On the Standard Linear Viscoelastic model

Transcription

1 On the Standard Linear Viscoelastic model M. Pellicer (Universitat de Girona) Work in collaboration with: J. Solà-Morales (Universitat Politècnica de Catalunya) (bounded problem) B. Said-Houari (Alhosn University, UAE) (problem in R N ) Trobada d EDP s i Aplicacions Universitat de Girona 10th June / 38

2 Outline 1 Introduction and modeling Mechanical approach: SLV model Acoustic approach: MGT equation 2 SLV in a bounded domain Description of the spectrum of A Main result 1: new scalar product and normality of A Main result 2: optimal exponential decay 3 SLV in R N The decay result The eigenvalues expansion method 2 / 38

3 Outline 1 Introduction and modeling Mechanical approach: SLV model Acoustic approach: MGT equation 2 SLV in a bounded domain Description of the spectrum of A Main result 1: new scalar product and normality of A Main result 2: optimal exponential decay 3 SLV in R N The decay result The eigenvalues expansion method 3 / 38

4 Mechanical approach: SLV model Deformation of a viscoelastic material (properties: viscous fluidity + elastic solidity) 4 / 38

5 Mechanical approach: SLV model Deformation of a viscoelastic material (properties: viscous fluidity + elastic solidity) Rheological approach (1D): 4 / 38

6 Rheological approach (1D): relation between σ (stress, force) and ε (strain, deformation) combination of mechanical tests: creeping and stress relaxation 5 / 38

7 Simplest models: 6 / 38

8 The Standard Linear Viscoelastic model a more realistic approach the most simple exhibiting creeping and stress relaxation 7 / 38

9 The Standard Linear Viscoelastic model a more realistic approach the most simple exhibiting creeping and stress relaxation 2 rheological approaches spring + Maxwell in parallel (3-parameter, generalized Maxwell,...) spring + Kelvin-Voigt in series (SL solid, Zener, Kelvin,...) 7 / 38

10 Constitutive equation: σ + ασ = E (ε + βε ) (same equation with both approaches, but different relation of the coefficients with the physical parameters) 8 / 38

11 Constitutive equation: σ + ασ = E (ε + βε ) (same equation with both approaches, but different relation of the coefficients with the physical parameters) The continuous model (Standard Linear Viscoelastic model): αu ttt + u tt a 2 (u xx + βu txx ) = 0, x [0, L], t 0 where u(x, t) = deformation, α, β > 0 8 / 38

12 Acoustic approach: MGT equation Propagation of sound Acoustic velocity potential in viscous thermally relaxing fluids Medical and industrial application of high intensity focused ultrasound (lithotripsy, thermotherapy, ultrasound cleaning,... ) 9 / 38

13 Conservation of mass + momentum + energy Second order approximations Fourier law for heat conduction: q = K θ Kuznetsov equation (classic nonlinear Acoustics) u tt c 2 u b u t = ( ) 1 B t c 2 2A (u t) 2 + u 2 (others: Westervelt equation, KZK equation,...) Problem: infinite speed of propagation!!! (no good for previous applications) 10 / 38

14 Alternative Maxwell-Cattaneo law: αq t + q = K θ (α: relaxation time of heat flux thermal inertia therm) Jordan-Moore-Gibson-Thompson equation: αu ttt +u tt c 2 u b u t = ( ) 1 B t c 2 2A (u t) 2 + u 2 11 / 38

15 Alternative Maxwell-Cattaneo law: αq t + q = K θ (α: relaxation time of heat flux thermal inertia therm) Jordan-Moore-Gibson-Thompson equation: αu ttt +u tt c 2 u b u t = ( ) 1 B t c 2 2A (u t) 2 + u 2 Linear part Moore-Gibson-Thompson equation: αu ttt +u tt c 2 u b u t = 0 11 / 38

16 References (M) G. C. Gorain, (Proc. Indian Acad. Sci. (Math. Sci.), 2010) (M) M.S. Alves, C. Buriol, M.V. Ferreira, J.E. Muñoz Rivera, M. Sepúlveda, O. Vera (JMAA, 2013) (M) J.A. Conejero, C. Lizama, F. Ródenas (App. Math. Inf. Sciences, 2015) (A) B. Kaltenbacher, I. Lasiecka, R. Marchand (Control Cybernet., 2011) (A) B. Kaltenbacher, I. Lasiecka, M. K Pospieszalska (M 3 ASc, 2012) (A) R. Marchand, T. McDevitt, R. Triggiani (M 3 ASc, 2012) 12 / 38

17 Outline 1 Introduction and modeling Mechanical approach: SLV model Acoustic approach: MGT equation 2 SLV in a bounded domain Description of the spectrum of A Main result 1: new scalar product and normality of A Main result 2: optimal exponential decay 3 SLV in R N The decay result The eigenvalues expansion method 13 / 38

18 The problem Consider αu ttt + u tt a 2 u a 2 β u t = 0, x Ω R N with homogeneous Dirichlet b.c. 14 / 38

19 The problem Consider αu ttt + u tt a 2 u a 2 β u t = 0, x Ω R N with homogeneous Dirichlet b.c. In general: α u ttt + u tt + L (u + βu t ) = 0, x Ω R N Ω bounded, regular domain L self-adjoint, > 0, compact resolvent operator in H (Hilbert) (eigenvalues: 0 < µ 1 µ 2..., semi-simple eigenfunctions: φ n orthonormal) several possible functional spaces H i (see [M-McD-T, 2012]) 14 / 38

20 Important remarks α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Although linear, this equation displays a variety of physical behaviors depending on the parameter values (non-existence, existence but chaotic, exponential decay) 15 / 38

21 Important remarks α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Although linear, this equation displays a variety of physical behaviors depending on the parameter values (non-existence, existence but chaotic, exponential decay) α > 0 hyperbolic type equation 15 / 38

22 Important remarks α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Although linear, this equation displays a variety of physical behaviors depending on the parameter values (non-existence, existence but chaotic, exponential decay) α > 0 hyperbolic type equation Strong damping term (β): α = 0 (wave equation with strong damping): regularizing effect (hyperbolic to parabolic behavior) α > 0 (SLV equation): WELL-POSEDNESS 15 / 38

23 What is known (existence and decay)? (previous references) α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N 16 / 38

24 What is known (existence and decay)? (previous references) α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Well-posedness (β) β = 0 (and L unbounded): C 0 semigroup β > 0: C 0 group. 16 / 38

25 What is known (existence and decay)? (previous references) α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Well-posedness (β) β = 0 (and L unbounded): C 0 semigroup β > 0: C 0 group. Decay (α/β) α < β (dissipative case): exponentially stable group α = β (conservative case): constant energy α > β: chaotic behaviour 16 / 38

26 What is known (existence and decay)? (previous references) α u ttt + u tt a 2 u a 2 β u t = 0, x Ω R N Well-posedness (β) β = 0 (and L unbounded): C 0 semigroup β > 0: C 0 group. Decay (α/β) α < β (dissipative case): exponentially stable group α = β (conservative case): constant energy α > β: chaotic behaviour Remark: third hidden parameter a 2 through the spectrum of L = a 2 (eigenvalues: µ n ) 16 / 38

27 What we do? We focus in the dissipative case α < β 17 / 38

28 What we do? We focus in the dissipative case α < β Using a careful analysis of the spectrum of A we... Improve the description of the spectrum of [MMcDT, 2012] 17 / 38

29 What we do? We focus in the dissipative case α < β Using a careful analysis of the spectrum of A we... Improve the description of the spectrum of [MMcDT, 2012] Find a new and explicit scalar product that makes A normal (almost always) 17 / 38

30 What we do? We focus in the dissipative case α < β Using a careful analysis of the spectrum of A we... Improve the description of the spectrum of [MMcDT, 2012] Find a new and explicit scalar product that makes A normal (almost always) Give the optimal exponential decay rate of solutions (previous work for wave equation with strong damping: M.P., J. Solà-Morales in JDE 2009) 17 / 38

31 We write it as a first order system: du dt Description of the spectrum of A = AU, U D(A), with AU = v w 1 α L(u + βv) 1 α w The eigenvalues: Characteristic equation: αλ 3 + λ 2 + βµ n λ + µ n = 0 Cubic equation 3 solutions for each n: λ n 1, λ n 2, λ n 3 (type depending on the sign of Cardano s discriminant, whose zeroes are m 1, m 2 ) 18 / 38

32 In general, 1 real sequence λ n 1 and 2 nonreal ones, λn 2 = λn 3 19 / 38

33 In general, 1 real sequence λ n 1 and 2 nonreal ones, λn 2 = λn 3 EXCEPT... if α/β < 1/9 (overdamping) and µ n [m 1, m 2 ] (finite number!): 3 real (if µ n (m 1, m 2 )) 1 real, 2 real double (if µ n = m 1 or µ n = m 2 ) (actually triple root if α/β = 1/9 and µ n = m 1 = m 2 ) Spectrum when α/β < 1/9 19 / 38

34 In general, 1 real sequence λ n 1 and 2 nonreal ones, λn 2 = λn 3 EXCEPT... if α/β < 1/9 (overdamping) and µ n [m 1, m 2 ] (finite number!): 3 real (if µ n (m 1, m 2 )) 1 real, 2 real double (if µ n = m 1 or µ n = m 2 ) (actually triple root if α/β = 1/9 and µ n = m 1 = m 2 ) Spectrum when α/β < 1/9 The cases m 1 µ n m 2 are not considered in [MMcDT, 2012]. In the discrete case, they can happen or not (depending on the parameters) (but they DO as α 0!!) In the continuous case (unbounded domain) they DO happen. 19 / 38

35 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

36 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

37 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

38 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

39 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

40 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

41 Characteristic polynomial αλ 3 + λ 2 + βµ n λ + µ n =0 as n increases: Case 0 < α β < / 38

42 They fulfill: λ n k R λn k 1 β (essential spectrum) 21 / 38

43 They fulfill: λ n k R λn k 1 β (essential spectrum) ( ) λ n k C \ R 0 > Re(λn k ) α 1 β 21 / 38

44 They fulfill: λ n k R λn k 1 β (essential spectrum) ( ) λ n k C \ R 0 > Re(λn k ) α 1 β ( ) α 1 β < 1 β if α β < 1 3 (resp. =, >) α β < 1 3 α β > / 38

45 Dominant spectrum Two possibilities 1 (essential spectrum): no oscillations in the dominant part β (overdamped case) {λ 1 2, λ1 3 } (the first two nonreal eigenvalues) α β > 1 3 σ max = Re(λ 1 2 ) 22 / 38

46 Dominant spectrum Two possibilities 1 (essential spectrum): no oscillations in the dominant part β (overdamped case) {λ 1 2, λ1 3 } (the first two nonreal eigenvalues) α β < 1 different possibilities 3 (still not explicit condition: solve the characteristic equation for µ n = µ 1 ) 22 / 38

47 Main result 1: new scalar product and normality of A Theorem 1 If µ n m 1, m 2 : we construct a new and explicit scalar product, G where A is a normal operator the eigenfunctions are orthonormal and complete with G 23 / 38

48 Main result 1: new scalar product and normality of A Theorem 1 If µ n m 1, m 2 : we construct a new and explicit scalar product, G where A is a normal operator the eigenfunctions are orthonormal and complete with G If any µ n = m 1, m 2, A not normal in any scalar product (multiplicity of eigenvalues). 23 / 38

49 Main result 1: new scalar product and normality of A Theorem 1 If µ n m 1, m 2 : we construct a new and explicit scalar product, G where A is a normal operator the eigenfunctions are orthonormal and complete with G If any µ n = m 1, m 2, A not normal in any scalar product (multiplicity of eigenvalues). Remarks A normal if AA = A A ( diagonable, orthonormal eigenfunctions) 23 / 38

50 Main result 1: new scalar product and normality of A Theorem 1 If µ n m 1, m 2 : we construct a new and explicit scalar product, G where A is a normal operator the eigenfunctions are orthonormal and complete with G If any µ n = m 1, m 2, A not normal in any scalar product (multiplicity of eigenvalues). Remarks A normal if AA = A A ( diagonable, orthonormal eigenfunctions) A normal is the best we can expect (A is not self-adjoint as it has non-real eigenvalues) 23 / 38

51 Main result 1: new scalar product and normality of A Theorem 1 If µ n m 1, m 2 : we construct a new and explicit scalar product, G where A is a normal operator the eigenfunctions are orthonormal and complete with G If any µ n = m 1, m 2, A not normal in any scalar product (multiplicity of eigenvalues). Remarks A normal if AA = A A ( diagonable, orthonormal eigenfunctions) A normal is the best we can expect (A is not self-adjoint as it has non-real eigenvalues) Main difficulty: proving the equivalence of the new scalar product and the natural one 23 / 38

52 The idea... If A normal in, G A diagonable orthonormal eigenfunctions in this norm. So, it is natural to define G G G = G n where G n = (Cn 1 ) T Cn 1, with C n = col(ψ n 1, Ψn 2, Ψn 3 ) (eigenfunctions) 24 / 38

53 The idea... If A normal in, G A diagonable orthonormal eigenfunctions in this norm. So, it is natural to define G G G = G n where G n = (Cn 1 ) T Cn 1, with C n = col(ψ n 1, Ψn 2, Ψn 3 ) (eigenfunctions) Remark 1 Observe that u, v Gn =... = Cn 1 u, Cn 1 v R 3 24 / 38

54 The idea... If A normal in, G A diagonable orthonormal eigenfunctions in this norm. So, it is natural to define G G G = G n where G n = (Cn 1 ) T Cn 1, with C n = col(ψ n 1, Ψn 2, Ψn 3 ) (eigenfunctions) Remark 2, Gn, uniformly in n G equivalent to the natural product 24 / 38

55 The idea... If A normal in, G A diagonable orthonormal eigenfunctions in this norm. So, it is natural to define G G G = G n where G n = (Cn 1 ) T Cn 1, with C n = col(ψ n 1, Ψn 2, Ψn 3 ) (eigenfunctions) Remark 3 Only possible when non (algebraically) multiple eigenvalues!!! (in which case A not normal because not diagonable) 24 / 38

56 Example in H = H0 1(Ω) H1 0 (Ω) L2 (Ω): ( ) µn 0 0 O n = 0 µ n αβ 2 +3α+β µ 2αβ 2 n + o(µ n) G n = α+β α+β µn + o(µn) 2αβ 4α 2 β 2 +β 2 +3α 2 4αβ o(1) α+β 2αβ µn + o(µn) 4α 2 β 2 +β 2 +3α o(1) 4αβ 3 2α µn + o(µn) (α+β) o(1) 4αβ 2 (α+β) 2 4αβ o(1) α+β 2β 1 + o(1) 25 / 38

57 Theorem 2 Main result 2: optimal exponential decay If A normal in, G : U(t) G e σmax t U(0) G σ max = Re(λ 1 2 ), 1 β < 0, dominant spectrum, optimal. 26 / 38

58 Theorem 2 Main result 2: optimal exponential decay If A normal in, G : U(t) G e σmax t U(0) G σ max = Re(λ 1 2 ), 1 β < 0, dominant spectrum, optimal. If A not normal, ALSO true (but in another norm) (multiple eigenvalues, but not dominant) 26 / 38

59 Theorem 2 Main result 2: optimal exponential decay If A normal in, G : U(t) G e σmax t U(0) G σ max = Re(λ 1 2 ), 1 β < 0, dominant spectrum, optimal. If A not normal, ALSO true (but in another norm) (multiple eigenvalues, but not dominant) Remarks In [MMcDT 2012]: exponential decay with the growth bound inf{ω R; e At M ω e ωt t 0} (usual norm) We improve it: M ω = 1, and the infimum is a minimum (also when σ max is the essential spectrum) 26 / 38

60 Idea... If A normal in, G... Eigenfunctions are orthonormal in G and complete 27 / 38

61 Idea... If A normal in, G... Eigenfunctions are orthonormal in G and complete In this basis: U(t) 2 G = n,j d n j 2 e 2Re(λn j )t n,j d n j 2 e 2σmax t (t > 0). 27 / 38

62 Idea... If A normal in, G... Eigenfunctions are orthonormal in G and complete In this basis: U(t) 2 G = n,j d n j 2 e 2Re(λn j )t n,j d n j 2 e 2σmax t (t > 0). Optimality: Clear if σ max = Re(λ 1 2 ) Also if σ max = 1 β, as we have λn 1 1 β 27 / 38

63 If A not normal... Multiple eigenvalues are not a problem because they are not dominant we can essentially do the same as before 28 / 38

64 If A not normal... Multiple eigenvalues are not a problem because they are not dominant we can essentially do the same as before ( ) A0 0 A = (A 0 A 1 the part of the multiple eigenvalues) 1 For A 0 the same as before applies: e A0t G0 e σmax (A 0) t σmax (A) t e For A 1 we can define, G1,ε such that (for a certain ε > 0) e A1t G1,ε e (σmax (A 1)+ε) t e 1 β t σmax (A) t e G defined as the orthogonal extension of G 0, G 1 28 / 38

65 Outline 1 Introduction and modeling Mechanical approach: SLV model Acoustic approach: MGT equation 2 SLV in a bounded domain Description of the spectrum of A Main result 1: new scalar product and normality of A Main result 2: optimal exponential decay 3 SLV in R N The decay result The eigenvalues expansion method 29 / 38

66 The problem We consider αu ttt + u tt u β u t = 0 in R N with α < β (a = 1 without loss of generality) What we do? Well-posed (not done before) decay rate (using eigenvalues expansion method) 30 / 38

67 The decay result Theorem: For N + j 3, j x u (t) L 2 C( u 0 L 1 + u 1 L 1 + u 2 L 1)(1 + t) 1/2 N/4 j/2 +C( x j u 0 L 2 + x j u 1 L 2 + x j u 2 L 2)e ct { } with c = min 1 β, Re(λ 2,3(ξ ν1 )) Decay: for all N 3 and all j 0 (similar formula for all j. In particular, N = 1, 2 decay only if j > 3/2, 1/2) Regularity: same as the initial conditions (no regularity-loss type) 31 / 38

68 Idea: eigenvalues expansion method Behaviour of the eigenvalues in the Fourier space Observe that same description of the spectrum as in the bounded domain, but ξ ( µ n ) being a continuous parameter 32 / 38

69 Idea: eigenvalues expansion method Behaviour of the eigenvalues in the Fourier space Observe that same description of the spectrum as in the bounded domain, but ξ ( µ n ) being a continuous parameter Why? Plancherel Theorem: j xu (t) 2 L 2 = R N ξ 2j û (ξ, t) 2 dξ Roughly speaking... ODE in the Fourier space for each ξ û(ξ, t) = C 1(ξ)e λ 1(ξ)t +e Re(λ 2(ξ))t [C 2(ξ) cos(im(λ 2(ξ))t) + C 3(ξ) sin(im(λ 2(ξ))t)] (C i (ξ) using initial conditions) 32 / 38

70 Split the previous integral in Υ L, Υ M, Υ H : 33 / 38

71 Split the previous integral in Υ L, Υ M, Υ H : Υ L = { ξ < ν 1 << 1} decay (as û e Re(λ(ξ))t ) λ 1(ξ) 1 α, λ 2(ξ) i ξ 1 2 (β α) ξ 2, Υ M = {ν 1 ξ ν 2 }: λ 3(ξ) i ξ 1 (β α) ξ 2 2 Re(λ n 1,2,3(ξ)) < min { } 1 β, Re(λ 2,3(ξ ν1 )) Υ H = { ξ > ν 2 >> 1} regularity (as û ξ m e c ξ ĉt ) λ 1(ξ) 1 β, λ 2(ξ) β α β 2βα + i ξ α, λ 3(ξ) β α β 2βα i ξ α 33 / 38

72 Imposing initial conditions (and taking the dominant terms)... ξ small: ) û(ξ, t) C L ( ξ 2 û 0 + ξ 2 û 1 + û 2 e α 1 t ) +C L ( û 0 + ξ 2 û 1 + û 2 e β α ξ 2t 2 cos( ξ t) ( +C L ξ û ξ û1 + 1 ) ξ û2 e β α ξ 2t 2 sin( ξ t) (some integral bounds for N + j 3 or for all N 1) ( ) 1/2 ξ 2j û (ξ, t) 2 dξ C( u 0 2 L + u L + u L )(1 + t) N 4 j 2 Υ L 34 / 38

73 ξ medium (and ξ m 1, m 2 ): û(ξ, t) C M (1 û 0 (ξ) + 1 û 1 (ξ) + 1 û 2 (ξ) ) e c 4t { } with c 4 = min 1 β, Re(λ 2,3(ξ ν1 )) > 0 (some integral bounds) ( ) 1/2 ξ 2j û (ξ, t) 2 dξ C( x j u 0 2 L + 2 x j u 1 2 L + 2 x j u 2 2 L )e c 4t 2 Υ M 35 / 38

74 (( C H ξ + 1 ) û 0(ξ) + ξ 2 ξ large: û(ξ, t) ( 1 ξ + 1 ξ 2 ) û 1(ξ) + )} with c 3 = min { 1 β, 1 2 ( 1 α 1 β ( 1 ξ + 1 ) ) û 2(ξ) e c 3t 2 ξ 3 > 0 (some integral bounds) ( ) 1/2 ξ 2j û (ξ, t) 2 dξ C( x j u 0 L 2+ x j u 1 L 2+ x j u 2 L 2)e c 3t Υ H 36 / 38

75 Joining the previous inequalities we arrive at... for N + j 3, x j u (t) L 2 C( u 0 L 1 + u 1 L 1 + u 2 L 1)(1 + t) 1/2 N/4 j/2 +C( x j u 0 L 2 + x j u 1 L 2 + x j u 2 L 2)e ct { } with c = min 1 β, Re(λ 2,3(ξ ν1 )) 37 / 38

76 References Bounded domain: M. P., J. Solà-Morales. Optimal scalar products in a standard linear viscoelastic model (Submitted) Unbounded domain: M. P., B. Said-Houari. Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound (Submitted) 38 / 38

77 References Bounded domain: M. P., J. Solà-Morales. Optimal scalar products in a standard linear viscoelastic model (Submitted) Unbounded domain: M. P., B. Said-Houari. Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound (Submitted) Thank you! 38 / 38