In this paper, we investigate the decay rate of stabilization of the solution of the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as a function of the order parametr. We prove that for appropriate constants, which appear in the above-mentioned model, we can decide upon the exponencial decrease of the solution to its attractor for time tending to infinity

Vodák, Rostislav

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On a decay rate for a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys

Weierstra-Institutfur Angewandte Analysis und Stochastikim Forschungsverbund Berlin e.V.Preprint ISSN 0946 { 8633On a decay rate for a Landau-Ginzburg systemwith viscosity for martensitic phase transitions inshape memory alloysRostislav Vodak1submitted: 27th June 20031Weierstrass Institutefor Applied Analysisand StochasticsMohrenstrasse 3910117 BerlinGermanyemail: vodak@wias-berlin.deNo. 856Berlin 2003WIAS2000 Mathematics Subject Classication. 35Q72, 73B30, 35B40.Key words and phrases. nonlinear thermo-viscoelasticity, shape memory alloys, phase transi-tions, asymptotic behaviour, Landau-Ginzburg theory.Supported by the DFG Research Center \Mathematics for key technologies" (FZT 86) inBerlin.Edited byWeierstra-Institut fur Angewandte Analysis und Stochastik (WIAS)Mohrenstrae 39D | 10117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractIn this paper, we investigate the decay rate of stabilization of the solu-tion of the system of partial dierential equations governing the dynamics ofmartensitic phase transitions in shape memory alloys under the presence of aviscous stress. The corresponding free energy is assumed in Landau-Ginzburgform and nonconvex as a function of the order parameter. We prove that forappropriate constants, which appear in the above-mentioned model, we candecide upon the exponential decrease of the solution to its attractor for timetending to innity.1 IntroductionIn the present paper, we continue the study of the asymptotic behaviour of thesolutions to a system that arises in the thermo-mechanical developments in a one-dimensional heat-conducting viscous solid of constant mass density  (assumed  =1). The solid is subjected to heating and loading. We think of metallic solids thatnot only respond to a change of the strain  by an elastic stress  = (), but alsoto a change of the curvature of their metallic lattice by a couple stress  = (x).We assume that the Helmholtz free energy density F is a potential of Landau-Ginzburg form, i.e.F = F (; x; ); (1.1)where  denotes the absolute temperature. To cover systems modelling rst-order,stress-induced and temperature-induced solid-solid phase transitions accompaniedby hysteresis phenomena, we do not assume that F is a convex function of the orderparameter .A particular class of materials, in which both temperature-induced and stress-induced rst-order phase transitions leading to a rather spectacular hysteretic be-haviour occur, are the so-called shape memory alloys. For an account of the physicalproperties of shape memory alloys, we refer the reader to Chapter 5 in the mono-graph [1]. The model we investigate has the following formutt  uxxt @@x(f1(ux) + f2(ux)) + Æuxxxx= 0; (1.2)CVt  kxx  f1(ux)uxt  u2xt= 0 (1.3)for (x; t) 2 (0; 1) (0;1), with positive physical constants i, i = 1; 2; 3, k, , Æ, 1,and CV, and wheref1(z) = F01(z); with F1(z) = 1z2;(1.4)f2(z) = F02(z); with F2(z) = 3z6  2z4  11z2:1The system is endowed with the boundary conditionsujx=0= uxxjx=0= 0; uxjx=1= (uxt  Æuxxx+ 1)jx=1= 0; (1.5)with1= f1(ux) + f2(ux); (1.6)andxjx=0;1= 0: (1.7)We prescribe the initial state by the initial datau(x; 0) = u0(x); ut(x; 0) = u1(x); (x; 0) = 0(x) > 0; x 2 [0; 1]: (1.8)We can divide the existing literature on the system (1.2)-(1.7) into three parts.The rst part covers the case when  > 0 and Æ > 0. For the existence of thesolution of the model (1.2)-(1.7), we refer the reader to Homann and Zochowski[5]. Sprekels, Zheng and Zhu in [10] investigated the asymptotic behaviour of thesolution of the system (1.2)-(1.7). In [9], the existence of the maximal compactattractor was derived. The second case Æ = 0 and  > 0 was studied in Racke andZheng [7] from the point of view of the global existence, uniqueness and asymptoticbehaviour of weak solutions if both ends of the rod are insulated and if at least oneend is stress-free. The third part covers the case when Æ > 0 and  = 0. For theglobal existence and uniqueness, we refer the reader to Sprekels and Zheng [8]. Wealso refer to the works [2] and [3].But one question still unanswered is whether we can obtain additional informationabout the decay rate for the model (1.2)-(1.7). The main theorem of this paperprovides an answer for the case Æ > 0 and  > 0. But in Section 4, we give suÆcientconditions that also cover the case Æ = 0. The main result of this paper is thefollowing.Theorem 1.1 Suppose that the solution u of the system (1:2)-(1:7) satises thecompatibility conditions utjx=0= (u0)xxjx=0= uxtjx=1= 0. Let the constants 1, Æ,i, i = 1; 2; 3, CV, k,  be positive. In addition, let there exist constants  > 0 andK > 1 such that +k(t)k1K+12K<2; + 11+223<Æ2(1.9)and4223+2k(t)k2121+ 2k(t)  1k11+ k(t)k41221K< Æ (1.10)for a.a. t 2 (0;1). Thenkut(t)k22+ kuxx(t)k22+ k(t)  (t)k22 k1e k2t; (1.11)where the constants k1, k2> 0 only depend on the initial state.2Remark 1.2 From the main result in [10] it may be concluded that kx(t)k2! 0for t ! 1 and kkL1(0;1;L1(0;1)) c0. Combining these two facts, we can assertthatk(t)k1 kx(t)k1+ kkL1(0;1;L1(0;1)) c1(t) + c0 c2; (1.12)where the constant c0only depends on the initial state, and where c1(t) ! 0 fort ! 1. Moreover, the constant c2is of the form c2= c(1 + 21), which is aconsequence of Lemma 2.5 in [10].It is worth pointing out that the method used here also provides similar propertiesfor the solutions to the 1-D Navier-Stokes equations cf. [11].In Section 2, we will look more closely at the energy conservation of the equations(1.2)-(1.7). Section 3 is devoted to the proof of our main result. Section 4 providesother conditions which lead to an exponential decrease in connection with the paper[4].The notation in this paper will be as follows: Lp(0; 1), 1  p  1, and Lp(0; T ;Lq(0; 1)),1  p; q  1, 0 < T  1, respectively, denote the standard Lebesgue and Bochnerspaces. By k  kpand k  kLp(0;T ;Lq(0;1)), we denote the corresponding norms. We alsouse the denotation (t) for the integralR10(x; t) dx.2 Energy inequalitiesIn this section, we proceed with the study of the energy identity, and we derive newforms of the energy inequality for the solution to the system (1.2)-(1.7).Lemma 2.1 Let the couple (; u) be the classical solution to the model (1:2)-(1:7).Thenddt"Z10 u2t2+Æu2xx2+ F2(ux) + CV!dx#= 0; (2.13)ddt"Z10 u2t2+Æu2xx2+ F2(ux)!dx# k(t)k21221Z10u2xdx+2Z10u2xtdx  0; (2.14)andddt"Z10 u2t2+Æu2xx2+ F2(ux) +CV(   )22K!dx#+kKZ10(   )2xdx 1KZ10f1(ux)(   )uxtdx KZ10u2xt(   ) dx k(t)k21221Z10u2xdx+2Z10u2xtdx  0: (2.15)3P r o o f: To deduce (2.13), take utas a test function for the equation (1.2). Thisgivesddt"Z10 u2t2+Æu2xx2+ F2(ux)!dx#+Z10f1(ux)uxtdx + Z10u2xtdx = 0 (2.16)because the boundary terms satisfy the identity [uxtut]10  [f1(ux)ut+ f2(ux)ut]10+ Æ[uxxxut]10  Æ[uxxuxt]10= 0:This is a consequence of (1.5)-(1.8). Integrating (1.3), and adding this equation tothe identity (2.16), we get (2.13).Now, we can derive the estimateZ10f1(ux)uxtdx=Z1021uxuxtdx 2k(t)k1Z10j1uxuxtj dx k(t)k21221Z10u2xdx +2Z10u2xtdx: (2.17)Combining (2.16) and (2.17), we deduce (2.14).Using Kas a test function in (1.3) yields the identityddtZ10CV(   )22Kdx+kKZ10(   )2xdx 1KZ10f1(ux)(   )uxtdx KZ10u2xt(   ) dx = 0 (2.18)for some K. After adding to (2.14) we conclude (2.15). 2From Lemma 2.1, we obtain the following result.Consequence 2.1 Under the assumptions of Theorem 1.1, we havekutk2L1(0;1;L2(0;1))+ kkL1(0;1;L1(0;1)) c0; (2.19)as well as Æ2  11 223!kuxxk2L1(0;1;L2(0;1))+ kF3(ux)kL1(0;1;L1(0;1)) c0; (2.20)where the constant c0only depends on the initial state, and where the function F3(z),F3(z)  0, is dened byF3(z) :=8<:3z6  2z4if jzj q233z6if jzj <q23:(2.21)4P r o o f: This is an obvious consequence of (2.13) and the inequality kuxk2 kuxxk2,because we can easily deduce thatF2(zi) = 0 for z1=s23; z2=  s23; z3= 0: 23 Proof of Theorem 1.1First we complete the form of the energy inequality. Using u as a test function forthe equation (1.2) we derive the identityddt"Z10utu+u2x2dx# Z10u2tdx+Z10(f1(ux)ux+ f2(ux)ux) dx+ ÆZ10u2xxdx = 0; (3.22)since [uxtu]10  [f1(ux)u+ f2(ux)u]10+ Æ[uxxxu]10  Æ[uxxux]10= 0:This is a consequence of the boundary and compatibility conditions.We multiply the equation (3.22) with ,  > 0, and we add this expression to (2.15).This leads to the inequalityddtV(t) +W(t)  0; t 2 (0;1); (3.23)whereV(t) :=Z10 u2t2+Æu2xx2+ F2(ux) +CV(   )22K+ utu+ u2x2!dx; (3.24)andW(t) :=  k(t)k21221Z10u2xdx+2Z10u2xtdx+kKZ10(   )2xdx 1KZ10f1(ux)(   )uxtdx KZ10u2xt(   ) dx Z10u2tdx+ Z1021(   1)u2x+ 63u6x  42u4xdx+ ÆZ10u2xxdx: (3.25)The only point remaining concerns the behaviour of the functionals W(t) and V(t).If we can prove thatW0(t)  k(0)V0(t)  0 (3.26)for some constant k(0) > 0 and all t 2 (0;1), then the inequality (1.11) follows.5First we estimate the integrals contained in V(t) one by one. It is easy to checkthatZ10jutuj dx   Z10u2t2dx+Z10u22dx! Z10u2t2dx + Z10u2xx2dx (3.27)and2Z10u2xdx  2Z10u2xxdx: (3.28)This follows immediately from the Young inequality and from the fact that kuk2kuxk2 kuxxk2. Hence, we have thatZ10 (1  )u2t2+ (Æ   )u2xx2+ F2(ux) +CV(   )22K!dx  V(t)Z10 (1 + )u2t2+ (Æ +  + )u2xx2+ F3(ux) +CV(   )22K!dx: (3.29)In the next step, we investigate the functional W(t). We conclude from the com-patibility conditions thatZ10u2tdx  Z10u2xtdx: (3.30)It is obvious that the inequality21Z10u2x(   1) dx  2k(t)  1k11Z10u2xxdx (3.31)is fullled. It is also easy to check thatf2(z)z = 63z6  42z4 4(3z6  2z4) + 23z6 4F3(z) 4223F4(z) + 23z6; (3.32)withF4(z) :=8<:z2if z q230 if z >q23:(3.33)According to the inequality above, we haveZ10f2(ux)uxdx  4Z10F3(ux) dx  4223Z10u2xxdx: (3.34)We easily deduce that the estimatek(t)k21221Z10u2xdx  k(t)k21221Z10u2xxdx (3.35)6holds.The Poincare inequality provides thatkKZ10(   )2xdx kKZ10(   )2dx: (3.36)Applying the Young inequality, we obtain the estimate1KZ10jf1(ux)(   )uxtj  k(t)k2121KZ10juxuxtj dx k(t)k41221KZ10u2xxdx+12KZ10u2xtdx: (3.37)In the same manner, we can derive thatKZ10u2xtj   j dx k(t)k1KZ10u2xtdx: (3.38)Combining the above estimates, we haveW(t)  2   k(t)k1K 12K!Z10u2xtdx +kKZ10(   )2dx+ Æ  4223 2k(t)k2121  2k(t)  1k11  k(t)k41221K!Z10u2xxdx+ 4Z10F3(ux) dx: (3.39)We conclude from (3.29) and (3.39) that there exists a constant k(0) such that(3.26) is fullled, and thereforeddtV(t) + k(0)V(t)  0: (3.40)An easy computation shows that V(t)  0, and this nishes the proof. 2Consequence 3.1 Suppose that the constants from Theorem 1.1 satisfy+k(t)k1K+12K<2; + 11+223<2(3.41)and4223+2k(t)k2121+ 2k(t)  1k11+ k(t)k41221K< Æ (3.42)for a.a. t 2 (0;1). Thenkut(t)k22+ kuxx(t)k22+ k(t)  (t)k22 k3e k4t; (3.43)where the constants k3, k4> 0 depend only on the initial state.7P r o o f: We can use the term2R10u2xdx, which appears in the functional V(t),to verify (3.43). 2Consequence 3.2 If the relations <2; + 11+223<Æ2(3.44)and4223+2k(t)k2121+ 2k(t)  1k11< Æ (3.45)hold for a.a. t 2 (0;1), thenkut(t)k22+ kuxx(t)k22 k5e k6t; (3.46)where the constants k5, k6> 0 depend only on the initial state.P r o o f: We can easily construct the functionalseV(t) andfW(t) adding (2.14) to(3.22). The rest of the proof proceeds in a similar way as above. 24 RemarksNow, we show possible modications of the proof, which lead to simpler forms of(1.9) and (1.10). We recommend the reader to compare these new results with [4].There it was proved that the only solution to the steady state equation is u = 0 ifinequality (4.49) bellow holds.Lemma 4.1 Suppose positive constants Æ, , i, i = 1; 2; 3, 1, and constants  > 0and K > 1 are given such that+k(t)k1K+12K<2; +223<Æ2: (4.47)If, in addition, the inequalities2k(t)k2121+ k(t)k41221K< Æ (4.48)and(x; t) > 1+22331(4.49)hold for all x 2 [0; 1] and all t  t0, thenkut(t)k22+ kuxx(t)k22+ k(t)  (t)k22 k7e k8t(4.50)for t  t0, where the constants k7, k8> 0 depend only on the initial state.8P r o o f: We rewrite (3.31) and (3.32) as63z6  42z4+ 2((x; t)  1)1z2 4F3(z) + F5(z); (4.51)whereF5(z) :=8<:63z6  42z4+ 2((x; t)  1)1z2if jzj q230 if jzj >q23It is easily seen that42z4= 42p12342z342p123z  63z6+22233z2: 2Remark 4.2 If we look more closely at the functionalF1(1; ux; uxx) :=Z10F6(1; ux; uxx) dx;whereF6(1; ux; uxx) :=  11u2x  42u4x+ 63u2x+ Æu2xx;we can see that, using the condition (1.9) and the inequality kuxk22 kuxxk22, wecan assert that the functional F1has only global minimum for ux= uxx= 0.Remark 4.3 If we dene the functionF8(; ux; uxx) := 1(   1)u2x  2u2x+ 3u6x+ Æu2xxand the functionalF2(; ux; uxx) :=Z10F8(; ux; uxx) dx;then the conditions (4.47)-(4.49) imply that this functional has only one globalminimum for ux= uxx= 0 using the estimate kuxk22 kuxxk22.Lemma 4.4 Suppose that there exist constants  and K such that that the estimates(3.41) and(x; t) > max(1+222331;2k(t)k211+ k(t)k4121K)(4.52)hold for all x 2 [0; 1] and all t  t0. Thenkut(t)k22+ kux(t)k22+ k(t)  (t)k22 k9e k10t: (4.53)This result also holds for the case Æ = 0.P r o o f: It is easy to check the assertion of this remark using (3.35), (3.37) and amodication of the estimate (4.51). 29References[1] Brokate M., Sprekels J.: Hysteresis and Phase Transitions, Appl. Math. Sci.121, Springer, New York, 1996.[2] Dafermos C.M.: Global smooth solutions top the initial boundary value problemfor the equations of one-dimensional nonlinear thermo-viscoelasticity , SIAM J.Math. Anal., 13 (1982), pp. 397-408.[3] Dafermos C. M., Hsiao L.: Global smooth thermo-mechanical process in one-dimensional nonlinear thermo-viscoelasticity , Nonlinear Anal., 6 (1982), pp.435-454.[4] Friedman A., Sprekels J.: Steady states of austenitic-martensitic domains inthe Ginzburg-Landau theory of shape memory alloys, Continuum Mach. Ther-modyn., 2 (1990), pp. 199-213.[5] Homan K. H., Zochowski A.: Existence of solutions to some non-linear ther-moelastic system with viscosity , Math. Meth. Appl. Sci., 15 (1992), pp. 187-204.[6] Luo T.: Qualitative Behaviour to Nonlinear Evolution Equations with Dissipa-tion, Ph. D. thesis, Institute of Mathematics, Academy of Sciences of China,Beijing, 1994.[7] Racke R., Zheng S.: Global existence and asymptotic behaviour in nonlinearthermo-viscoelasticity , J. Dierential Equations, 134 (1997), pp. 46-67.[8] Sprekels J., Zheng S.: Global solutions to the equations of a Ginzburg-Landautheory for structural phase transitions in shape memory alloys, Physica D, 39(1989), pp. 59-76.[9] Sprekels J., Zheng S.: Maximal attractor for the system of a Landau-Ginzburgtheory for structural phase transitions in shape memory alloys, Physica D, 121(1998), pp. 252-262.[10] Sprekels J., Zheng S., Zhu P.: Asymptotic behaviour of the solutions to aLandau-Ginzburg system with viscosity for martensitic phase transitions inshape memory alloys, SIAM J. Math. Anal., 29 (1998), pp. 69-84.[11] Straskraba I., Zlotnik A.: On a decay rate for 1D-viscous compressiblebarotropic uid equations. J. Evol. Equ., 2 (2002), pp. 69-96.10

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On a decay rate for a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys

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