliary result concerning traces of functions in H2(0, ℓ): Lemma 2.3. Let L∗>0be given. Then ⏐⏐⏐⏐ df dx(0)⏐⏐⏐⏐≤C(L∗) ℓ3/2∥f∥H2(0,ℓ) for any f ∈H2(0, ℓ)and any ℓwith 0< ℓ ≤L∗. The proof is elementary. It can be found in [1]. Theorem 2.4. Assume that 0< ℓ ≤L≤L∗,0<T0<T, uℓ x(0,t)=uL x(0,t)in (0,T),∥u0∥L2(0,L)≤M0, |uℓ x(x,t)| ≤ M in (0, ℓ)×(T0,T)and |uL x(x,t)| ≤ M in (0,L)×(T0,T), where L∗, M0and M are some positive constants. There exists δ0>0(only depending on L∗, T0, T, M0and M) such that, if ∫T T0|η(t)|2dt ≥δ0,(12) one necessarily has ℓ=L. Proof. In this proof, we will denote by Athe one-dimensional Dirichlet Laplacian in (ℓ, L), with the associated eigenvalues 0< ζ1< ζ2<···. Let us assume that ℓ < L. Then, arguing as in the proof of Theorem 2.1, we deduce that uL(ℓ, t)=uL(L,t)=0 in (0,T).(13) Therefore, from well known energy estimates, one has ∥uL(·,t)∥L2(ℓ,L)= ∥e−tAuL(·,0)∥L2(ℓ,L)≤M0e−ζ1t∀t∈(T0,T), where ζ1is the first eigenvalue of A, that is, ζ1=π2(L−ℓ)−2. 4 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Let us put uL=uh+zfor t∈(T0,T), with uh(·,t):= e−(t−T0)AuL(·,T0). Then, we have: ∥uh(·,t)∥H2(ℓ,L)≤M0 T0 e−ζ1T0in (T0,T). On the other hand, z(·,t)=∫t T0 e−(t−s)A(uLuL x)(·,s)ds and the standard parabolic regularity estimates and the fact that |uL x| ≤ Myield: ∥z∥L2(T0,T;H2(ℓ,L)) ≤C(M)∥uL∥L2(T0,T;L2(ℓ,L)) ≤C(T,M0,M)e−ζ1T0. Therefore, ∥uL∥L2(T0,T;H2(ℓ,L)) ≤C(T,M0,M)(1+1 T0)e−ζ1T0 and, from Lemma 2.3, we get: ∥uL x(ℓ, ·)∥L2(T0,T)≤C(L∗,T,M0,M) (L−ℓ)3/2(1+1 T0)exp (−π2T0 (L−ℓ)2).(14) Maximizing the right hand side with respect to L−ℓ, we obtain: ∥uL x(ℓ, ·)∥L2(T0,T)≤1 T3/4 0(1+1 T0)C(L∗,T,M0,M). Now, we can continue exactly as in the proof of Theorem 2.7 in [1] and deduce that, if δ0is large enough, we get a contradiction. □ 3. The Burgers equation with heat effects The system is now ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−uxx +uux=kθ, 0<x< ℓ, t>0, θt−θxx +uθx=0,0<x< ℓ, t>0, u(0,t)=η(t),u(ℓ, t)=0,t>0, θ(0,t)=λ(t), θ(ℓ, t)=0,t>0, u(x,0) =u0(x), θ(x,0) =θ0(x),0<x< ℓ. (15) Here, k∈Ris given. As before, ucan be interpreted as the velocity of the fluid particles in a one-direction flow. This time, we assume that heat effects are important and, consequently, the evolution of a temperature θ=θ(x,t) must also be taken into account. We will deal with the following inverse problem: IP-2: Fix (u0, θ0)and (η, λ)in (15) in appropriate spaces and assume that β:= ux|x=0and α:= θx|x=0are known. Then, find ℓ. The uniqueness property to analyze is as follows: Uniqueness for IP-2: Let (uℓ, θℓ)and (uL, θL)be the solutions to (15) associated to the spatial intervals (0, ℓ)and (0,L), respectively. Assume that the corresponding observations (uℓ x(0,·), θℓ x(0,·)) and (uL x(0,·), θL x(0,·)) coincide, that is, uℓ x(0,t)=uL x(0,t)and θℓ x(0,t)=θL x(0,t)in (0,T).(16) Then, do we have ℓ=L? If (u0, θ0)≡(0,0), we have again uniqueness: Theorem 3.1. Assume that 0< ℓ ≤L<T, η, λ ∈L∞(0,T)satisfy (η, λ)≡ (0,0) and (u0, θ0)≡(0,0). Let (uℓ, θℓ)and (uL, θL)be the solutions to (15) respectively corresponding to ℓand L and let us assume that, for some M >0,|uℓ x|+|θℓ x| ≤ M and |uL x|+|θL x| ≤ M respectively in (0, ℓ)×(0,T)and (0,L)×(0,T), furthermore, (16) is satisfied. Then, ℓ=L. The proof is very similar to the proof of Theorem 2.1. Thus, if we assume that ℓ < Land we set v:= uℓ−uL and ψ:= θℓ−θL, it is clear from unique continuation that (v, ψ)=(0,0) in (0, ℓ)×(0,T). From energy estimates, we deduce that (uL, θL)=(0,0) in (ℓ, L)×(0,T) and finally, again from unique continuation, (uL, θL)≡(0,0), which is impossible. On the other hand, it is obvious that any solution to (1) is a particular solution to (15), corresponding to θ0≡0 and λ≡0. Consequently, the counter-example considered in Section 2.1.2 is also a counter-example to uniqueness for IP-2 when we allow u0to be nonzero. 5 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 To our knowledge, it is unknown if a counter-example to uniqueness can also be found with θ0≡ 0. As before, we can deduce a uniqueness result for (15) for large η. More precisely, the following holds: Theorem 3.2. Assume that 0< ℓ ≤L≤L∗,0<T0<T, ∥(u0, θ0)∥L2(0,L)≤M0,|uℓ x(x,t)|+|θℓ x(x,t)| ≤ M in (0, ℓ)×(T0,T), |uL x(x,t)|+|θL x(x,t)| ≤ M in (0,L)×(T0,T)and (16) holds. There exists δ1>0(only depending on L∗, T0, T, M0and M) such that, if ∫T T0|η(t)|2dt ≥δ1,(17) one necessarily has ℓ=L. Proof. It is similar to the proof of Theorem 2.4. Thus, let us assume that ℓ < L. As before, this implies uL(ℓ, t)=uL(L,t)=0 and θL(ℓ, t)=θL(L,t)=0 in (0,T). The following estimates for (uL, θL) hold: ∥uL(·,T0)∥L2(ℓ,L)=M0e−ζ1T0and ∥θL(·,T0)∥L2(ℓ,L)=M0e−ζ1T0, ∥uL∥L2(T0,T;L2(ℓ,L)) =C(T,M0)e−ζ1T0and the same hold for θL. Let us put uL=w+z, with w(·,t):= e−(t−T0)AuL(·,T0). Then ∥w(·,t)∥H2(ℓ,L)≤C T0 e−ζ1T0and z(·,t)=∫t T0 e(t−s)A(−uLuL x+kθL)(·,s)ds in (T0,T), whence ∥z∥L2(T0,T;L2(ℓ,L)) ≤C[M∥uL∥L2(T0,T;L2(ℓ,L)) +k∥θL∥L2(T0,T;L2(ℓ,L))]. Consequently, ∥uL∥L2(T0,T;H2(ℓ,L)) ≤C(T,L∗,M,M0)(1+1 T0)e−ζ1T0 and ∥uL x(ℓ, ·)∥L2(T0,T)≤C(T,L∗,M,M0) (L−ℓ)3/2(1+1 T0)e−π2T0 (L−ℓ)2. At this point, we can continue as in the proof of Theorem 2.4 and deduce that, for δ1large enough, (17) leads to a contradiction. □ It is interesting to note that, in this result, the size of λ(that is, θ|x=0) is not relevant at all. Remark 3.3. A simplified version of (15) can be obtained if we skip the transport terms. We find the linear system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−uxx =kθ, 0<x< ℓ, t>0, θt−θxx =0,0<x< ℓ, t>0, u(0,t)=η(t),u(ℓ, t)=0,t>0, θ(0,t)=λ(t), θ(ℓ, t)=0,t>0, u(x,0) =u0(x), θ(x,0) =θ0(x),0<x< ℓ (18) It is not difficult to check that the assertions on uniqueness/nonuniqueness in Section 2can be extended to this system with very similar (and in fact simpler) arguments. □ Similar inverse problems can be considered for coupled Burgers-heat systems where the heat flux is given and the temperature is observed at x=0. These are the following: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−uxx +uux=kθ, 0<x< ℓ, t>0, θt−θxx +uθx=0,0<x< ℓ, t>0, u(0,t)=u(t),u(ℓ, t)=0,t>0, θx(0,t)=χ(t), θx(ℓ, t)=0,t>0, u(x,0) =u0(x), θ(x,0) =θ0(x),0<x< ℓ (19) 6 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−uxx =kθ, 0<x< ℓ, t>0, θt−θxx =0,0<x< ℓ, t>0, u(0,t)=u(ℓ, t)=0,t>0, θx(0,t)=χ(t), θx(ℓ, t)=0,t>0, u(x,0) =u0(x), θ(x,0) =θ0(x),0<x< ℓ. (20) Now, the problems for (19) and (20) are as follows: (u0, θ0), u,χand the additional observations β:= ux|x=0 and ζ:= θ|x=0are known and, again, we try to find ℓ. The same questions above are in order. Results similar to Theorems 3.1 and 3.2 can be proved in this context. 4. The case of the variable density Burgers equation This is more interesting, but also more difficult. We consider a non-homogeneous (or variable density) one-dimensional fluid, modelled as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ(ut+uux)−uxx =0,0<x< ℓ, t>0, ρt+uρx=0,0<x< ℓ, t>0, u(0,t)=u(t),u(ℓ, t)=0,t>0, ρ(0,t)=ρ(t),t∈R+∩{t:u(t)>0}, u(x,0) =u0(x), ρ(x,0) =ρ0(x),0<x< ℓ. (21) Of course, this can be viewed as a toy model for the variable density Navier–Stokes system. The corresponding inverse problem is the following: IP-3: Fix (u0, ρ0)and (u, ρ)in (21) in appropriate spaces and assume that β:= ux|x=0and γ:= ρ|x=01{t:u(t)≤0}are known. Then, find ℓ. This is the uniqueness question we are interested in: Uniqueness for IP-3: Let (uℓ, ρℓ)and (uL, ρL)be the solutions to (21) respectively associated to (0, ℓ)and (0,L). Assume that the corresponding (uℓ x(0,·), ρℓ(0,·)) and (uL x(0,·), ρL(0,·)) coincide. Then, do we have ℓ=L? 4.1. A result for zero initial data When the initial data vanish, we have a positive uniqueness result for this problem: Theorem 4.1. Assume that 0< ℓ ≤L, T >0and (u0, ρ0)and (u, ρ)satisfy {u, ρ ∈L∞(0,T),u≡ 0, ρ ≥0, u0≡0, ρ0∈L∞(0,L), ρ0≥a0>0. Let (uℓ, ρℓ)and (uL, ρL)be the solutions to (21) for 0<t<T respectively corresponding to ℓand L. Let us assume that |uℓ t|+|uℓ x|+|ρℓ x| ≤ M and |uL t|+|uL x|+|ρL x| ≤ M respectively in (0, ℓ)×(0,T)and (0,L)×(0,T)and uℓ x(0,·)=uL x(0,·)and ρℓ(0,·)=ρL(0,·). Then, ℓ=L. For the proof, we will use a unique continuation property satisfied by the solutions to systems of the form {a(x,t)vt−vxx +b(x,t)vx+c(x,t)v+d(x,t)p=0,(x,t)∈Q, pt+m(x,t)px+r(x,t)v=0,(x,t)∈Q,(22) where we assume that Q:= (0, ℓ)×(0,T), b,c,d,m,r∈C0(Q),a∈C1(Q) and a≥a0>0 in Q.(23) More precisely, we have the following: Proposition 4.2. Assume that (23) is satisfied and (v, p)solves (22), with v, vx, vxx,p,px∈C0(Q). Also, assume that {v(0,t)=0, vx(0,t)=0,p(0,t)=0,0<t<T, v(x,0) =0,p(x,0) =0,0<x< ℓ. (24) Then, one has v≡0and p ≡0. The proof of this Proposition relies on appropriate local Carleman estimates for the solutions to (22) and is postponed to Section 4.2. 7 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Proof of Theorem 4.1.Note that uℓ∈L∞((0, ℓ)×(0,T)) and uL∈L∞((0,L)×(0,T)). If we set v:= uℓ−uLand p:= ρℓ−ρL, one has ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ρℓvt−vxx +ρℓvuℓ x+ρℓuLvx+(uL t+uLuL x)p=0,0<x< ℓ, t>0, pt+uLpx+vρℓ x=0,0<x< ℓ, t>0, v(0,t)=0, vx(0,t)=0,p(0,t)=0,t>0, v(x,0) =0,p(x,0) =0,0<x< ℓ. Consequently, vand psatisfies (22) with a=ρℓ,b=ρℓuL,c=ρℓuℓ x,d=uL t+uLuL x,m=uLand r=ρℓ xand (24). In view of Proposition 4.2, one has v=0 and p=0 in (0, ℓ)×(0,T). This yields uL(x,t)=0 in (ℓ, L)×(0,T). Since the equations satisfied by uLand ρLalso possess the unique continuation property, we find that uL≡0, which is impossible. □ It would be interesting to find nonzero initial data (u0, ρ0) such that uniqueness fails. On the other hand, it would also be interesting to prove a result similar to Theorem 3.2 asserting that, if the boundary data are large enough (with respect to the other data in the system), uniqueness is satisfied. However, to our knowledge these questions are open. A still more complex situation is found when we deal with a variable density fluid where thermal effects are relevant. For example, we can consider the variable density Boussinesq-like system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ(ut+uux)−uxx =θ, 0<x< ℓ, t>0, ρ(θt+uθx)−θxx =0,0<x< ℓ, t>0, ρt+uρx=0,0<x< ℓ, t>0, u(0,t)=u(t),u(ℓ, t)=0,t>0, ρ(0,t)=ρ(t),t∈R+∩{t:u(t)>0}, θx(0,t)=θx(ℓ, t)=0,t>0, ρ(x,0) =ρ0(x),u(x,0) =u0(x), θ(x,0) =θ0(x),0<x< ℓ. (25) This is the related inverse problem: (u0, θ0, ρ0) and (u, ρ) are given and the additional observations β:= ux|x=0and ζ:= θ|x=0are known for t∈(0,T) and we try to find ℓ. A result similar to Theorem 4.1 can also be established in this case. 4.2. Proof of Proposition 4.2 The proof of Proposition 4.2 can be obtained by combining two Carleman inequalities that can be deduced for the solutions to the first and the second equation in (22). The main steps are the following: •To choose a suitable weight function (the same in both inequalities); •To argue as in [18] and [12] deduce appropriate estimates for vand p. •Finally, to add and eliminate all undesirable terms on the right hand sides. Step 1: Let us first recall some known Carleman estimates for the solutions to equations like those in (22). Thus, assume that a,band care as in Proposition 4.2 and set Lv:= avt−vxx +bvx+cvfor any suitable v. For any λ > 0, β > 0, x0> ℓ,δ > 0 and T>0 (to be definitively fixed below), we take ϕ(x,t):= eλψ(x,t),with ψ(x,t):= |x−x0|2−4δ2β T2|t−T/2|2.(26) Note that ϕcan be used in the proof of the Carleman inequality in [18]. Consequently, the following holds: Theorem 4.3. There exists λ0>0with the following property: for any λ≥λ0, there exist constants s0=s0(λ)>0and C0=C0(λ)such that ∫∫Q(1 sϕ(|vt|2+|vxx|2)+sλ2ϕ|vx|2+s3λ4ϕ3|v|2)e2sϕdx dt ≤C0(∫∫Q|Lv|2e2sϕdx dt +∫T 0(s3λ3ϕ3|v|2+sλϕ|vx|2+|vt|2)e2sϕdt⏐⏐⏐x=0,ℓ +s2λ2eC0λ∫ℓ 0(|v|2+|vx|2)e2sϕdx⏐⏐⏐t=0,T) (27) for all s ≥s0and any v∈H2,1(Q). Now, let mbe as in (23) and let us set B:= ϕt+mϕxand Ep := pt+mpxfor any p. We can also adapt the proof of the Carleman estimate for transport equations in [4] and deduce the following result: 8 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Theorem 4.4. Assume that min(x,t)∈Q|B(x,t)| ≥ B0>0. Then, there exist constants s0>0and C >0such that s2∫∫Q|p|2e2sϕdx dt ≤C∫∫Q|Ep|2e2sϕdx dt +s∫T 0 mB|p|2e2sϕdt⏐⏐⏐ x=ℓ x=0+s∫ℓ 0 B|p|2e2sϕdx⏐⏐⏐ t=T t=0 (28) for all s ≥s0and any p ∈H1(Q). Step 2: Let us assume that t0∈(0,T) and δ > 0 is such that 0 <t0−δ < t0+δ < Tand let us set Qδ:= (0, ℓ)×(t0−δ, t0+δ). Let us introduce the new variable ˜ twith ˜ t=t0−δ+2δ Ttand the new function ˜ϕwith ˜ϕ(x,˜ t):= eλ˜ ψ(x,˜ t)and ˜ ψ(x,˜ t):= ψ(x,˜ t)≡ |x−x0|2−β|˜ t−t0|2. Then, (27) can be rewritten as an estimate in Qδ. By denoting ˜ t(resp. ˜ϕ) again by t(resp. ϕ), the following is found: ∫∫Qδ(1 sϕ(|vt|2+|vxx|2)+sλ2ϕ|vx|2+s3λ4ϕ3|v|2)e2sϕdx dt ≤C(∫∫Qδ|p|2e2sϕdx dt +K1+K2), (29) where K1:= ∫t0+δ t0−δ(s3λ3ϕ3|v|2+sλϕ|vx|2+|vt|2)e2sϕdt⏐⏐⏐x=0,ℓ ≤Cs3λ3eCλ∫t0+δ t0−δ(|v(0,t)|2+|vx(0,t)|2+|vt(0,t)|2)e2sϕ(0,t)dt +Cs3λ3eCλM2∫t0+δ t0−δ e2sϕ(ℓ,t)dt (30) and K2:= Cs2λ2eCλ∫ℓ 0(|v|2+|vx|2)e2sϕdx⏐⏐⏐t=t0−δ,t0+δ≤Cs2λ2eCλM2e2seλ(|x0|2−βδ2).(31) On the other hand, the estimate (28) applied to the second equation of (22) in Qδgives: s2∫∫Qδ|p|2e2sϕdx dt ≤C∫∫Qδ|v|2e2sϕdx dt +s∫t0+δ t0−δ mB|p|2e2sϕ⏐⏐⏐ x=ℓ x=0 dt +s∫ℓ 0 B|p|2e2sϕdx⏐⏐⏐ t=t0+δ2 t=t0−δ and we find that s2∫∫Qδ|p|2e2sϕdx dt ≤C∫∫Qδ|v|2e2sϕdx dt +R1+R2,(32) where R1:= CseCλM2∫t0+δ t0−δ|p|2e2sϕdt⏐⏐⏐ x=ℓ x=0 ≤CseCλM2∫t0+δ t0−δ|p(0,t)|2e2sϕ(0,t)dt +CseCλM4∫t0+δ t0−δ e2sϕ(ℓ,t)dt (33) and R2:= CseCλM∫ℓ 0|p|2e2sϕdx⏐⏐⏐ t=t0+δ t=t0−δ≤CseCλM3e2seλ(|x0|2−βδ2).(34) In (30),(31),(33) and (34), we have used that |v|+|vx|+|vt|+|p| ≤ Min Q. It is not restrictive to assume that M≥1. 9 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 11. Burgers equation, η=0, fixed u0(x). The computed boundary observation ux(0,·) for L1 c=5.996562049. Fig. 12. Burgers equation, η=0, fixed u0(x). The computed boundary observation ux(0,·) for L2 c=4.007345905. As before, two different situations will be analyzed for this problem. In both cases, respectively corresponding to zero initial data and nonzero initial data and sufficiently large η, we will check that uniqueness holds. Case 2.3: Burgers-heat system with (u0, θ0)=(0,0) and η= 0. We observe that this case is reduced to the Burgers single equation. We take T=5, η(t)=5 sin3tin (0,T) and (u0(x), θ0(x)) ≡(0,0). Starting from Li=1, our goal is to recover the desired value of the length Ld=2. The computed length is Lc=1.999999964, the cost is J(Lc)<10−16 is reached in the iterate 10 of the optimization algorithm. The corresponding solution to (15) is displayed in Figs. 23 and 24. The evolution of the iterates and the cost in the minimization process in the absence of the random noise appear in Figs. 25 and 26, respectively. Case 2.4: Burgers-heat system with (u0, θ0)= (0,0) and large η. We take T=5, η(t)=5 sin3tin (0,T) and u0(x)=0.1x(2 −x), θ0(x)=0.1(1 +x2(x−3)). Starting from Li=1.4, our goal is to recover the desired value of the length Ld=2. 16 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 13. Burgers equation, η=0, fixed u0(x). The computed solution corresponding to L1 c=5.998083259. Fig. 14. Burgers equation, η=0, fixed u0(x). The computed solution corresponding to L2 c=4.000601673. Fig. 15. Burgers equation with heat effect with (u0, θ0)=(0,0) and (η, λ)= (0,0) with two observations ux(0,t) and θx(0,t). The computed solution u. The computed length is Lc=2.001874913, the cost is J(Lc)<10−6is reached in the iterate 9 of the optimization algorithm. The corresponding solution to (15) is displayed in Figs. 27 and 28. The evolution of the iterates and the cost in the minimization process in the absence of the random noise appear in Figs. 29 and 30, respectively. 17 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 16. Burgers equation with heat effect with (u0, θ0)=(0,0) and (η, λ)= (0,0) with two observations ux(0,t) and θx(0,t). The computed solution θ. Fig. 17. Burgers equation with heat effect with (u0, θ0)=(0,0) and (η, λ)= (0,0) with two observations ux(0,t) and θx(0,t). The iterates in active-set algorithm. Declaration of competing interest Jone Apraiz was supported by the Spanish Government’s Ministry of Science, Innovation and Universities (MICINN), under grant PGC2018- 094522-B-I00 and the Basque Government, under grant IT12247-19. Anna Doubova and Enrique Fernández-Cara were partially supported by MICINN, under grant PID2020-114976GB-I00. Masahiro Yamamoto was supported by Grant-in-Aid for Scientific Research (A) 20H00117 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303). This work has been supported by the RUDN University Strategic Academic Leadership Program. Acknowledgments The first author was supported by the Spanish Government’s Ministry of Science, Innovation and Universities (MICINN), under grant PGC2018-094522-B-I00 and the Basque Government, under grant IT12247-19. The second and third authors were partially supported by MICINN, under grant PID2020-114976GB-I00. Lastly, the fourth author was supported by Grant-in-Aid for Scientific Research (A) 20H00117 of Japan Society for the Promotion of Science and by The National Natural Science Foundation of China (no. 11771270, 91730303). This work has been supported by the RUDN University Strategic Academic Leadership Program. 18 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 18. Burgers equation with heat effect with (u0, θ0)=(0,0) and (η, λ)= (0,0) with two observations ux(0,t) and θx(0,t). Evolution of the cost. Fig. 19. Burgers equation with heat effect with (u0, θ0)= (0,0) and large (η, λ) with two observations ux(0,t) and θx(0,t). The computed solution u. Fig. 20. Burgers equation with heat effect with (u0, θ0)= (0,0) and large (η, λ) with two observations ux(0,t) and θx(0,t). The computed solution θ. 19 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 21. Burgers equation with heat effect with (u0, θ0)= (0,0) and large (η, λ) with two observations ux(0,t) and θx(0,t). The iterates in active-set algorithm. Fig. 22. Burgers equation with heat effect with (u0, θ0)= (0,0) and large (η, λ) with two observations ux(0,t) and θx(0,t). Evolution of the cost. Fig. 23. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The computed solution u. 20 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 24. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The computed solution θ. Fig. 25. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The iterates in active-set algorithm. Fig. 26. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). Evolution of the cost. 21 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 27. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The computed solution u. Fig. 28. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The computed solution θ. Fig. 29. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). The iterates in active-set algorithm. 22 J. Apraiz, A. Doubova, E. Fernández-Cara et al. Communications in Nonlinear Science and Numerical Simulation 107 (2022) 106113 Fig. 30. Burgers equation with heat effect with (u0, θ0)=(0,0) and η= 0 with one observation ux(0,t). Evolution of the cost. References [1] Apraiz J, Cheng J, Doubova A, Fernández-Cara E, Yamamoto M. 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