· Aim and methods Aim and methods: To show the sharp decay estimate (n≥ 1) for linearized system for Hs ∩L1 initial data: Energy method in the Fourier space To show the global

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. ..

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Mathematical Analysis for Systems ofViscoelasticity and Viscothermoelasticity

Shuichi Kawashima

Faculty of Mathematics, Kyushu University

Joint work with

P.M.N. Dharmawardane, J.E. Munoz Rivera and T. Nakamura

ERC-NUMERIWAVES Seminar

BCAM, Bilbao, Basque Country, Spain

March 11, 2013

Shuichi Kawashima (Kyushu University) Mathematical Analysis for Viscoelasticity March 11, 2013 1 / 45

Outline

.

. .1 Introduction

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. .

2 Known results

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. .

3 Linear decay property

.

. .

4 Global existence

.

. .

5 Global existence and decay

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. .

6 Global existence and sharp decay

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. .

7 Viscothermoelasticity


1. Introduction

Elastic system:

utt −∑j

σj(∂xu)xj = 0,

u(x, t) : m-vector function of x ∈ Rn and t ≥ 0, displacement

σj(v) : m-vector functions of v = (v1, · · · , vn) ∈ Rmn, stress

σj(v) = Dvjϕ(v), ϕ(v) : free energy

Elastic system (in quasilinear form):

utt −∑j,k

Bjk(∂xu)uxjxk= 0,

Bjk(v) = Dvkσj(v), Bjk(v)T = Bkj(v) for j, k and v.


Viscoelastic system

Viscoelastic system: We consider viscoelastic system

utt −∑j

σj(∂xu)xj +∑j,k

Kjk ∗ uxjxk+ Lut = 0, (1)

u(x, 0) = u0(x), ut(x, 0) = u1(x), (2)

where

Kjk(t) : smooth m×m real matrix functions of t ≥ 0,

Kjk(t)T=Kkj(t) for j, k and t ≥ 0,

“ ∗ ” denotes the convolution with respect to t,

L : m×m real constant matrix.


Viscoelastic system

Viscoelastic system (in quasilinear form):

utt −∑j,k

Bjk(∂xu)uxjxk+

∑j,k

Kjk ∗ uxjxk+ Lut = 0.

Introduce the symbols of differential operators

Bω(v) =∑j,k

Bjk(v)ωjωk, Kω(t) =∑j,k

Kjk(t)ωjωk,

where ω = (ω1, · · · , ωn) ∈ Sn−1.

Linearized system:

utt −∑j,k

Bjk(0)uxjxk+

∑j,k



Structural assumptions

[A1] Bω(0) > O for ω ∈ Sn−1: real symmetric and positive definite,

Kω(t) ≥ O for ω ∈ Sn−1 and t ≥ 0: real symmetric and

nonnegative definite,

L ≥ O: real symmetric and nonnegative definite.

[A2] Bω(0)−Kω(t) > O for ω ∈ Sn−1 uniformly in t ≥ 0, where

Kω(t) =∫ t0 Kω(s)ds.

[A3] Kω(0) + L > O for ω ∈ Sn−1.

[A4] Kω(t) decays exponentially for t→ ∞:

−C0Kω(t) ≤ Kω(t) ≤ −c0Kω(t), −C0Kω(t) ≤ Kω(t) ≤ C0Kω(t)

for ω ∈ Sn−1 and t ≥ 0, where

Kω(t) = ∂tKω(t), Kω(t) = ∂2tKω(t).









[A3] Kω(0) + L > O for ω ∈ Sn−1.













[A3] Kω(0) + L > O for ω ∈ Sn−1.













[A3] Kω(0) + L > O for ω ∈ Sn−1.






Aim and methods

Aim and methods:

To show the sharp decay estimate (n ≥ 1) for linearized system for

Hs ∩ L1 initial data: Energy method in the Fourier space

To show the global existence and asymptotic stability (n ≥ 1) for

Hs initial data: Energy method

To show the decay estimate (n ≥ 2) for Hs initial data:

Time-weighted energy method

To show the sharp decay estimate (n ≥ 1) for Hs ∩ L1 initial data:

Time-weighted energy method and Semigroup argument.


2. Known results

Viscoelastic models decribed by a single equation:

utt −n∑

j,k=1

bjkuxjxk+ γ

n∑j,k=1

gjk ∗ uxjxk+ αut = 0.

Simplified equation:

utt −∆u+ γg ∗∆u+ αut = 0.

(1) γ = 0 and the damping term is effective α > 0:

Matsumura, RIMS, 12 (1976),

Todorova-Yordanov, J. Diff. Eqs., 174 (2000),

Japanese contribution: Nakao, Nishihara, Ikehata, Narazaki.

Standard decay estimate:

∥u(t)∥L2 ≤ C(1 + t)−n/4.


2. Known results


utt −n∑

j,k=1

bjkuxjxk+ γ

n∑j,k=1









∥u(t)∥L2 ≤ C(1 + t)−n/4.


2. Known results


utt −n∑

j,k=1

bjkuxjxk+ γ

n∑j,k=1









∥u(t)∥L2 ≤ C(1 + t)−n/4.


2. Known results


utt −n∑

j,k=1

bjkuxjxk+ γ

n∑j,k=1









∥u(t)∥L2 ≤ C(1 + t)−n/4.


Known results


(2) α = 0 and the memory term is effective γ > 0:

Fabrizio-Lazzari, Arch. Rational Mech. Anal., 116 (1991),

Rivera, Quart. Appl. Math., 52 (1994),

Liu-Zheng, Quart. Appl. Math., 54 (1996)

The solution verifies the standard decay estimate of the form

∥(ut, ∂xu)(t)∥L2

≤ C(1 + t)−n/4∥(u1, ∂xu0)∥L1 + Ce−ct∥(u1, ∂xu0)∥L2 .


Known results







∥(ut, ∂xu)(t)∥L2



Known results







∥(ut, ∂xu)(t)∥L2



Known results

utt −∆u− γg ∗ (−∆)σu+ αut = 0.

(3) α = 0 and the memory term is effective γ > 0 with 0 < σ < 1:

Rivera-Naso-Vegni, JMAA, 286 (2003)

Decay structure: weak, regularity-loss type.

In bounded domain, the energy decays polynomially but

NOT exponentially.

Similar weak decay structure:

Dissipative Timoshenko system:

Rivera-Racke (2003), Ide-Haramoto-Kawashima (2008)

Dissipative plate equation with inertial effect:

da Luz-Charao (2009), Sugitani-Kawashima (2010)


Known results

utt −∆u− γg ∗ (−∆)σu+ αut = 0.





NOT exponentially.







Known results

utt −∆u− γg ∗ (−∆)σu+ αut = 0.





NOT exponentially.







3. Linear decay property

Linear viscoelastic system: We consider

utt −n∑

j,k=1

Bjkuxjxk+

n∑j,k=1

Kjk ∗ uxjxk+ Lut = 0, (3)

u(x, 0) = u0(x), ut(x, 0) = u1(x). (4)

where

u = u(x, t) : m-vector function of x ∈ Rn and t ≥ 0,

Bjk = Bjk(0) : m×m real constant matrices,




Linear decay property

Linear viscoelastic system:

utt −n∑

j,k=1

Bjkuxjxk+

n∑j,k=1


.

Theorem 1 (Sharp decay estimate for linear problem)Dharmawardane, Rivera & S.K. (2010)

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. ..

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Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs ∩ L1 for s ≥ 0.Then the linear solution satisfies the sharp decay estimate:

∥(∂kxut, ∂k+1x u)(t)∥L2

≤ C(1 + t)−n/4−k/2∥(u1, ∂xu0)∥L1 + Ce−ct∥(∂kxu1, ∂k+1x u0)∥L2 ,

(5)

where 0 ≤ k ≤ s.


Linear viscoelastic system in the Fourier space

Linear viscoelastic system:

utt −n∑

j,k=1

Bjkuxjxk+

n∑j,k=1


Linear viscoelastic system in the Fourier space:

Taking the Fourier transform, we get

utt + |ξ|2Bωu− |ξ|2Kω ∗ u+ Lut = 0,

where ω = ξ/|ξ| ∈ Sn−1, and

Bω =

n∑j,k=1

Bjkωjωk, Kω(t) =

n∑j,k=1

Kjk(t)ωjωk.


Pointwise estimate in the Fourier space


utt + |ξ|2Bωu− |ξ|2Kω ∗ u+ Lut = 0.

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Proposition 1 (Pointwise estimate in the Fourier space)Dharmawardane, Rivera & S.K. (2010)

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.

.

. ..

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.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs for s ≥ 0. Then theFourier image of the solution satisfies the pointwise estimate:

|ut(ξ, t)|2 + |ξ|2|u(ξ, t)|2 ≤ Ce−cρ(ξ)t(|u1(ξ)|2 + |ξ|2|u0(ξ)|2), (6)

where ρ(ξ) = |ξ|2/(1 + |ξ|2).


Energy method in the Fourier space


utt + |ξ|2Bωu− |ξ|2Kω ∗ u+ Lut = 0.

Construction of Lyapunov function:

d

dtE + F ≤ 0.

E : Lyapunov function, F : Dissipative term.

Claim: cE0 ≤ E ≤ CE0, F ≥ cρ(ξ)E,

where

E0 = |ut|2 + |ξ|2|u|2 + |ξ|2Kω[u, u], ρ(ξ) = |ξ|2/(1 + |ξ|2)

Kω[u, u](t) =

∫ t

0⟨Kω(t− τ)(u(t)− u(τ)), u(t)− u(τ)⟩dτ.



Step 1. Take the inner product with ut to get

1

2

d

dtE1 + F1 + ⟨Lut, ut⟩ = 0,

where

E1 = |ut|2 + |ξ|2⟨(Bω −Kω)u, u⟩+ |ξ|2Kω[u, u],

F1 =1

2|ξ|2(−Kω[u, u] + ⟨Kωu, u⟩).

Here ⟨ , ⟩ denotes the inner product in Cm, and

Kω[u, u](t) =

∫ t

0⟨Kω(t− τ)(u(t)− u(τ)), u(t)− u(τ)⟩dτ.



Step 2. Take the inner product with −(Kω ∗ u)t to get

1

2

d

dtE2 + ⟨Kω(0)ut, ut ⟩ = R2,

where R2 is an error term and

E2 = |ξ|2|Kω ∗ u|2 − 2Re ⟨ut, (Kω ∗ u)t ⟩.

Step 3. Take the inner product with u to have

1

2

d

dtE3 + |ξ|2⟨(Bω −Kω)u, u⟩ = R3,

where R3 is an error term and

E3 = ⟨Lu, u⟩+ 2Re ⟨ut, u⟩.



Step 4. Combining all these equalities, we arrive at

1

2

d

dtE + F = R,

where we put

E = E1 + ρ(ξ)(αE2 + βE3),

F = ρ(ξ){α⟨(Kω(0) + L)ut, ut⟩+ β|ξ|2⟨(Bω −Kω)u, u⟩

}+ F1 + (1− αρ(ξ))⟨Lut, ut⟩,

R = ρ(ξ)(αR2 + βR3)

with ρ(ξ) = |ξ|2/(1 + |ξ|2).



Claim: For suitably small α and β, we have

cE0 ≤ E ≤ CE0, F ≥ cρ(ξ)E, |R| ≤ F/2.

Here E0 = |ut|2 + |ξ|2|u|2 + |ξ|2Kω[u, u]. Thus we obtain

d

dtE + F ≤ 0,

and hence we haved

dtE + cρ(ξ)E ≤ 0.

This E is our Lyapunov function. Applying Gronwall’s ineqaulity, we get

|ut(ξ, t)|2 + |ξ|2|u(ξ, t)|2 ≤ Ce−cρ(ξ)t(|u1(ξ)|2 + |ξ|2|u0(ξ)|2).


4. Global existence

Viscoelastic system: We consider viscoelastic system

utt −∑j

σj(∂xu)xj +∑j,k

Kjk ∗ uxjxk+ Lut = 0,

u(x, 0) = u0(x), ut(x, 0) = u1(x),

where

u(x, t) : m-vector function of x ∈ Rn and t ≥ 0,

σj(v) : m-vector functions of v ∈ Rmn such that σj(v) = Dvjϕ(v),

where ϕ(v) is the free energy,




Global existence

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Theorem 2 (Global existence and asymptotic decay)Dharmawardane, Nakamura & S.K. (2011)

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.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs for s ≥ [n/2] + 2.Suppose E0 = ∥(u1, ∂xu0)∥Hs is small. Then we have the global solutionsatisfying the energy estimate:

∥(ut, ∂xu)(t)∥2Hs +

∫ t

0∥(∂xut, ∂2xu)(τ)∥2Hs−1dτ ≤ CE2

0 . (7)

Moreover, we have the asymptotic decay for t→ ∞:

∥(∂jxut, ∂j+1x u)(t)∥L∞ → 0,

where 0 ≤ j ≤ s− [n/2]− 2.


Energy method

Equation of energy: Take the inner product with ut to get{ϕ(∂xu) +

1

2|ut|2

}t−

∑j

{⟨σj(∂xu), ut⟩}xj

+∑j,k

⟨Kjk ∗ uxjxk, ut⟩+ ⟨Lut, ut⟩ = 0.

Energy form:

H = ϕ(∂xu)− ϕ(0)−∑j

⟨σj(0), uxj ⟩+1

2|ut|2.

Equation of energy form:

Ht −∑j

{⟨σj(∂xu)− σj(0), ut⟩}xj

+∑j,k

⟨Kjk ∗ uxjxk, ut⟩+ ⟨Lut, ut⟩ = 0.


Energy method

Modified equation of energy form:{H+

1

2

∑j,k

(− ⟨Kjkuxj , uxk

⟩+Kjk[uxj , uxk])}

t

−∑j

{⟨σj(∂xu)− σ(0), ut⟩

}xj

+∑j,k

{⟨Kjk ∗ uxk

, ut⟩}xj

+1

2

∑j,k

(− Kjk[uxj , uxk

] + ⟨Kjkuxj , uxk⟩)+ ⟨Lut, ut⟩ = 0.

This comes from the equality∑j,k

⟨Kjk ∗ uxjxk, ut⟩ =

1

2

∑j,k


⟩+Kjk[uxj , uxk])t

+∑j,k

{⟨Kjk ∗ uxk

, ut⟩}xj

+1

2

∑j,k

(− Kjk[uxj , uxk

] + ⟨Kjkuxj , uxk⟩).


Energy method

Dissipation from memory term:

QK [∂xu] := Q♯K [∂xu] +Q♭

K [∂xu],

where

Q♯K [∂xu] :=

∑j,k

∫Rn

Kjk[uxj , uxk] dx,

Q♭K [∂xu] :=

∑j,k

∫Rn

⟨Kjkuxj , uxk⟩ dx,

Kjk[ψj , ψk](t) =

∫ t

0⟨Kjk(t− τ)(ψj(t)− ψj(τ)), ψk(t)− ψk(τ) ⟩ dτ.


Energy method

Energy norm:

E0(t)2 = sup

0≤τ≤t

(∥(ut, ∂xu)(τ)∥2Hs +

s∑l=0

QK [∂l+1x u](τ)

).

Dissipation norm:

D0(t)2 = D0(t)

2 +

∫ t

0∥(∂xut, ∂2xu)(τ)∥2Hs−1dτ,

where

D0(t)2 =

∫ t

0

(∥(I − P )ut(τ)∥2Hs +

s∑l=0

QK [∂l+1x u](τ)

)dτ,

and P is the orthogonal projection onto Ker(L).


Energy method

.

Proposition 2 (Energy estimate)

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.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs for s ≥ [n/2] + 2.Assume that N0(t) = sup0≤τ≤t ∥(∂xu, ∂xut, ∂2xu)(τ)∥L∞ is small. Thenwe have the energy inequality:

E0(t)2 +D0(t)

2 ≤ CE20 + CN0(t)D0(t)

2. (8)

Proof of Theorem 2: Since N0(t) ≤ CE0(t), we have

E0(t)2 +D0(t)

2 ≤ CE20 + CE0(t)D0(t)

2,

which gives

E0(t)2 +D0(t)

2 ≤ CE20

for small E0. This a priori estimate combined with the local existence

result proves Theorem 2.Shuichi Kawashima (Kyushu University) Mathematical Analysis for Viscoelasticity March 11, 2013 26 / 45

5. Global existence and decay

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Theorem 3 (Global existence and decay)Dharmawardane, Nakamura & S.K. (2012)

.

.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 2 and (u1, ∂xu0) ∈ Hs for s ≥ [n/2] + 2.Suppose E0 = ∥(u1, ∂xu0)∥Hs is small. Then we have the global solutionsatisfying the time-weighted energy estimate:

E(t) +D(t) ≤ CE0. (9)

In particular, we have the decay estimate:

∥(∂mx ut, ∂m+1x u)(t)∥Hs−m ≤ CE0(1 + t)−m/2,

where 0 ≤ m ≤ s.



Time-weighted norms:

E(t)2 =s∑

m=0

Em(t)2, D(t)2 =s∑

m=0

Dm(t)2 +s−1∑m=0

Dm(t)2,

where

Em(t)2 = sup0≤τ≤t

(1+ τ)m(∥(∂mx ut, ∂m+1

x u)(τ)∥2Hs−m +s∑

l=m

QK [∂l+1x u](τ)

),

Dm(t)2 =

∫ t

0(1 + τ)m

(∥(I − P )∂mx ut(τ)∥2Hs−m +

s∑l=m

QK [∂l+1x u](τ)

)dτ,

Dm−1(t)2=

∫ t

0(1+τ)m−1

(∥(∂mx ut, ∂m+1

x u)(τ)∥2Hs−m+

s∑l=m

QK [∂l+1x u](τ)

)dτ.



.

Proposition 3 (Time-weighted energy estimate)

.

.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs for s ≥ [n/2] + 2.Assume that N0(t) = sup0≤τ≤t ∥(∂xu, ∂xut, ∂2xu)(τ)∥L∞ is small. Thenwe have the time-weighted energy inequality:

E(t)2 +D(t)2 ≤ CE20 + CN(t)D(t)2. (10)

Time-weighted L∞ norm:

N(t) = sup0≤τ≤t

{∥∂xu(τ)∥L∞ + (1 + τ)∥(∂xut, ∂2x)(τ)∥L∞

}.

Note that for s ≥ [n/2] + 2,

∥∂xu(t)∥L∞ ≤ CE(t)(1 + t)−n/4,

∥(∂xut, ∂2xu)(t)∥L∞ ≤ CE(t)(1 + t)−n/4−1/2.



Proof of Theorem 3: When n ≥ 2 and s ≥ [n/2] + 2, we have

N(t) ≤ CE(t). Consequently, we obtain

E(t)2 +D(t)2 ≤ CE20 + CE(t)D(t)2,

which gives

E(t)2 +D(t)2 ≤ CE20 ,

provided that E0 is small. This a priori estimate combined with the local

existence result proves Theorem 3.


6. Global existence and sharp decay

.

Theorem 4 (Global existence and sharp decay)Dharmawardane, Nakamura & S.K. (2012)

.

.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs ∩ L1 fors ≥ [n/2] + 3. Suppose E1 = ∥(u1, ∂xu0)∥Hs∩L1 is small. Then we havethe global solution satisfying:

E(t) +D(t) +M(t) ≤ CE1. (11)

In particular, we have the sharp decay estimate:

∥(∂mx ut, ∂m+1x u)(t)∥Hs−m−1 ≤ CE1(1 + t)−n/4−m/2,

where 0 ≤ m ≤ s− 1.

M(t) =s−1∑m=0

sup0≤τ≤t

(1 + τ)n/4+m/2∥(∂mx ut, ∂m+1x u)(τ)∥Hs−m−1 .



.

Proposition 3 (Time-weighted energy estimate)

.

.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs for s ≥ [n/2] + 2.Assume that N0(t) = sup0≤τ≤t ∥(∂xu, ∂xut, ∂2xu)(τ)∥L∞ is small. Thenwe have the time-weighted energy inequality:

E(t)2 +D(t)2 ≤ CE20 + CN(t)D(t)2.

Time-weighted L∞ norm:

N(t) = sup0≤τ≤t

{∥∂xu(τ)∥L∞ + (1 + τ)∥(∂xut, ∂2x)(τ)∥L∞

}.

Note that for n ≥ [n/2] + 3,

∥∂xu(t)∥L∞ ≤ CM(t)(1 + t)−n/2,

∥(∂xut, ∂2xu)(t)∥L∞ ≤ CM(t)(1 + t)−n/2−1/2.


Semigroup argument

Time-weighted norm for sharp decay:

M(t) =

s−1∑m=0

sup0≤τ≤t

(1 + τ)n/4+m/2∥(∂mx ut, ∂m+1x u)(τ)∥Hs−m−1 .

Semigroup argument:

.

Proposition 4 (Sharp decay estimate)

.

.

.

. ..

.

.

Assume [A1] - [A4]. Let n ≥ 1 and (u1, ∂xu0) ∈ Hs ∩ L1 fors ≥ [n/2] + 2. Assume that N0(t) = sup0≤τ≤t ∥(∂xu, ∂xut, ∂2xu)(τ)∥L∞

is small. Then we have:

M(t) ≤ CE1 + CE(t)M(t) + CM(t)2. (12)


Time-weighted energy method and semigroup argument

Proof of Theorem 4: When n ≥ 1 and s ≥ [n/2] + 3, we have

N(t) ≤ CM(t). Consequently, we obtain

E(t)2 +D(t)2 ≤ CE20 + CM(t)D(t)2,

M(t) ≤ CE1 + CE(t)M(t) + CM(t)2.

This gives

E(t) +D(t) +M(t) ≤ CE1,

provided that E1 is small. This a priori estimate combined with the local

existence result proves Theorem 4.


Semigroup argument

Linearized system:

utt −∑j,k

Bjk(0)uxjxk+

∑j,k


Fundamental solutions: G(x, t) and H(x, t)

Here u = G(x, t) and u = H(x, t) are m×m matrix solutions satisfying

the initial conditions

u(x, 0) = δ(x) I, ut(x, 0) = O,

u(x, 0) = O, ut(x, 0) = δ(x) I,

where δ(x) denotes the Dirac delta function and I is the identity matrix.


Semigroup argument

Linear solution formula:

u(t) = G(t) ∗ u0 +H(t) ∗ u1.

Decay estimate for solution operators:

Dharmawardane, Rivera & S.K (2010)

∥∂k+1x G(t) ∗ ϕ∥L2 + ∥∂kxGt(t) ∗ ϕ∥L2

≤ C(1 + t)−n/4−(k+1−j)/2∥∂jxϕ∥L1 + Ce−ct∥∂k+1x ϕ∥L2 ,

∥∂k+1x H(t) ∗ ψ∥L2 + ∥∂kxHt(t) ∗ ψ∥L2

≤ C(1 + t)−n/4−(k−j)/2∥∂jxψ∥L1 + Ce−ct∥∂kxψ∥L2 ,

where 0 ≤ j ≤ k + 1 and 0 ≤ j ≤ k, respectively.


Semigroup argument

Nonlinear system:

utt −∑j,k

Bjk(0)uxjxk+

∑j,k

Kjk ∗ uxjxk+ Lut =

∑j

gj(∂xu)xj ,

where gj(∂xu) = σj(∂xu)− σj(0)−∑

k Bjk(0)uxk

.

Nonlinear solution formula:

u(t) = G(t) ∗ u0 +H(t) ∗ u1 +∫ t

0H(t− τ) ∗ ∂xg(∂xu)(τ) dτ.

Differentiate with respect to t to get

ut(t) = Gt(t) ∗ u0 +Ht(t) ∗ u1 +∫ t

0Ht(t− τ) ∗ ∂xg(∂xu)(τ) dτ.


7. Viscothermoelasticity

Viscothermoelastic system: We consider viscothermoelastic system

utt −∑j

σj(∂xu, θ)xj +∑j,k


{e(∂xu, θ) +

1

2|ut|2

}t−

∑j

{⟨σj(∂xu, θ), ut⟩

}xj

+∑j,k


∑j,k

{bjk(∂xu, θ)θxk

}xj.

(13)

u : m-vector function of x ∈ Rn and t ≥ 0, displacement

θ : scalar function of x ∈ Rn and t ≥ 0, temperature

σj(v, θ) : m-vector functions of v ∈ Rmn and θ > 0, stress

e(v, θ) : internal energy, bjk(v, θ) : coefficients of heat conduction.


Viscothermoelasticity

Constitutive relation:

de =∑j

σjdvj + θds,

e = e(v, θ) : internal energy, σj = σj(v, θ) : stress,

s = s(v, θ) : entropy.

Helmholtz free energy: ϕ = ϕ(v, θ), ϕ = e− θs.

This gives

dϕ =∑j

σjdvj − sdθ.



Equation of internal energy:

et −∑j

⟨σj , uxjt⟩ =∑j,k

(bjkθxk)xj + ⟨Lut, ut⟩.

Equation of temperature:

eθθt −∑j

⟨θσjθ, uxjt⟩ =∑j,k


Equation of entropy:

st =∑j,k

(1θbjkθxk

)xj

+1

θ2

∑j,k

bjkθxjθxk+

1

θ⟨Lut, ut⟩.

Here e, eθ, s, σj , σjθ and bjk are evaluated at (∂xu, θ).



Viscothermoelastic system (in quasilinear form):

utt −∑j,k

Bjkuxjxk−

∑j

σjθθxj +∑j,k


eθθt −∑j



Bjk(v, θ) = Dvkσj(v, θ).

Here Bjk, eθ, σjθ and bjk are evaluated at (∂xu, θ).

Linearized system:

utt −∑j,k

Bjkuxjxk−

∑j

σjθθxj +∑j,k


eθθt −∑j


bjkθxjxk.

Here Bjk, eθ, σjθ and bjk are evaluated at constant state (0, θ).Shuichi Kawashima (Kyushu University) Mathematical Analysis for Viscoelasticity March 11, 2013 41 / 45

Energy form

Energy form:

H = e− e−∑j

⟨σj , uxj ⟩ − θ(s− s) +1

2|ut|2.

e = e(0, θ), s = s(0, θ), σj = σj(0, θ) are constant state.

Equation of energy form:

Ht −∑j

{⟨σj − σj , ut⟩}xj +∑j,k

⟨Kjk ∗ uxjxk, ut⟩

+θ

θ⟨Lut, ut⟩+

θ

θ2

∑j,k

bjkθxjθxk=

∑j,k

(bjk

θ − θ

θθxk

)xj

.


Modified equation of energy form:{H+

1

2

∑j,k


⟩+Kjk[uxj , uxk])}

t

−∑j

{⟨σj − σ, ut⟩

}xj

+∑j,k

{⟨Kjk ∗ uxk

, ut⟩}xj

+1

2

∑j,k

(− Kjk[uxj , uxk

] + ⟨Kjkuxj , uxk⟩)

+θ

θ⟨Lut, ut⟩+

θ

θ2

∑j,k

bjkθxjθxk=

∑j,k

(bjk

θ − θ

θθxk

)xj

.

This comes from the equality∑j,k


1

2

∑j,k


⟩+Kjk[uxj , uxk])t

+∑j,k

{⟨Kjk ∗ uxk

, ut⟩}xj

+1

2

∑j,k

(− Kjk[uxj , uxk

] + ⟨Kjkuxj , uxk⟩).



Research plan:

To show the sharp decay estimate (n ≥ 1) for linearized system for

Hs ∩ L1 initial data: Energy method in the Fourier space

To show the global existence and asymptotic stability (n ≥ 1) for

Hs initial data: Energy method

To show the decay estimate (n ≥ 2) for Hs initial data:


To show the sharp decay estimate (n ≥ 1) for Hs ∩ L1 initial data:

Time-weighted energy method and Semigroup argument.

This will be a joint work with Dharmawardane and Nakamura.


Thank You Very Much

for

Your Attention


Documents

· Aim and methods Aim and methods: To show the sharp decay estimate (n≥ 1) for linearized system for Hs ∩L1 initial data: Energy method in the Fourier space To show the global