Hölder stability in type III thermoelastodynamics

H¨older stability in Type III thermoelastodynamics M.C. Leseduarte and R. Quintanilla Matem`atica Aplicada 2, ETSEIAT, Universitat Polit`ecnica de Catalunya Colom, 11. Terrassa (08222). Barcelona. Spain E-mail addresses: [email protected], [email protected] To G.A. Maugin in his 70th birthday Abstract This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We here assume that the mass density is positive and the thermal conductivity tensor is positive definite. However, we do not assume the positivity of any other tensor. In this situation, we obtain H¨older continuous dependence results on the supply terms. We also sketch how to prove the continuous dependence on the initial data. keywords: Type III thermoelastodynamics, H¨older stability, Continuous dependence on initial data and supply terms, Lagrange identities method 1 Introduction The thermoelastic theory proposed by Green and Naghdi [5–7] has deserved an intense investigation in the last years. Three sub-theories, where an entropy balance law replaces the customary entropy inequality, have been proposed. These theories were labelled as Type I, II and III. The linear version of Type I agrees with the usual classical theory of thermoelasticity. For Type II, the energy of the system is constant with respect to the time. For this reason it is also known as “thermoelasticity without energy dissipation”. Type III is the more general theory and it contains the other two as limiting cases. We can recall several papers (see Iesan [8,9]; Iesan and Quintanilla [10]; Lazzari and Nibbi [13]; Leseduarte et al. [14,15]; Liu and Quintanilla [16,17]; Liu and Lin [18]; Messaoudi and Soufyane [19]; Puri and Jordan [20]; Qin et al. [21]; Quintanilla [22–27]; Quintanilla and Racke [28]; Quintanilla and Straughan [29–31]; Yang and Wang [34], among others), where existence, uniqueness, continuous dependence, spatial and time behavior have been studied. We recall that the linear system for the centrosymmetric Type III thermoelastodyamics can be written as (ρ¨ui=aijkhuk,h −aijθ,j +ρfi, c˙ θ=−aij ˙ui,j +kijθ,i,j +bijα,i,j +ρr, (1.1) where uiis the displacement vector, θis the temperature, ρis the mass density, cis the thermal capacity, (aijkh) is the elasticity tensor, aij is the coupling tensor, kij is the thermal 1 conductivity tensor and bij denotes a tensor which is typical for the Types II and III theories. The constitutive tensors ρ,c,aijkh,aij,bij and kij are smooth functions of the position. They satisfy the symmetries aijkh =akhij (1.2) and kij =kji, bij =bji.(1.3) The thermal displacement αis defined by α(x x x, t) = Zt 0 θ(x x x, s)ds +α0(x x x),(1.4) and fiand rare the supply terms. To guarantee the well-posedness of the Type III thermoelasticity, we need to assume that the mass density and the heat capacity are positive as well as the elasticity tensor and the thermal conductivity. Lyapunov stability of the solutions is implied when bij is also positive definite. In the case where we do not assume the positivity of the elasticity tensor aijkh, the problem determined by the system (1.1) with usual initial and boundary conditions becomes ill-posed. It is worth recalling that this condition can be present in the case of prestressed solids. It is worth noting several references for this situation. For instance, results concerning uniqueness and growth of solutions have been obtained at [25, 29] under the condition that the tensors kij and bij are positive. It is also worth mentioning that the only contribution concerning the case when aijkh and bij are not definite was obtained recently (see [15]). There, it was showed the uniqueness of solutions. We here want to show how to obtain the continuous dependence with respect to the supply terms and initial data under similar restrictions. In 1960, John [11] showed how to obtain continuous dependence results in the sense of H¨older. This concept is weaker than the usual definition of the continuous dependence. The basic idea consists to impose that the solutions belong to a suitable constraint class. Since this contribution, many investigations have been directed to this kind of results. We may cite the works of Ames and Payne [1] and Knops and Payne [12] concerning isothermal elastodynamics. We can also recall that H¨older stability results can be found in different frameworks from the thermoelasticity (see Cimmelli and dell’Isola [3] and dell’Isola [4]). For the classical thermoelastodynamics, we may recall the contributions of Wilkes [33], Ames and Straughan [2] and Rionero and Chirita [32]. It is worth recalling and comparing several related results. In [15], we proposed a uniqueness result under similar assumptions to the ones considered here and in [24] the author proposed a H¨older stability result by using the logarithmic convexity argument under the assumption that bij is positive definite. We here obtain a new result proving the H¨older stability of the solutions when we do not assume that the elasticity tensor neither the bij tensor are positive definite. It is known that two usual techniques to study ill-posed problems in thermoelasticity are the logarithmic convexity and the Lagrange identities method. We here consider the second one. We want to emphasize that since the tensor bij is not positive definite, it does not seem possible to apply the logarithmic convexity arguments. However, we here are able to use the Lagrange identities method. In fact, Lagrange identity method allows us to obtain equality (3.5) that implies the inequality (3.12). This inequality is the starting point to develop our approach. Therefore, Lagrange identity is the fundamental ingredient to obtain our results. 2 The plain of this note is the following. In the next section we recall the basic assumptions and the conditions defining the problem. A basic inequality is obtained in Section 3. H¨older’s continuous dependence with respect to the supply terms is obtained in Section 4. In the last section we sketch how to extend the argument to prove continuous dependence with respect to the initial data. 2 Preliminaries In this section we propose the basic assumptions where we are going to work with. We study smooth solutions of (1.1) on B×I, where Iis a bounded time interval and Bis a regular domain with boundary Γ smooth enough to apply the Divergence Theorem. In what follows we suppose that the constitutive tensors are bounded about and have the following properties: (A1) The mass density ρand the heat capacity care positive functions. That is, ρ(x x x)≥ρ0>0, c(x x x)≥c0>0, x x x∈B. (A2) The thermal conductivity tensor kij is positive definite. That is, there exists a positive constant Csuch that kijξiξj≥Cξiξi,(2.1) for every vector (ξi). The physical meaning of condition (A1) is obvious. Condition (A2) guarantees that the dissipation of the system is not negative and then, the energy does not increase. It is an usual assumption in the thermomechanical studies. We also note that from (2.1), the following inequality |bijξiξj| ≤ C1kijξiξj(2.2) is satisfied for every vector (ξi), where C1is a calculable constant1which depends on the tensors kij and bij. To the field equations we adjoin the boundary conditions ui(x x x, t) = ¯ui(x x x, t), α(x x x, t) = ¯α(x x x, t), x x x∈Γ, t ∈I, (2.3) together with the initial conditions ui(x x x, 0) = u0 i(x x x),˙ui(x x x, 0) = v0 i(x x x), α(x x x, 0) = α0(x x x),˙α(x x x, 0) = θ0(x x x), x x x∈B. (2.4) To study the continuous dependence of the solutions with respect to the supply terms, we denote by u(1) i, α(1)the solution corresponding to the external data f(1) i, r(1)and by u(2) i, α(2), the solution corresponding to the supply terms f(2) i, r(2). We introduce the notation ui=u(2) i−u(1) i, α =α(2) −α(1), Fi=f(2) i−f(1) i, q =r(2) −r(1).(2.5) 1We note that C1=b∗/k∗, where b∗is the maximum of the absolute values of the eigenvalues of the matrix bij and k∗is the minimum of the eigenvalues of kij . 3 It follows that (ui, α) satisfies the problem determined by the system (ρ¨ui=aijkhuk,h −aijθ,j +ρFi, c˙ θ=−aij ˙ui,j +kijθ,i,j +bijα,i,j +ρq, (2.6) with the homogeneous boundary conditions ui(x x x, t) = α(x x x, t)=0, x x x∈Γ, t ∈I, (2.7) and the null initial conditions ui(x x x, 0) = ˙ui(x x x, 0) = α(x x x, 0) = ˙α(x x x, 0) = 0, x x x∈B. (2.8) In our studies it will be useful the following inequality Zt 0 kijα,iα,j ds ≤4C2 2t2 π2Zt 0 kij ˙α,i ˙α,j ds, (2.9) which is satisfied for every function αsuch that α(0) = 0 and where C2is a calculable constant2. 3 A basic inequality In this section we obtain an inequality satisfied by the solutions of the problem defined by the system (2.6) with boundary and initial conditions (2.7) and (2.8), respectively. This inequality will be relevant to obtain our results. As we assume null Dirichlet boundary conditions and null initial conditions, we obtain that the energy equality ZBρ˙ui˙ui+cθ2+aijkhui,juk,h +bijα,iα,jdv + 2 Zt 0ZB kijθ,iθ,j dv ds −2Zt 0ZB (ρFi˙ui+ρqθ)dv ds = 0 (3.1) is satisfied for every solution of our problem. On the other hand, from the equalities d ds [ρ˙ui(s) ˙ui(2t−s)] = ρ¨ui(s) ˙ui(2t−s)−ρ˙ui(s)¨ui(2t−s),(3.2) d ds [cθ(s)θ(2t−s)] = c˙ θ(s)θ(2t−s)−cθ(s)˙ θ(2t−s) (3.3) and the null initial and boundary conditions, we obtain ZBρ˙ui˙ui+bijα,iα,j −aijkhui,juk,h −cθ2dv =Zt 0ZB ρ(Fi(s) ˙ui(2t−s)−Fi(2t−s) ˙ui(s)) dv ds −Zt 0ZB ρ(q(s)θ(2t−s)−q(2t−s)θ(s)) dv ds. (3.4) 2We note that C2 2=k∗/k∗, where k∗is the maximum of the eigenvalues of the symmetric matrix kij and k∗ is the minimum of the eigenvalues of kij . 4 From the equality (3.1) and the relation (3.4), we conclude that ZB (ρ˙ui˙ui+bijα,iα,j)dv +Zt 0ZB kijθ,iθ,j dv ds =Zt 0ZB (ρFi˙ui+ρqθ)dv ds +1 2Zt 0ZB (ρFi(s) ˙ui(2t−s)−ρFi(2t−s) ˙ui(s)) dv ds +1 2Zt 0ZB (ρq(2t−s)θ(s)−ρq(s)θ(2t−s)) dv ds. (3.5) We now consider several estimates: Zt 0ZB [ρFi˙ui+ρqθ]dv ds ≤Zt 0ZB ρFiFidv ds1/2Zt 0ZB ρ˙ui˙uidv ds1/2 +Zt 0ZB ρq2dv ds1/2Zt 0ZB ρθ2dv ds1/2 ≤Zt 0ZBρFiFi+ρq2dv ds1/2Zt 0ZBρ˙ui˙ui+ρθ2dv ds1/2 , (3.6) where we have used the inequality √a√b+√c√d≤√a+c√b+d. (3.7) In a similar way, we have that Zt 0ZB [ρFi(s) ˙ui(2t−s)−ρFi(2t−s) ˙ui(s)] dv ds ≤Zt 0ZB ρFiFidv ds1/2Z2t tZB ρ˙ui˙uidv ds1/2 +Z2t tZB ρFiFidv ds1/2Zt 0ZB ρ˙ui˙uidv ds1/2 ≤Z2t 0ZB ρFiFidv ds1/2Z2t 0ZB ρ˙ui˙uidv ds1/2 . (3.8) We can also obtain that Zt 0ZB [ρq(2t−s)θ(s)−ρq(s)θ(2t−s)] dv ds ≤Z2t 0ZB ρq2dv ds1/2Z2t 0ZB ρθ2dv ds1/2 . (3.9) Therefore, we see that ZB (ρ˙ui˙ui+bijα,iα,j)dv +Zt 0ZB kijθ,iθ,j dv ds ≤3 2Z2t 0ZBρFiFi+ρq2dv ds1/2Z2t 0ZBρ˙ui˙ui+ρθ2dv ds1/2 . (3.10) 5 Let us assume that sup t∈[0,T]ZBρ˙ui˙ui+ρθ2dv ≤N2 1(3.11) and that t≤T/2. We obtain that the inequality ZB (ρ˙ui˙ui+bijα,iα,j)dv +Zt 0ZB kijθ,iθ,j dv ds ≤3 2T1/2N1ZT 0ZBρFiFi+ρq2dv ds1/2(3.12) is satisfied for every t≤T/2. Estimate (3.12) is fundamental in our approach. However, as we do not assume that bij is a positive definite matrix, we do not have a positive quadratic form on the left side of the estimate. Thus, we will need to manipulate this term to work with a positive quadratic form. This will be the main aim of the next sections. 4 The main result The aim of this section is to obtain an estimate for the solutions of the problem determined by (2.6), (2.7) and (2.8). From the inequality (3.12) and the use of the Poincar´e type inequality (2.9) we will be able to get such kind of estimate. We are going to decompose the interval [0, T/2] in a finite sequence of subintervals I0,I1, I2, . . . , Imsuch that [0, T/2] ⊂ m [ j=0 Ijand such that each intersection Ii∩Ii+1 has a unique element, denoted by {ti}, for i= 1, . . . , m −1. Then, we will obtain that ZB ρ˙ui˙uidv +1 2Zt 0ZB kijθ,iθ,j dv ds ≤En,(4.1) whenever t∈In∩[0, T/2] where the estimate Enis defined by the recurrence E0=3 2T1/2N1Zt 0ZBρFiFi+ρq2dv ds1/2 (4.2) and En+1 =16C1C2T πEn+3 2T1/2N1Zt 0ZBρFiFi+ρq2dv ds1/2 .(4.3) Our first step is to obtain that ZB ρ˙ui˙uidv +1 2Zt 0ZB kijθ,iθ,j dv ds ≤E0,(4.4) whenever t∈I0=h0,π 16C1C2i. 6 From (2.2), the arithmetic-geometric mean inequality and (2.9), we know that ZB bijα,iα,j dv +Zt 0ZB kijθ,iθjdv ds ≥ −C1ZB kijα,iα,j dv +Zt 0ZB kijθ,iθ,j dv ds =−2C1Zt 0ZB kijα,iθ,j dv ds +Zt 0ZB kijθ,iθ,j dv ds ≥ −2C1Zt 0ZB kijα,iα,j dv ds1/2Zt 0ZB kijθ,iθ,jdv ds1/2 +Zt 0ZB kijθ,iθ,j dv ds ≥1−4C1C2t πZt 0ZB kijθ,iθ,jdv ds. (4.5) If we assume that t < π 16C1C2 , we see that ZB bijα,iα,j dv +Zt 0ZB kijθ,iθ,jdv ds ≥1 2Zt 0ZB kijθ,iθ,jdv ds. (4.6) Thus, in view of the estimate (3.12), we obtain the inequality (4.4). Let us assume that we have obtained an estimate of the type ZB ρ˙ui˙uidv +1 2Zt 0ZB kijθ,iθ,j dv ds ≤En,(4.7) for nπ 16C1C2 < t < (n+ 1)π 16C1C2 <T 2, and we want to get a similar bound for (n+ 1)π 16C1C2 < t < min (n+ 2)π 16C1C2 ,T 2. The analysis starts by considering the relations ZB ρ˙ui˙uidv +Zt 0ZB kijθ,iθ,j dv ds =ZB ρ˙ui˙uidv +ZB bijα,iα,j dv +Zt 0ZB kijθ,iθ,j dv ds −ZB bijα,iα,j dv ≤3 2T1/2N1ZT 0ZBρFiFi+ρq2dv ds1/2 +C1ZB kijα,iα,j dv. (4.8) Here we have applied the estimates (2.2) and (3.12). We note that C1ZB kijα,iα,j dv ≤2C1ZB kij (α,i(t)−α,i(tn+1)) (α,j(t)−α,j(tn+1)) dv + 2C1ZB kijα,i(tn+1)α,j(tn+1)dv, (4.9) 7 where tn+1 =(n+ 1)π 16C1C2 . We have that C1ZB kijα,iα,j dv ≤4C1Zt tn+1 ZB kij (α,i(s)−α,i(tn+1)) θ,j dv ds + 4C1Ztn+1 0ZB kijα,iθ,j dv ds ≤4C1Zt tn+1 ZB kij (α,i(s)−α,i(tn+1)) (α,j(s)−α,j(tn+1)) dv ds1/2Zt tn+1 ZB kijθ,iθ,j dv ds1/2 + 4C1Ztn+1 0ZB kijα,iα,j dv ds1/2Ztn+1 0ZB kijθ,iθ,j dv ds1/2 ≤8C1C2(t−tn+1) πZt tn+1 ZB kijθ,iθ,j dv ds +8C1C2tn+1 πZtn+1 0ZB kijθ,iθ,j dv ds. (4.10) From (4.8) and (4.9), it follows that ZB ρ˙ui˙uidv +Zt 0ZB kijθ,iθ,j dv ds ≤3 2T1/2N1ZT 0ZBρFiFi+ρq2dv ds1/2 +8C1C2(t−tn+1) πZt 0ZB kijθ,iθ,j dv ds +16C1C2tn+1 πEn. (4.11) If we assume that t−tn+1 <π 16C1C2 , we obtain the desired estimate (4.1)–(4.3). We note that from the recurrence (4.3), we have Em≤16C1C2T π16C1C2T πEm−2+E0+E0 ≤ ··· ≤ 16C1C2T πm E0+16C1C2T πm−1 E0+···+E0. (4.12) Hence, Em≤"m X k=0 16C1C2T πk#E0.(4.13) Thus, we can conclude that the estimate ZB ρ˙ui˙uidv +1 2Zt 0ZB kijθ,iθ,j dv ≤3 2T1/2N1"m X k=0 16C1C2T πk#ZT 0ZBρFiFi+ρq2dv ds1/2(4.14) is satisfied whenever mis the first natural number such that m > 8C1C2T π. Therefore, we have proved the following result. 8 Theorem 4.1 Let u(1) i, α(1)and u(2) i, α(2)be the solutions of the system (1.1) corresponding to the supply terms f(1) i, r(1)and f(2) i, r(2), respectively. Then, the difference denoted by (2.5) satisfies the estimate ZB ρ˙ui˙uidv +1 2Zt 0ZB kijθ,iθ,j dv ≤3 2T1/2N1     1−16C1C2T πm+1 1−16C1C2T π     ZT 0ZBρFiFi+ρq2dv ds1/2 , (4.15) where mis the first natural number such that m > 8C1C2T/π. 5 Continuous dependence on initial data The analysis proposed in Section 4 can be adapted to study the stability with respect to the initial data. Let us assume that we have two solutions u(1) i, α(1)and u(2) i, α(2)to the homogeneous version (fi= 0, r= 0) of the system (1.1) with the same boundary conditions, but with different initial conditions. Using the notation proposed previously, we denote by (ui, α) the solution of the homogeneous version of the system (2.6) with homogeneous boundary conditions (2.7) and the initial conditions ui(x x x, 0) = u(2) i(x x x, 0) −u(1) i(x x x, 0) = u∗ i(x x x), ˙ui(x x x, 0) = ˙u(2) i(x x x, 0) −˙u(1) i(x x x, 0) = v∗ i(x x x), α(x x x, 0) = α(2)(x x x, 0) −α(1)(x x x, 0) = α∗(x x x), ˙α(x x x, 0) = ˙α(2)(x x x, 0) −˙α(1)(x x x, 0) = θ∗(x x x). (5.1) In this case, the energy equation gives E(t) = ZBρ˙ui˙ui+cθ2+aijklui,juk,l +bijα,iα,jdv + 2 Zt 0ZB kijθ,iθ,j dv ds =E(0),(5.2) where E(0) = ZBρv∗ iv∗ i+c(θ∗)2+aijklu∗ i,ju∗ k,l +bijα∗ ,iα∗ ,jdv. (5.3) As we consider the homogeneous system, the Lagrange identities argument implies that ZBρ˙ui˙ui+bijα,iα,j −aijklui,juk,l −cθ2dv =ZBρv∗ i˙ui(2t) + bijα∗ ,iα,j(2t)−aijklu∗ i,juk,l(2t)−cθ∗θ(2t)dv. (5.4) From (5.2) and (5.4), we obtain that ZB (ρ˙ui˙ui+bijα,iα,j)dv +Zt 0ZB kijθ,iθ,j dv ds =E(0) 2+1 2ZBρv∗ i˙ui(2t) + bijα∗ ,iα,j(2t)−aijklu∗ i,juk,l(2t)−cθ∗θ(2t)dv. (5.5) 9