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Asymptotic stabilization of the hyperelastic-rod wave equation Giuseppe Maria Coclite Department of Mathematics University of Bari Via E. Orabona 4 70125 Bari (Italy) EMAIL: coclitegm@dm.uniba.it URL:

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Asymptotic stabilization

  • f the hyperelastic-rod wave equation

Giuseppe Maria Coclite

Department of Mathematics University of Bari Via E. Orabona 4 70125 Bari (Italy) EMAIL: coclitegm@dm.uniba.it URL: http://www.dm.uniba.it/Members/coclitegm/

Padova HYP 2012 joint work with Prof. F. Ancona (Padova)

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 1 / 32

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The Hyperelastic-Rod Wave Equation

∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • t ≥ 0 time

x ∈ R space (one-dimensional) u(t, x) ∈ R unknown (one-dimensional) γ > 0 is a given constant

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 2 / 32

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Physics

Hyperelastic-Rod Waves

Finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. γ depends on the material constants and the pre-stress of the rod. Dai (1998 - 1998) Dai & Huo (2002)

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 3 / 32

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Shallow Water Waves

Shallow Water Waves

  • Depth of the water <

< Length of the waves γ = 1 unidirectional shallow water waves u ≡ wave velocity above the bottom flat bottom Camassa-Holm equation Camassa & Holm (1993) Johnson (2002)

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 4 / 32

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The Hyperelastic-Rod Wave Equation ∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • is formally equivalent to the elliptic-hyperbolic system

∂tu + γu∂xu + ∂xP = 0, −∂2

xxP + P = 3 − γ

2 u2 + γ 2 (∂xu)2 . Since e−|x| 2 is the Green’s function of the Helmholtz operator −∂2

xx + 1

P(t, x) =1 2

  • R

e−|x−y|

3 − γ

2 u(t, y)2 + γ 2 (∂xu(t, y))2

  • dy,

∂xP(t, x) =1 2

  • R

e−|x−y|sign (y − x)

3 − γ

2 u(t, y)2 + γ 2 (∂xu(t, y))2

  • dy.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 5 / 32

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Asymptotic Stabilization

Problem

Find an operator f : H1(R) − → H−1(R) such that for every initial condition u0 ∈ H1(R) the solution of the Cauchy problem

  

∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • + f [u]

u(0, x) = u0(x) decays as t − → ∞, i.e., lim

t− →∞ u(t, x) = 0.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 6 / 32

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Physics

Damp the waves on hyperelastic rods f [u] ≡ external force

H1 Regularity

Weak solutions

u(t, ·) ∈ H1(R) u(t, ·) is bounded and continuous Blow-up of ∂xu

More regularity on the initial conditions gives more regularity of the solutions (there are several smooth solutions) Solitons and peakons are weak solutions

Literature

Glass (2008): source compactly sopported type feedback, H2 solutions Perrollaz (2010): boundary feedback, H2 solutions

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 7 / 32

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Our feedback law

f [u] = −λ(1 − ∂2

xx)u,

λ > 0 ∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • −λ(1 − ∂2

xx)u

  • ∂tu + γu∂xu + ∂xP = −λu

−∂2

xxP + P = 3−γ 2 u2 + γ 2 (∂xu)2

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 8 / 32

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Formal Energy Estimate

  • ∂tu + γu∂xu + ∂xP = −λu

−∂2

xxP + P = 3−γ 2 u2 + γ 2 (∂xu)2

⇓ ∂t

  • u2 + (∂xu)2

2

  • +∂x

γ

2u(∂xu)2 − 1 − γ 2 u3 + uP

  • = −λ
  • u2 + (∂xu)2

The total energy E(t) := u(t, ·)2

H1(R) =

  • R
  • u(t, x)2 + (∂xu(t, x))2

dx satisfies the following ordinary differential equation d dt E(t) = −2λE(t) and therefore E(t) = E(0)e−2λt, t ≥ 0.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 9 / 32

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Definition (Weak Solutions)

A function u : [0, ∞) × R − → R is a weak solution of the Cauchy problem

  

∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • u − λ(1 − ∂2

xx)u

u(0, x) = u0(x) if u is continuous; u(t, ·) ∈ H1(R) at every t ∈ [0, ∞); the map t − → u(t, ·) is Lipschitz continuous from [0, ∞) into L2(R) and satisfies the initial condition together with the following equality between functions in L2(R): d dt u = −γu∂xu − ∂xP − λu, for a.e. t ∈ [0, ∞).

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 10 / 32

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The Main Result

Let γ, λ > 0 be fixed. There exists a semigroup S : [0, ∞) × H1(R) − → H1(R) such that the following properties hold. For every u0 ∈ H1(R), u(t, x) = St(u0)(x) is a weak solution of

  

∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • u − λ(1 − ∂2

xx)u,

u(0, x) = u0(x). E(t) ≤ E(0)e−2λt, t ≥ 0. For every u0 ∈ H1(R) there exists a constant C > 0 such that ∂xSt(u0)(x) ≤ C

  • 1 + 1

t

  • ,

t > 0. For every {u0,n}n ⊂ H1(R) and u0 ∈ H1(R) u0,n − → u0 in H1(R) = ⇒ S(u0,n) − → S(u0) in L∞

loc((0, ∞) × R).

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 11 / 32

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Remark

Semigroup of solutions

no uniqueness of weak solutions solitons interaction conservative and dissipative solutions

Energy exponential decay Oleinik type estimate

∂xu is bounded from above ∂xu may go to −∞

S is not continuous as a map with values in H1.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 12 / 32

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Strategy

We introduce a new set of variable that solves a semilinear system of

  • rdinary differential equations.

Short time existence of solutions for the semilinear system. An energy estimate gives the existence of global in time solutions for the semilinear system. Continuous dependence with respect to the initial conditions for the semilinear system. We come back to the original variables and prove our result. Bressan & Constantin (Anal. Appl. - 2007)

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 13 / 32

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New variables

Let u be a smooth solution of

  

∂tu − ∂3

txxu + 3u∂xu = γ

  • 2∂xu∂2

xxu + u∂3 xxxu

  • u − λ(1 − ∂2

xx)u

u(0, x) = u0(x). Therefore (u, P) is a smooth solution of

      

∂tu + γu∂xu + ∂xP = −λu −∂2

xxP + P = 3−γ 2 u2 + γ 2 (∂xu)2

u(0, x) = u0(x).

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 14 / 32

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Energy variable ξ ∈ R. The map y ∈ R − →

y

  • 1 + (∂xu0)2

dx is continuous, increasing, and goes to ∞ and −∞ as y − → ∞ and y − → −∞, respectively. So we can define implicitly the function y0 = y0(ξ) by the relation

y0(ξ)

  • 1 + (∂xu0)2

dx = ξ, ξ ∈ R. ξ plays the role of a Lagrangian variable (it is constant along characteristics)

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 15 / 32

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Characteristic curve t − → y(t, ξ) ∂ty(t, ξ) = γu(t, y(t, ξ)), y(0, ξ) = y0(ξ). Notation u(t, ξ) := u(t, y(t, ξ)), P(t, ξ) := P(t, y(t, ξ)). New variables v = v(t, ξ) and q = q(t, ξ) v := 2 arctan(∂xu), q := (1 + (∂xu)2)∂ξy. v is bounded q ≥ 0

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 16 / 32

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The semilinear system for u = u(t, ξ), v = v(t, ξ), q = q(t, ξ)

                        

∂tu = −∂xP − λu ∂tv =

  • 3−γ

2 u2 − P

  • (1 + cos(v)) − γ sin2 v

2

− λ sin(v)

∂tq =

  • 3−γ

2 u2 − P + γ 2

  • sin(v)q − 2λ sin2 v

2

q

u(0, ξ) = u0(y0(ξ)) v(0, ξ) = 2 arctan(∂xu0(y0(ξ))) q(0, ξ) = 1

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 17 / 32

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The nonlocal term P = P(t, ξ) P(t, ξ) =1 2

  • R

e

−

  • ξ′

ξ

cos2 v(t,s)

2

  • q(t,s)ds
  • ×

×

3 − γ

2 u(t, ξ′)2 cos2

v(t, ξ′)

2

  • + γ

2 sin2

v(t, ξ′)

2

  • ×

× q(t, ξ′)dξ′, ∂xP(t, ξ) =1 2

  • R

e

−

  • ξ′

ξ

cos2 v(t,s)

2

  • q(t,s)ds
  • ×

× sign

ξ − ξ′ ×

×

3 − γ

2 u(t, ξ′)2 cos2

v(t, ξ′)

2

  • + γ

2 sin2

v(t, ξ′)

2

  • ×

× q(t, ξ′)dξ′.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 18 / 32

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In order to obtain global dissipative solutions, a modification of the system for u, v, q is needed. Assume that, along a given characteristic t − → y(t, ξ), the wave breaks at a first time t = τ(ξ). Arguing as for the Burgers equation and reminding that ∂xu satisfies an Oleinik type inequality, the wave break means ∂xu(t, ξ) − → −∞, as t − → τ(ξ)−. For all t ≥ τ(ξ) we then set v(t, ξ) ≡ −π and remove the values of u(t, ξ), v(t, ξ), q(t, ξ) from the computation of P and ∂xP.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 19 / 32

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The dissipative semilinear system for u = u(t, ξ), v = v(t, ξ), q = q(t, ξ)

                                            

∂tu = −∂xP − λu ∂tv =

  

  • 3−γ

2 u2 − P

  • (1 + cos(v)) − γ sin2 v

2

− λ sin(v)

if v > −π if v ≤ −π ∂tq =

  

  • 3−γ

2 u2 − P + γ 2

  • sin(v)q − 2λ sin2 v

2

q

if v > −π if v ≤ −π u(0, ξ) = u0(y0(ξ)) v(0, ξ) = 2 arctan(∂xu0(y0(ξ))) q(0, ξ) = 1

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 20 / 32

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The dissipative nonlocal term P = P(t, ξ) P(t, ξ) =1 2

  • {v(ξ′)>−π}

e

−

  • {ξ′≤ξ, v(ξ′)>−π} cos2 v(t,s)

2

  • q(t,s)ds
  • ×

×

3 − γ

2 u(t, ξ′)2 cos2

v(t, ξ′)

2

  • + γ

2 sin2

v(t, ξ′)

2

  • ×

× q(t, ξ′)dξ′, ∂xP(t, ξ) =1 2

  • {v(ξ′)>−π}

e

−

  • {ξ′≤ξ, v(ξ′)>−π} cos2 v(t,s)

2

  • q(t,s)ds
  • ×

× sign

ξ − ξ′ ×

×

3 − γ

2 u(t, ξ′)2 cos2

v(t, ξ′)

2

  • + γ

2 sin2

v(t, ξ′)

2

  • ×

× q(t, ξ′)dξ′.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 21 / 32

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Global existence and uniqueness for the semilinear system

Local existence and uniqueness General theorem on directionally continuous ordinary differential equations in functional spaces Bressan & Shen - 2006 Global existence Energy estimate Global bound on the total energy E(t) = u(t, ·)2

H1(R) =

  • R
  • u(t, x)2 + (∂xu(t, x))2

dx =

  • {v(t,ξ)>−π}
  • u2(t, ξ) cos2

v(t, ξ)

2

  • + sin2

v(t, ξ)

2

  • q(t, ξ)dξ

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 22 / 32

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Stability for the semilinear system

Claim

Let {u0,n}n ⊂ H1(R) and u0 ∈ H1(R). If u0,n − → u0 in H1(R), then un − → u in L∞((0, T) × R) for every T > 0, where un and u are the solutions of the semilinear dissipative system in correspondence of u0,n and u0, respectively.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 23 / 32

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Let (u, v, q) and ( u, v, q) be any two solutions of the semilinear dissipative system and T > 0. For every ξ ∈ {ξ ∈ R; v(T, ξ) = −π} ∪ {ξ ∈ R; v(T, ξ) = −π}. Let τ(ξ) be the first time at which one of the two solutions reaches the value −π, namely τ(ξ) = inf{t ∈ [0, T]; min{v(t, ξ), v(t, ξ)} = −π}. Since the map τ(·) is measurable, we can construct a measure-preserving, measurable map α − → ξ(α) from [0, α∗] onto Λ such that α ≤ α′ ⇐ ⇒ τ(ξ(α′)) ≤ τ(ξ(α)). The inverse mapping ξ − → α(ξ) from Λ into [0, α∗] is still measure-preserving.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 24 / 32

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Distance functional j(t) =J((u(t, ·), v(t, ·), q(t, ·)), ( u(t, ·), v(t, ·), q(t, ·))) =u(t, ·) − u(t, ·)L∞(R) + v(t, ·) − v(t, ·)L2(R) + q(t, ·) − q(t, ·)L2(R) + K0

α∗

eKα|v(t, ξ(α)) − v(t, ξ(α))|dα, where K and K0 are two positive constants.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 25 / 32

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Choosing suitably K and K0 d dt j(t) ≤ Mj(t), for some constant M > 0. Therefore j(t) ≤ eMtj(0), 0 ≤ t ≤ T, and in particular u(t, ·) − u(t, ·)L∞(R) ≤ ceMt u(0, ·) − u(0, ·)H1(R) .

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 26 / 32

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Global Dissipative Solutions in the Original Variables u = u(t, x), P = P(t, x)

Let (u, v, q) be the solution of the semilinear system. Define y(t, ξ) = y0(ξ) +

t

u(τ, ξ)dτ. For each fixed ξ, the function t − → y(t, ξ) solves ∂ty(t, ξ) = u(t, ξ), y(0, ξ) = y0(ξ). We set u(t, x) = u(t, ξ) if y(t, ξ) = x. Clearly u(0, x) = u0(x) x ∈ R.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 27 / 32

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The facts the energy estimate on u(t, ·)H1(R) the image of the singular set where v = −π has measure zero (in the x-variable), i.e., meas({y(t, ξ); v(t, ξ) = −π}) = 0 give that u is continuous t − → u(t, ·) ∈ L2(R) is Lipschitz continuous

d dt u = −γu∂xu − ∂xP − λu.

Therefore u is a weak solution of the hyperelastic rod wave equation.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 28 / 32

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The Semigroup

Given an initial data u0 ∈ H1(R), we denote by u(t, x) = St(u0)(x) the corresponding global solution of the hyperelastic-rod wave equation. We have to prove St(Sτ(u0)) = St+τ(u0), t, τ > 0. Let (u, v, q) be the solution of the problem in the auxiliary variables. Call

  • u = Sτ(u0). We choose ξ0 such that

y(τ, ξ0) = 0 and consider the new energy variable σ = σ(ξ) as a solution of d dξ σ(ξ) =

  • q(τ, ξ)

if v(τ, ξ) > −π, if v(τ, ξ) = −π, σ(ξ0) = 0. We have

y(τ,ξ)

  • 1 + (∂xu(τ, x))2

dx = σ(ξ).

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 29 / 32

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The map ξ = ξ(σ) ξ(σ) = sup{s; σ(s) ≤ σ} provides almost everywhere an inverse of σ(·). Define

  • u(t, σ) =u(τ + t, ξ(σ)),
  • v(t, σ) =v(τ + t, ξ(σ)),
  • q(t, σ) =q(τ + t, ξ(σ))

q(τ, ξ(σ)) . Since ( u, v, q) solves the same equations of (u, v, q). S is a semigroup.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 30 / 32

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The decay as t − → ∞

Given an initial data u0 ∈ H1(R), let u(t, x) be the corresponding global solution of the hyperelastic-rod wave equation and (u, v, q) be the solution

  • f the problem in the auxiliary variables. Consider

E(t) = u(t, ·)2

H1(R) =

  • R
  • u(t, x)2 + (∂xu(t, x))2

dx =

  • {v(t,ξ)>−π}
  • u2(t, ξ) cos2

v(t, ξ)

2

  • + sin2

v(t, ξ)

2

  • q(t, ξ)dξ.

We have d dt E ≤ −2λE(t). Therefore E(t) ≤ e−2λtE(0), t ≥ 0.

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 31 / 32

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The Oleinik type estimate

Since ∂tv

  • v=π =

3 − γ

2 u2 − P

  • (1+cos(v))−γ sin2

v

2

  • −λ sin(v)
  • v=π = −γ,

we can choose δ > 0 so that ∂tv(t, ξ) ≤ −γ 2, v ∈ [π − δ, π). As a consequence v(t, ξ) < min

  • π − δ, π − tγ

2

  • ,

v ∈ [π − δ, π). Hence, ∂xu ≤ C

  • 1 + 1

t

  • follows from the identity

∂xu = sin(v) 1 + cos(v).

Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 32 / 32