Inverse Problems for One-Dimensional Fluid-Solid Interaction Models

Communications on Applied Mathematics and Computation https://doi.org/10.1007/s42967-024-00437-3 ORIGINAL PAPER Inverse Problems for One-Dimensional Fluid-Solid Interaction Models J. Apraiz1·A. Doubova2·E. Fernández-Cara2·M. Yamamoto3 Received: 31 January 2024 / Revised: 6 May 2024 / Accepted: 4 June 2024 © The Author(s) 2024 Abstract We consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one endpoint of the spatial interval. In particular, we establish unique results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities. Keywords Burgers equation ·Fluid-solid interaction ·Free boundaries ·Inverse problems · Stability ·Uniqueness Mathematics Subject Classification 35K15 ·35R35 ·35R30 ·35B35 ·65M32 1 Introduction We will consider a nonlinear system that models the interaction of a one-dimensional fluid evolving in (−1,1)and a solid particle. It will be assumed that the velocity of the fluid is governed by the viscous Burgers equation at both sides of the point mass location y=p(t). For simplicity, it will be accepted that the fluid density is constant and equal to 1 and the solid particle has unit mass. BJ. Apraiz [email protected] A. Doubova doubov[email protected] E. Fernández-Cara [email protected] M. Yamamoto [email protected] 1Dpto. Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940 Leioa, Bizkaia, Spain 2Dpto. EDAN e IMUS, Universidad de Sevilla, Campus Reina Mercedes, 41012 Seville, Spain 3Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan 123 Communications on Applied Mathematics and Computation For any pat least in C0([0,T])satisfying |p(t)|⩽1forallt∈[0,T],let us introduce the open sets Q(p)={(x,t)∈R2:−1<x<1,x= p(t), 0<t<T}, Q(p)={(x,t)∈Q(p):x<p(t)},and Qr(p)={(x,t)∈Q(p):x>p(t)}. On the other hand, the jump of the function fat the point xwill be denoted in the sequel by [f](x), that is, [f](x):= lim s→0+f(x+s)−lim s→0−f(x+s). We will consider fluid-particle systems of the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ wt−wxx +wwx=0,(x,t)∈Q(p), w(p(t), t)=p(t), [wx](p(t), t)=p(t), t∈(0,T), w(−1,t)=α(t), w(1,t)=η(t), t∈(0,T), w(x,0)=w0(x), x∈(−1,1), p(0)=q0,p(0)=q1, (1) where (at least) w0∈L2(−1,1), α, η ∈C0([0,T]), |q0|<1, and q1∈R. Here, w(x,t)is the velocity of the fluid particle located at xat time t,p(t)is the position occupied by the particle at time t,andαand ηare Dirichlet data. It is assumed that w0, q0,andq1are initial data, respectively, for the fluid velocity, the particle position, and the particle velocity. The first condition at x=p(t)in (1) means that the velocity of the fluid and the solid mass coincide at this point. In the second condition, we state Newton’s law: the force exerted by the fluid on the particle equals the product of the particle mass and its acceleration. Thus, if we introduce the notation u:= w|Q(p)and v:= w|Qr(p),the jump condition at the points (p(t), t)can be written in the form (vx−ux)(p(t), t)=p(t), t∈(0,T). (2) The previous system can be viewed as a preliminary simplified version of other more complicated and more realistic models in higher dimensions that we plan to analyze in the future. For example, it is meaningful to consider a system governed by the Navier-Stokes equations around a moving sphere that interacts with the fluid. More precisely, let R0,R1,andRext be given with 0 <R0<Rext,and B(R0)be the open ball centered at 0 of radius R0.Set ext ={(x,t):|x|=Rext,0<t<T} and, for any R∈C0([0,T])with R0⩽R(t)⩽Rext,introduce Q(R)={(x,t):R(t)<|x|<Rext,0<t<T}, (R)={(x,t):|x|=R(t), 0<t<T}. 123 Communications on Applied Mathematics and Computation Then, it makes sense to search for functions u,p,andRsatisfying ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−νu+(u·∇)u+∇p=0,∇·u=0,(x,t)∈Q(R), u(x,t)=a(x,t), (x,t)∈ext, u(x,t)=˙ R(t) R(t)x,(x,t)∈(R), u(x,0)=u0(x), x∈B(R0), R(0)=R0,˙ R(0)=R1, (−pId.+2νDu)x R(t)=¨ R(t)x,(x,t)∈(R), where the constant ν>0 and the radially symmetric fields a=a(x,t)and u0=u0(x)are given, Id.is the identity matrix, and Dudenotes the symmetrized gradient of u,that is, Du=1 2(∇u+∇uT). A related question is whether we can determine the function Rfrom a,u0, and exterior boundary observations F=(−pId.+νDu)x Rext on (a part of) ext. This justifies the relevance of the analysis of inverse problems for (1). As far as we know, the first works where the simplified model (1) has been considered are Refs. [8,9]. There, the authors allowed the spatial variable to take any value in Rinstead of (−1,1). In particular, in Ref. [8], the authors proved the existence and uniqueness of a solution and described its large-time behavior for just one solid mass submerged in the fluid. In Ref. [9], similar results were established in the case of various rigid bodies immersed in the fluid. These results were later extended to a multi-dimensional framework in Ref. [7]. Let us also mention that the controllability properties of a system similar to (1) have been analyzed in Refs. [3,6]. Let us first identify the so called direct problem for (1). Direct problem Given the data T>0,w 0∈H1(−1,1), q0∈(−1,1), q1∈R,α ∈ C0([0,T]),andη∈C0([0,T]), find the solution (w, p)to (1). It can be shown that, if α(0)=w0(−1), η(0)=w0(1), and |q0|+|q1|+w0H1(−1,1)is sufficiently small, then there exists a solution (w, p)to (1) with w∈C0([0,T];H1(−1,1)) and p∈H2(0,T);see, for example, Ref. [6, Theorem 1.1]. In fact, the result in Ref. [6] only states that p∈C1([0,T]). However, the regularity of the restrictions of wto Q(p)and Qr(p)shows that the a.e. defined function t→[wx](p(t), t) is square-integrable and, consequently, p ∈L2(0,T). The inverse problem related to system (1) we are interested in is the following. Inverse problem Given the data T>0,q0∈(−1,1), q1∈R,α∈C0([0,T]),andthe observation βwith β(t)=wx(−1,t)for t∈(0,T), find η:= w(1,·). In what follows, we will frequently use solutions (w, p)with p∈H2(0,T), u:= w|Q(p)∈H2(Q(p)), and v:= w|Qr(p)∈H2(Qr(p)). Note that, for any such solution the traces of uand ux(and vand vx)onthecurvex=p(t) have a sense. In particular, the functions t→ux(p(t), t)and t→vx(p(t), t)belong to L2(0,T). In this paper, we will study related uniqueness and stability properties. In particular, we will give answers to questions like the following. 123 Communications on Applied Mathematics and Computation Global uniqueness Let (wi,pi)be a solution to (1) associated to some T,q0,q1,andα for i=1,2.Assume that the corresponding observations coincide at x=−1,that is, w1,x(−1,t)=w2,x(−1,t)for 0 <T1<t<T2<T.Then, do we have p1=p2 and w1=w2? Global stability Let (wi,pi)be as before and set βi:= wi,x(−1,·)and ηi=wi(1,·) for i=1,2.Is there any estimate of the kind η1−η2L∞(T1,T2)+p1−p2L∞(T1,T2)⩽φ(β1−β2L∞(0,T)) for some continuous function φ:R+→ R+satisfying lims→0+φ(s)=0? The paper is organized as follows. First, in Sect. 2, we prove a preliminary fundamental lemma that plays a key role in the proof of the conditional stability. It provides estimates of the traces on the interface x=p(t) of the difference of two solutions to (1) in terms of the boundary data and observations. In Sect. 3, we establish a stability estimate and then the uniqueness of the lateral inverse problem corresponding to the system satisfied in the left part Q(p)of the whole domain. By reflection, similar results are fulfilled by the solution to the system satisfied in Qr(p). Section 4is devoted to establishing a global stability and uniqueness result for the inverse problem in the whole domain Q(p). 2 Preliminaries As already said, the main result in this section is crucial for the proof of a local stability property that will be established in Sect. 3(see Proposition 1). Lemma 1 Let us assume that ut−uxx +aux+bu =0,(x,t)∈Q(p), u(−1,t)=α(t), ux(−1,t)=β(t), t∈(0,T)(3) with a,b∈L∞(Q(p)), α ∈H3/2(0,T), β ∈H1/2(0,T), u∈H2(Q(p)),and there exist constants M >0and δ∈(0,1)such that uH2(Q(p)) ⩽M,pH2(0,T)⩽M,and |p(t)|⩽1−δ, ∀t∈[0,T].(4) Then, (a) for any >0,there exist constants K>0and θ∈(0,1)such that |u(p(t), t)|⩽K log 1 kθ∀t∈[, T],(5) provided α, β,and k satisfy 0⩽αL2(0,T)+βL2(0,T)<k<1;(6) (b) in particular,if α≡0and β≡0in (0,T), then u ≡0in Q(p). Proof The proof of part (a) can be obtained by adapting some arguments in Ref. [10]that rely on appropriate Carleman estimates. Carleman estimates were first used in Ref. [2]to establish uniqueness and stability results for inverse problems; see also Refs. [4,5]. The argument is decomposed into three steps. 123 Communications on Applied Mathematics and Computation •Step 1 First, we introduce the change of variables y=x+1 p(t)+1,z(y,t)=u(x,t)for (x,t)∈Q(p). In this way, the values of yremain in the new spatial domain (0,1), Q(p)is transformed into Q:= (0,1)×(0,T),andzsatisfies the system ⎧ ⎨ ⎩ zt−1 (p(t)+1)2zyy +p(t) p(t)+1A(y,t)zy+1 p(t)+1B(y,t)z=0,(y,t)∈Q, z(0,t)=α(t), zy(0,t)=(p(t)+1)β(t), t∈(0,T), where A(y,t):= ax+1 p(t)+1and B(y,t):= bx+1 p(t)+1. Then, we perform a second change of variables x∗=2−2y,u∗(x∗,t)=z(y,t)for (y,t)∈Q and we now have ⎧ ⎪ ⎨ ⎪ ⎩ u∗ t−1 (p(t)+1)2u∗ x∗x∗−2p(t) p(t)+1A∗(x∗,t)u∗ x∗+1 p(t)+1B∗(x∗,t)u∗=0,(x∗,t)∈Q∗, u∗(2,t)=α(t), u∗ x∗(2,t)=−1 2(p(t)+1)β(t), t∈(0,T), (7) where Q∗:= (0,2)×(0,T). After skipping the stars in these variables, coefficients, and sets, we see that the task is reduced to prove the existence of Kand θsuch that |u(0,t)|⩽K log 1 kθ,∀t∈[, T].(8) Thus, the rest of the proof is devoted to establishing (8). •Step 2 In this step, we will prove the following intermediate estimate: |u(0,t)|⩽K0, log 1 Fθ0,∀t∈[, T],(9) where θ0∈(0,1)is independent of and F:= sup x∈[1,2]u(x,·)L2(,T)+ux(x,·)L2(,T). The plan is as follows. •First, we introduce a change of variables that reduces the task to the proof of (12). Let us assume (for instance) that t⩽T/2 and indicate our strategy. •For any small η>0,let χηbe the cutoff function in (15), where the sets Q(η) are defined in (13)–(14); see Figs. 1and 2. Then, we take vη:= ˆuχηand use a global Carleman inequality for vη.This way, we obtain (19)andthen(20). •At this point, we optimize the previous estimate with respect to s(which gives the L2 estimates (21)) and use the Sobolev interpolation and the assumptions (4) (which leads to similar L∞estimates). 123 Communications on Applied Mathematics and Computation Fig. 1 The set Q(η) when ¯ t⩽T/2 Fig. 2 The curves ψ=η, ψ=2η,andψ=3η •Then, by integrating with respect to η, we deduce an estimate in L2(0,1)of ˆu(·,T/2); using again the Sobolev interpolation the same function can be bounded in L∞(0,1)and this gives easily (12). Thus, let us fix ¯ t∈[, T]and introduce the change of variables: ˆ t=T 2¯ tt,ˆu(x,ˆ t)=u(x,t)for (x,t)∈(0,2)×(0,T). (10) Then, ˆuis defined in ˆ Q:= (0,2)×(0,T2/(2t)) and satisfies a system similar to (7) with coefficients ˆ D,ˆ A,and ˆ Bthat are uniformly bounded for t⩾>0: ˆuˆ t−ˆ D(ˆ t)ˆuxx −ˆ A(x,ˆ t)ˆux+ˆ B(x,ˆ t)u=0,(x,ˆ t)∈ˆ Q, ˆu(2,ˆ t)=ˆα(ˆ t), ˆux(2,ˆ t)=ˆ β(ˆ t), ˆ t∈(0,T2/(2t)). (11) Therefore, as already said, what we have to prove is (9)forˆ t=T/2,that is, |ˆu(0,T/2)|⩽K0, log 1 Fθ0.(12) 123 Communications on Applied Mathematics and Computation Fig. 3 The sets Q(2η), Q(3η),andQ(4η) In the sequel, we will distinguish two cases. Case 1: t⩽T/2. Let us introduce ψ(x,ˆ t):= x−ρ|ˆ t−T/2|2and ϕ(x,ˆ t):= eλψ(x,ˆ t),(13) where 4/(T−2)2<ρ<16/T2and λ>0 is sufficiently large, and the sets Q(η) := {(x,ˆ t)∈(0,2)×(0,T):0<x<1+η, ψ(x,ˆ t)>η}(14) for η∈(0,1);see Fig. 1. For any η∈(0,1/4), introduce a function χη∈C∞(R2)satisfying χη(x,ˆ t)=1 for ψ(x,ˆ t)⩾3η, χη(x,ˆ t)=0forψ(x,ˆ t)⩽2ηand, for instance, 0 ⩽χη⩽1.Then, χη(x,ˆ t)=1inQ(3η), 0inQ(η) \Q(2η);(15) see Figs. 2and 3. Moreover, |χη,x|+|χη,ˆ t|⩽C ηand |χη,xx|+|χη,xˆ t|+|χη,ˆ tˆ t|⩽C η2.(16) Then, set vη:= ˆuχη.At this point, we can use a global Carleman estimate for vηin Q(η) (see [10, Theorem 3.2]). Thus, rewrite the PDE in (11) in the form Eˆu=0.There exist s0,λ 0>0 such that, for any s⩾s0and any λ⩾λ0,one has s3λ4Q(η) e2sϕϕ3|vη|2dxdt⩽C0, Q(η) e2sϕ|H|2dxdt+C1,eCsG2 σ,(17) where C0, and C1, are constants stemming from the uniform bounds of the coefficients in (11)forallˆ t⩾>0andH:= Evη.Note that Hcan be nonzero only in Q(2η) \Q(3η). In (17), we can take G2 σ:= sup x∈[1,2]ˆu(x,·)2 L2(σ,T−σ) +ˆux(x,·)2 L2(σ,T−σ) 123 Communications on Applied Mathematics and Computation with σ=T/2−1/√ρ. Then, thanks to the choice of ρin (13) one has [σ, T−σ]⊂(, T). Without loss of generality, we assume that Gσ<1. In the sequel, we will denote by C(resp. C) a generic positive constant independent of s and (resp. independent of s). Of course, both constants can change from line to line. Taking into account the definition of Q(η) (see Fig. 3), we have from (17)that s3λ4Q(4η)∩Q(η) e2sϕϕ3|ˆu|2dxdt⩽CQ(2η)\Q(3η) e2sϕ|H|2dxdt+CeCsG2 σ (18) for any s⩾s0and any λ⩾λ0. Now, in view of (16), using the fact that ϕ⩾μ2:= e4λη in Q(4η)∩Q(η) and ϕ⩽μ1:= e3λη in Q(2η) \Q(3η), we obtain from (18)that e2sμ2Q(4η)∩Q(η) |ˆu|2dxdt⩽C η4e2sμ1Q(2η)\Q(3η) |ˆu|2+|ˆux|2dxdt+CeCsG2 σ and, therefore, Q(4η)∩Q(η) |ˆu|2dxdt⩽C η4e−2s(μ2−μ1)ˆu2 H1(Q)+CeCsG2 σ,∀s⩾s0.(19) From the definition of Gσand assumption (4), one has ˆu2 L2(Q(4η)) ⩽2ˆu2 L2(Q(4η)∩Q(η)) +2ˆu2 L2(Q(4η)\Q(η)) ⩽C η4e−2s(μ2−μ1)ˆu2 H1(Q)+CeCsG2 σ ⩽C η4e−2sλη M2+CeCsG2 σ. Thus, if we fix λ, take s=s+s0, and rename sas s,then we deduce that ˆuL2(Q(4η)) ⩽C1 η2e−sλη M+GσeCs,∀s⩾0.(20) Now, in order to get the best estimate, we minimize this quantity by choosing s=1 C+λη log Mλ CGση and we deduce that ˆuL2(Q(4η)) ⩽C MC/(C+λη) η1+C/(C+λη) Gλη/(C+λη) σfor 0 <η< 1 4.(21) Then, using the classical Sobolev embedding and an interpolation inequality, for any ∈ (1,2), we obtain ˆuL∞(Q(4η)) ⩽CˆuH(Q(4η)) ⩽Cˆu/(+1) H+1(Q(4η))ˆu1/(+1) L2(Q(4η)) ⩽CM/(+1)MC/(C+λη) η1+C/(C+λη) 1/(+1) Gλη/((C+λη)(+1)) σ 123 Communications on Applied Mathematics and Computation Fig. 4 The set Q(η) when ¯ t>T/2 whenever 0 <η<1/4.In particular, after a choice of in (1,2), one has for some a∈(1,2) independent of ηthat |ˆu(4η, T/2)|⩽C η1−a/2MGCη σ,∀η∈(0,1/4). After integration with respect to ηand a variable change x=4η, we obtain 1 0|ˆu(x,T/2)|2dx=41/4 0ˆu(4η, T/2)|2dη⩽CM2+∞ 0 1 η2−aG2Cη σdη =CM2+∞ 0 1 η2−ae−2Clog (1/Gσ)η dη=CM2(log (1/Gσ))−(a−1). Thus, using again interpolation, we see that, for any m∈(1/2,1), ˆu(·,T/2)L∞(0,1)⩽ˆu(·,T/2)Hm(0,1)⩽ˆu(·,T/2)m H1(0,1)ˆu(·,T/2)1−m L2(0,1) ⩽CMmˆu(·,T/2)1−m L2(0,1)⩽CM1 |log (1/Gσ)|θ0, where θ0=1 2(a−1)(1−m). We note that θcan be taken arbitrarily close to 0 and is independent of . It is clear that Gσ⩽F.Therefore, if ¯ t⩽T/2,we have (12), which gives (9). Case 2: ¯ t>T/2. Let us introduce again the function ψ, given by (13). In this case, we consider the sets Q(η) ={(x,ˆ t)∈(0,2)×(0,T2/(2¯ t)):0<x<1+η, ψ(x,t)>η} for 0 <η<1/4;see Fig. 4. Using once more vη=ˆuχηand arguing as before, we get s3λ4Q(4η)∩Q(η) e2sϕϕ3|ˆu|2dxdt⩽CQ(2η)\Q(3η) e2sϕ|H|2dxdt+CeCsG2 σ, (22) wherewehaveset G2 σ:= sup x∈[1,2]ˆu(x,·)2 L2(σ,T2/(2¯ t)) +ˆux(x,·)2 L2(σ,T2/(2¯ t)). 123