Post on 23-Jun-2020
On some nonlocal nonlinear wave equations
Albert Erkip (Sabanci University, Istanbul, Turkey)
In collaboration with
Ceni Babaoglu (ITU), Handan Borluk (Isik Univ.),
Nilay Duruk (Sabanci Univ.), Husnu A. Erbay (Ozyegin Univ.),
Saadet Erbay (Ozyegin Univ.), Gulcin M. Muslu (ITU)
This work has been supported by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under the project TBAG-110R002.
5-7 November 2012
A. Erkip, Nonlocal Models and Peridynamics
Outline
Nonlocal models
Locally nonlinear, nonlocal elastic model
Peridynamic model
Double dispersive model
The Cauchy problem
Local well posedness
Global existence vs. finite time blow up
Travelling wave solutions.
Further questions
A. Erkip, Nonlocal Models and Peridynamics
Locally nonlinear, nonlocal elastic model
Equation of Motion
∂2
∂t2v(x , t) = ∇ ·
∫β(x − y)f (∇v(y , t))dy
x : space variable,
v (x , t) : displacement at time t,
∇v : strain,
f (p) = DpW (p) : stress,a general smooth nonlinear function with f (0) = 0
W (∇v) : strain energy function
The convolution with β causes the nonlocal effect.
When f is linear, one obtains Eringen’s nonlocal model.
A. Erkip, Nonlocal Models and Peridynamics
Locally nonlinear, nonlocal elastic model
Equation of Motion
∂2
∂t2v(x , t) = ∇ ·
∫β(x − y)f (∇v(y , t))dy
x : space variable,
v (x , t) : displacement at time t,
∇v : strain,
f (p) = DpW (p) : stress,a general smooth nonlinear function with f (0) = 0
W (∇v) : strain energy function
The convolution with β causes the nonlocal effect.
When f is linear, one obtains Eringen’s nonlocal model.
A. Erkip, Nonlocal Models and Peridynamics
Locally nonlinear, nonlocal elastic model
Equation of Motion
∂2
∂t2v(x , t) = ∇ ·
∫β(x − y)f (∇v(y , t))dy
x : space variable,
v (x , t) : displacement at time t,
∇v : strain,
f (p) = DpW (p) : stress,a general smooth nonlinear function with f (0) = 0
W (∇v) : strain energy function
The convolution with β causes the nonlocal effect.
When f is linear, one obtains Eringen’s nonlocal model.
A. Erkip, Nonlocal Models and Peridynamics
Locally nonlinear, nonlocal elastic model
1-d longitudal motion: with u = vx :
utt = [β ∗ f (u)]xx
Duruk, HA Erbay, Erkip: Nonlinearity 2010
1-d transverse motion:
u1tt = [β1 ∗ f1(u1, u2)]xxu2tt = [β2 ∗ f2(u1, u2)]xx
Duruk, HA Erbay, Erkip: JDE 2011
2-d anti-plane shear motion:
wtt = [β ∗ f1(wx ,wy )]x + [β ∗ f2(wx ,wy )]y
HA Erbay, S Erbay, Erkip: Nonlinearity 2011
A. Erkip, Nonlocal Models and Peridynamics
Examples for the kernel
Triangular kernel
β(x) =
1− |x | |x | ≤ 10 |x | ≥ 1.
[β ∗ v ]xx (x) = v (x + 1)− 2v (x) + v (x − 1) = ∆2v (x) ,
one gets the differential-difference equation
utt = ∆2 (u + g (u)) .
Exponential kernel
β(x) =1
2e−|x |.
[β ∗ v ]xx (x) = (1− ∂2x )−1v ,
one gets the Improved Boussinesq equation
utt − uxx − uttxx = g (u)xx .
Gaussian Kernel β(x) = e−x2
gives an integro-differentialequation.
A. Erkip, Nonlocal Models and Peridynamics
Preidynamic model
Formulation of elasticity allowing for discontinuitiesSilling: JMPS 2000.
Analysis of the linear Cauchy problemEmmrich, Weckner: Comm.Math Sci. 2007,Du, Zhou, M2AN: 2011
Analysis of the nonlinear Cauchy problem on R
utt =
∫α (x − y) w(u(y)− u(x))dy .
H.A.Erbay, Erkip, Muslu: JDE 2012
Well posedness of the nonlinear Cauchy problem
utt =
∫Ω
f (u(y)− u(x), x − y)dy .
Emmrich, Puhst: 2012
A. Erkip, Nonlocal Models and Peridynamics
Double dispersive model
Babaoglu, H. A. Erbay, Erkip: Nonlinear Anal. TMA 2013
utt − Luxx = B(g(u))xx
where L and B are linear pseudodifferential operators (in x)defined via Fourier transform
F (Lv) (ξ) = l(ξ)F(v)(ξ), F (Bv) (ξ) = b(ξ)F(v)(ξ),
of orders ρ and −r ≤ 0 respectively (i.e. |l(ξ)| = O |ξ|r as r →∞.)
The more familiar Boussinesq family,
utt − Luxx + Mutt = g(u)xx ,
is of the above form with
B = (1 + M)−1, L = L((1 + M)−1.
A. Erkip, Nonlocal Models and Peridynamics
Double dispersive model
Babaoglu, H. A. Erbay, Erkip: Nonlinear Anal. TMA 2013
utt − Luxx = B(g(u))xx
where L and B are linear pseudodifferential operators (in x)defined via Fourier transform
F (Lv) (ξ) = l(ξ)F(v)(ξ), F (Bv) (ξ) = b(ξ)F(v)(ξ),
of orders ρ and −r ≤ 0 respectively (i.e. |l(ξ)| = O |ξ|r as r →∞.)
The more familiar Boussinesq family,
utt − Luxx + Mutt = g(u)xx ,
is of the above form with
B = (1 + M)−1, L = L((1 + M)−1.
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Locally nonlinear nonlocal model
utt = [β ∗ f (u)]xx , x ∈ R, t > 0
u (x , 0) = ϕ (x) , ut (x , 0) = ψ (x) , x ∈ R.
Assumption: b(ξ) = β(ξ) ≤ C (1 + ξ2)−r2 . Regard the convolution
as a pseudodifferential operator with symbol b(ξ) of order −r .
For r ≥ 2; D2x B is of negative order; hence maps Hs → Hs . For
s > 1/2, f (u) is locally Lipschitz on Hs . Then we have an Hs
valued ODE.
Theorem: Let X be a Banach space; and let T : X → X belocally Lipschitz. Then there is some T > 0 so that initial valueproblem for the X valued ODE U ′′ = T (U), is well posed withsolution in C 2([0,T ],X ) for initial data U0,U1 ∈ X .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Locally nonlinear nonlocal model
utt = [β ∗ f (u)]xx , x ∈ R, t > 0
u (x , 0) = ϕ (x) , ut (x , 0) = ψ (x) , x ∈ R.
Assumption: b(ξ) = β(ξ) ≤ C (1 + ξ2)−r2 . Regard the convolution
as a pseudodifferential operator with symbol b(ξ) of order −r .
For r ≥ 2; D2x B is of negative order; hence maps Hs → Hs . For
s > 1/2, f (u) is locally Lipschitz on Hs . Then we have an Hs
valued ODE.
Theorem: Let X be a Banach space; and let T : X → X belocally Lipschitz. Then there is some T > 0 so that initial valueproblem for the X valued ODE U ′′ = T (U), is well posed withsolution in C 2([0,T ],X ) for initial data U0,U1 ∈ X .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Locally nonlinear nonlocal model
utt = [β ∗ f (u)]xx , x ∈ R, t > 0
u (x , 0) = ϕ (x) , ut (x , 0) = ψ (x) , x ∈ R.
Assumption: b(ξ) = β(ξ) ≤ C (1 + ξ2)−r2 . Regard the convolution
as a pseudodifferential operator with symbol b(ξ) of order −r .
For r ≥ 2; D2x B is of negative order; hence maps Hs → Hs . For
s > 1/2, f (u) is locally Lipschitz on Hs . Then we have an Hs
valued ODE.
Theorem: Let X be a Banach space; and let T : X → X belocally Lipschitz. Then there is some T > 0 so that initial valueproblem for the X valued ODE U ′′ = T (U), is well posed withsolution in C 2([0,T ],X ) for initial data U0,U1 ∈ X .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Locally nonlinear nonlocal model
utt = [β ∗ f (u)]xx , x ∈ R, t > 0
u (x , 0) = ϕ (x) , ut (x , 0) = ψ (x) , x ∈ R.
Assumption: b(ξ) = β(ξ) ≤ C (1 + ξ2)−r2 . Regard the convolution
as a pseudodifferential operator with symbol b(ξ) of order −r .
For r ≥ 2; D2x B is of negative order; hence maps Hs → Hs . For
s > 1/2, f (u) is locally Lipschitz on Hs . Then we have an Hs
valued ODE.
Theorem: Let X be a Banach space; and let T : X → X belocally Lipschitz. Then there is some T > 0 so that initial valueproblem for the X valued ODE U ′′ = T (U), is well posed withsolution in C 2([0,T ],X ) for initial data U0,U1 ∈ X .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Peridynamic model
Theorem: Let s > 12 and r ≥ 2. There is some T > 0 such that
the Cauchy problem is well-posed with solutionu ∈ C 2([0,T ],Hs(R)) for initial data ϕ,ψ ∈ Hs(R).
Extensions:For bounded data, smoothness can be pulled down to s ≥ 0 ifbeta is more regular; i.e.:
when r > r2
when βxx is a finite measure.
when βxx is a finite measure, also works for W k,p(R)∩ L∞(R)and C k
b (R), with interger k .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Peridynamic model
Theorem: Let s > 12 and r ≥ 2. There is some T > 0 such that
the Cauchy problem is well-posed with solutionu ∈ C 2([0,T ],Hs(R)) for initial data ϕ,ψ ∈ Hs(R).
Extensions:For bounded data, smoothness can be pulled down to s ≥ 0 ifbeta is more regular; i.e.:
when r > r2
when βxx is a finite measure.
when βxx is a finite measure, also works for W k,p(R)∩ L∞(R)and C k
b (R), with interger k .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Peridynamic model
utt =
∫α(x − y)w(u(y)− u(x))dy , x ∈ R, t > 0
Under integrability conditions on α the peridynamic equation is aBanach space valued ODE.
Theorem: Assume that α ∈ L1. Then there is some T > 0 suchthat the Cauchy problem is well posed with solution inC 2([0; T ]; X ) for initial data in X , where X is any of the spaces
W k,p(R),C kb (R), with interger k , if w ∈ C k+1.
Hs(R), s > 0 if w is a polynomial.
With suitable smoothness and integrability conditions on f ,extends to the general bond based (?) problem on Rn and on adomain Ω. (Emmrich-Pusht)
The Lipschitz condition seems essential (Emmrich-Pusht)
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Peridynamic model
utt =
∫α(x − y)w(u(y)− u(x))dy , x ∈ R, t > 0
Under integrability conditions on α the peridynamic equation is aBanach space valued ODE.
Theorem: Assume that α ∈ L1. Then there is some T > 0 suchthat the Cauchy problem is well posed with solution inC 2([0; T ]; X ) for initial data in X , where X is any of the spaces
W k,p(R),C kb (R), with interger k , if w ∈ C k+1.
Hs(R), s > 0 if w is a polynomial.
With suitable smoothness and integrability conditions on f ,extends to the general bond based (?) problem on Rn and on adomain Ω. (Emmrich-Pusht)
The Lipschitz condition seems essential (Emmrich-Pusht)
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Peridynamic model
utt =
∫α(x − y)w(u(y)− u(x))dy , x ∈ R, t > 0
Under integrability conditions on α the peridynamic equation is aBanach space valued ODE.
Theorem: Assume that α ∈ L1. Then there is some T > 0 suchthat the Cauchy problem is well posed with solution inC 2([0; T ]; X ) for initial data in X , where X is any of the spaces
W k,p(R),C kb (R), with interger k , if w ∈ C k+1.
Hs(R), s > 0 if w is a polynomial.
With suitable smoothness and integrability conditions on f ,extends to the general bond based (?) problem on Rn and on adomain Ω. (Emmrich-Pusht)
The Lipschitz condition seems essential (Emmrich-Pusht)
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Double dispersive model
utt − Luxx = Bg(u)xx ,
Recall: L is of order ρ, B is of order −r ≤ 0.
When ρ+ 2 > 0; the equation is not an ODE.
When L is coercive, there is hyperbolic behavior due to the
semigroup action S(t) = (−D2x L)−
12 sin((−D2
x L)12 t).
S(t) has a smoothing effect of order 1 + ρ2
Theorem: Local Existence Let L be coercive, ρ+ 2 > 0,ρ2 + 1 + r ≥ 2 and s > 1
2 . There is some T > 0 such that theCauchy problem
utt − Luxx = Bg(u)xx , −∞ < x <∞, t > 0
u(x , 0) = ϕ(x), ut(x , 0) = ψ(x) −∞ < x <∞,
is well posed with solution in C ([0; T ]; Hs) ∩ C 1([0; T ]; Hs−1− ρ2 )
for initial data (ϕ,ψ) ∈ Hs × Hs−1− ρ2 .
A. Erkip, Nonlocal Models and Peridynamics
Local well posedness: Double dispersive model
utt − Luxx = Bg(u)xx ,
Recall: L is of order ρ, B is of order −r ≤ 0.
When ρ+ 2 > 0; the equation is not an ODE.
When L is coercive, there is hyperbolic behavior due to the
semigroup action S(t) = (−D2x L)−
12 sin((−D2
x L)12 t).
S(t) has a smoothing effect of order 1 + ρ2
Theorem: Local Existence Let L be coercive, ρ+ 2 > 0,ρ2 + 1 + r ≥ 2 and s > 1
2 . There is some T > 0 such that theCauchy problem
utt − Luxx = Bg(u)xx , −∞ < x <∞, t > 0
u(x , 0) = ϕ(x), ut(x , 0) = ψ(x) −∞ < x <∞,
is well posed with solution in C ([0; T ]; Hs) ∩ C 1([0; T ]; Hs−1− ρ2 )
for initial data (ϕ,ψ) ∈ Hs × Hs−1− ρ2 .
A. Erkip, Nonlocal Models and Peridynamics
Local well-posedness: Double dispersive model
When ρ+ 2 ≤ 0; the equation
utt − Luxx = Bg(u)xx ,
is an ODE.
Theorem: Local Existence Let L be coercive, ρ+ 2 ≤ 0, r ≥ 2and s > 1
2 . There is some T > 0 such that the Cauchy problem
utt − Luxx = Bg(u)xx , −∞ < x <∞, t > 0
u(x , 0) = ϕ(x), ut(x , 0) = ψ(x) −∞ < x <∞,
is well posed with solution in C 1([0; T ]; Hs) for initial dataϕ,ψ ∈ Hs
There is no loss of derivatives.
A. Erkip, Nonlocal Models and Peridynamics
Local well-posedness: Double dispersive model
When ρ+ 2 ≤ 0; the equation
utt − Luxx = Bg(u)xx ,
is an ODE.
Theorem: Local Existence Let L be coercive, ρ+ 2 ≤ 0, r ≥ 2and s > 1
2 . There is some T > 0 such that the Cauchy problem
utt − Luxx = Bg(u)xx , −∞ < x <∞, t > 0
u(x , 0) = ϕ(x), ut(x , 0) = ψ(x) −∞ < x <∞,
is well posed with solution in C 1([0; T ]; Hs) for initial dataϕ,ψ ∈ Hs
There is no loss of derivatives.
A. Erkip, Nonlocal Models and Peridynamics
Local well-posedness: Double dispersive model
When ρ+ 2 ≤ 0; the equation
utt − Luxx = Bg(u)xx ,
is an ODE.
Theorem: Local Existence Let L be coercive, ρ+ 2 ≤ 0, r ≥ 2and s > 1
2 . There is some T > 0 such that the Cauchy problem
utt − Luxx = Bg(u)xx , −∞ < x <∞, t > 0
u(x , 0) = ϕ(x), ut(x , 0) = ψ(x) −∞ < x <∞,
is well posed with solution in C 1([0; T ]; Hs) for initial dataϕ,ψ ∈ Hs
There is no loss of derivatives.
A. Erkip, Nonlocal Models and Peridynamics
Global Existence vs. Blow-up
In all three models, global existence / blow-up is controlled by thenonlinearity; in turn by the L∞ norm of u (t).
Lemma: For sufficiently smooth f , and v ∈ Hs ∩ L∞, s ≥ 0,
‖v‖Hs ≤ C (‖v‖L∞)‖v‖Hs
Theorem: (Global existence criterion) The solution of theCauchy problem exists for all times if and only if for any T > 0
lim supt→T−
‖u (t)‖L∞ <∞.
The L∞ control is achieved via energy indentities.
We can define several ”energy” terms E (t), with E ′(t) = 0.
A. Erkip, Nonlocal Models and Peridynamics
Global Existence vs. Blow-up
In all three models, global existence / blow-up is controlled by thenonlinearity; in turn by the L∞ norm of u (t).
Lemma: For sufficiently smooth f , and v ∈ Hs ∩ L∞, s ≥ 0,
‖v‖Hs ≤ C (‖v‖L∞)‖v‖Hs
Theorem: (Global existence criterion) The solution of theCauchy problem exists for all times if and only if for any T > 0
lim supt→T−
‖u (t)‖L∞ <∞.
The L∞ control is achieved via energy indentities.
We can define several ”energy” terms E (t), with E ′(t) = 0.
A. Erkip, Nonlocal Models and Peridynamics
Energy identities (conserved quantities)
For the locally nonlinear nonlocal equation
Recall, f = DW ,
E1(t) =1
2‖ut(t)‖2
L2 +
∫ ∫βxx(x − y)W (u(y , t))dydx .
In terms of displacement v , (u = vx) vtt = [β ∗ f (vx)]x ,
E2(t) =1
2‖vt(t)‖2
L2 +
∫ ∫β(x − y)W (u(y , t))dydx .
When β(ξ) > 0; via Ph = F−1(β(ξ)−1/2h(ξ)),
E3(t) =1
2‖Pvt(t)‖2
L2 +
∫W (u(x , t))dx .
For peridynamic equation, utt =∫α(x − y)w(u(y)− u(x))dy ,
E1(x , t) =1
2‖ut(t)‖2
L2 +
∫ ∫α(x − y)W (u(y , t)− u (x , t))dydx .
A. Erkip, Nonlocal Models and Peridynamics
Energy identities (conserved quantities)
For the locally nonlinear nonlocal equation
Recall, f = DW ,
E1(t) =1
2‖ut(t)‖2
L2 +
∫ ∫βxx(x − y)W (u(y , t))dydx .
In terms of displacement v , (u = vx) vtt = [β ∗ f (vx)]x ,
E2(t) =1
2‖vt(t)‖2
L2 +
∫ ∫β(x − y)W (u(y , t))dydx .
When β(ξ) > 0; via Ph = F−1(β(ξ)−1/2h(ξ)),
E3(t) =1
2‖Pvt(t)‖2
L2 +
∫W (u(x , t))dx .
For peridynamic equation, utt =∫α(x − y)w(u(y)− u(x))dy ,
E1(x , t) =1
2‖ut(t)‖2
L2 +
∫ ∫α(x − y)W (u(y , t)− u (x , t))dydx .
A. Erkip, Nonlocal Models and Peridynamics
Energy identities (conserved quantities)
For the locally nonlinear nonlocal equation
Recall, f = DW ,
E1(t) =1
2‖ut(t)‖2
L2 +
∫ ∫βxx(x − y)W (u(y , t))dydx .
In terms of displacement v , (u = vx) vtt = [β ∗ f (vx)]x ,
E2(t) =1
2‖vt(t)‖2
L2 +
∫ ∫β(x − y)W (u(y , t))dydx .
When β(ξ) > 0; via Ph = F−1(β(ξ)−1/2h(ξ)),
E3(t) =1
2‖Pvt(t)‖2
L2 +
∫W (u(x , t))dx .
For peridynamic equation, utt =∫α(x − y)w(u(y)− u(x))dy ,
E1(x , t) =1
2‖ut(t)‖2
L2 +
∫ ∫α(x − y)W (u(y , t)− u (x , t))dydx .
A. Erkip, Nonlocal Models and Peridynamics
Energy identities (conserved quantities)
For the locally nonlinear nonlocal equation
Recall, f = DW ,
E1(t) =1
2‖ut(t)‖2
L2 +
∫ ∫βxx(x − y)W (u(y , t))dydx .
In terms of displacement v , (u = vx) vtt = [β ∗ f (vx)]x ,
E2(t) =1
2‖vt(t)‖2
L2 +
∫ ∫β(x − y)W (u(y , t))dydx .
When β(ξ) > 0; via Ph = F−1(β(ξ)−1/2h(ξ)),
E3(t) =1
2‖Pvt(t)‖2
L2 +
∫W (u(x , t))dx .
For peridynamic equation, utt =∫α(x − y)w(u(y)− u(x))dy ,
E1(x , t) =1
2‖ut(t)‖2
L2 +
∫ ∫α(x − y)W (u(y , t)− u (x , t))dydx .
A. Erkip, Nonlocal Models and Peridynamics
Global Existence vs. Blow-up
The locally nonlinear nonlocal equation
Theorem: Suppose r > 3 and initial data is sufficiently smooth. Ifthere is some k > 0 so that W (r) ≥ −kr 2. Then the Cauchyproblem has a global solution.
Theorem: Let the kernel satisfiy βxx ∗ v = h ∗ v − λv for someλ ≥ 0 and for some h ∈ L1 ∩ L∞. Suppose initial data issufficiently smooth. If there is some C > 0 and q > 1 so that|f (r)|q ≤ CW (r). Then the Cauchy problem has a globalsolution.
Peridynamic equation
Theorem: Suppose w (η) = |η|υ−1 η with υ ≤ 3 and α ∈ L1 ∩ L∞
with α ≥ 0 a.e.. Then the Cauchy problem has a global solution.
A. Erkip, Nonlocal Models and Peridynamics
Global Existence vs. Blow-up
The locally nonlinear nonlocal equation
Theorem: Suppose r > 3 and initial data is sufficiently smooth. Ifthere is some k > 0 so that W (r) ≥ −kr 2. Then the Cauchyproblem has a global solution.
Theorem: Let the kernel satisfiy βxx ∗ v = h ∗ v − λv for someλ ≥ 0 and for some h ∈ L1 ∩ L∞. Suppose initial data issufficiently smooth. If there is some C > 0 and q > 1 so that|f (r)|q ≤ CW (r). Then the Cauchy problem has a globalsolution.
Peridynamic equation
Theorem: Suppose w (η) = |η|υ−1 η with υ ≤ 3 and α ∈ L1 ∩ L∞
with α ≥ 0 a.e.. Then the Cauchy problem has a global solution.
A. Erkip, Nonlocal Models and Peridynamics
Global Existence vs. Blow-up
The locally nonlinear nonlocal equation
Theorem: Suppose r > 3 and initial data is sufficiently smooth. Ifthere is some k > 0 so that W (r) ≥ −kr 2. Then the Cauchyproblem has a global solution.
Theorem: Let the kernel satisfiy βxx ∗ v = h ∗ v − λv for someλ ≥ 0 and for some h ∈ L1 ∩ L∞. Suppose initial data issufficiently smooth. If there is some C > 0 and q > 1 so that|f (r)|q ≤ CW (r). Then the Cauchy problem has a globalsolution.
Peridynamic equation
Theorem: Suppose w (η) = |η|υ−1 η with υ ≤ 3 and α ∈ L1 ∩ L∞
with α ≥ 0 a.e.. Then the Cauchy problem has a global solution.
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
HA Erbay, S Erbay, Erkip (in preparation)
We assume L and B are coercive. Consider the double dispersiveequation
utt − Luxx = B(g(u))xx , g(u) = ±|u|p−1u.
A travelling wave solution u(x , t) = ϕ(x − ct) with velocity csatisfies
(L− c2I )B−1ϕ+ g(ϕ) = 0.
Variational problem: Extremal points of:
I (v) =1
2(‖L1/2B−1/2v‖2
L2 − c2‖B−1/2v‖2L2)
subject to ‖v‖Lp+1 = 1 , are travelling waves
We use Lion’s ”concentration compactness” principle.
In general travelling waves are not unique even up totranslation.
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
Theorem: Let L and B be coercive of orders ρ and −r ≤ 0respectively. Then there are constants C ,D (depending on thesymbols l(ξ), b(ξ)) so that the double dispersive equation hastravelling wave solutions with velocity c ,
for all c with for all c2 ≤ C 2, when ρ ≥ 0,
for all c with for all c2 ≥ D2, when ρ ≤ 0.
When ρ ≥ 0, orbitally stability depends on the concavity of acertain ”degree function”. In particular, travelling wave solutionsare orbitally stable for sufficiently small c .
When ρ ≤ 0 and c2 ≥ D2, this is not the case; extreme points arenot ground states of the energy.
Stubbe, Portugaliae Mathematica, 1989.
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
Theorem: Let L and B be coercive of orders ρ and −r ≤ 0respectively. Then there are constants C ,D (depending on thesymbols l(ξ), b(ξ)) so that the double dispersive equation hastravelling wave solutions with velocity c ,
for all c with for all c2 ≤ C 2, when ρ ≥ 0,
for all c with for all c2 ≥ D2, when ρ ≤ 0.
When ρ ≥ 0, orbitally stability depends on the concavity of acertain ”degree function”. In particular, travelling wave solutionsare orbitally stable for sufficiently small c .
When ρ ≤ 0 and c2 ≥ D2, this is not the case; extreme points arenot ground states of the energy.
Stubbe, Portugaliae Mathematica, 1989.
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
In the ”Boussineq form”: utt − Luxx + Mutt = g(u)xx ,
with L, M of orders sL, sM ≥ 0, respectively.
B = (1 + M)−1, L = L((1 + M)−1., we have ρ = sL − sM . So:
When sL − sM > 0, there are travelling waves for c2 < C 2,possible orbital stability.
When sL − sM < 0, there are travelling waves for c2 < D2, noorbital stability.
When sL − sM = 0, both regimes may occur.
Examples:
The good Boussinesq equation L = 1−D2x , M = 0, sL = 2, sM = 0.
Improved Boussinesq equation L = 1, M = 1− D2x , sL = 0, sM = 2.
Stubbe observed this for the case L = a0 + a|Dx |µ, M = 1 + p|Dx |µ
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
In the ”Boussineq form”: utt − Luxx + Mutt = g(u)xx ,
with L, M of orders sL, sM ≥ 0, respectively.
B = (1 + M)−1, L = L((1 + M)−1., we have ρ = sL − sM . So:
When sL − sM > 0, there are travelling waves for c2 < C 2,possible orbital stability.
When sL − sM < 0, there are travelling waves for c2 < D2, noorbital stability.
When sL − sM = 0, both regimes may occur.
Examples:
The good Boussinesq equation L = 1−D2x , M = 0, sL = 2, sM = 0.
Improved Boussinesq equation L = 1, M = 1− D2x , sL = 0, sM = 2.
Stubbe observed this for the case L = a0 + a|Dx |µ, M = 1 + p|Dx |µ
A. Erkip, Nonlocal Models and Peridynamics
Travelling wave solutions
In the ”Boussineq form”: utt − Luxx + Mutt = g(u)xx ,
with L, M of orders sL, sM ≥ 0, respectively.
B = (1 + M)−1, L = L((1 + M)−1., we have ρ = sL − sM . So:
When sL − sM > 0, there are travelling waves for c2 < C 2,possible orbital stability.
When sL − sM < 0, there are travelling waves for c2 < D2, noorbital stability.
When sL − sM = 0, both regimes may occur.
Examples:
The good Boussinesq equation L = 1−D2x , M = 0, sL = 2, sM = 0.
Improved Boussinesq equation L = 1, M = 1− D2x , sL = 0, sM = 2.
Stubbe observed this for the case L = a0 + a|Dx |µ, M = 1 + p|Dx |µ
A. Erkip, Nonlocal Models and Peridynamics
Further Questions, Nonlocal problems on a domain
An example The Improved Boussineq equation on R can bewritten as utt = [
∫R
12 e−|x−y |f (u(y))dy ]xx .
On an interval [a,b] one can consider either
utt = [
∫ b
a
1
2e−|x−y |f (u(y))dy ]xx , (I )
or
utt = [
∫ b
a(
1
2e−|x−y | + k(x , y))f (u(y))dy ]xx , (II )
where G (x , y) = 12 e−|x−y | + k(x , y) is the Green’s function for
1− D2x with suitable boundary conditions; both relate to the
Improved Boussinesq equation on [a, b].
The interpretation in (I ) does not allow consideration of smoothsolutions; whereas for (II ) one can use eigenfunctions of 1− D2
x
with the boundary conditions for a full analysis.
A. Erkip, Nonlocal Models and Peridynamics
Further Questions, Nonlocal problems on a domain
An example The Improved Boussineq equation on R can bewritten as utt = [
∫R
12 e−|x−y |f (u(y))dy ]xx .
On an interval [a,b] one can consider either
utt = [
∫ b
a
1
2e−|x−y |f (u(y))dy ]xx , (I )
or
utt = [
∫ b
a(
1
2e−|x−y | + k(x , y))f (u(y))dy ]xx , (II )
where G (x , y) = 12 e−|x−y | + k(x , y) is the Green’s function for
1− D2x with suitable boundary conditions; both relate to the
Improved Boussinesq equation on [a, b].
The interpretation in (I ) does not allow consideration of smoothsolutions; whereas for (II ) one can use eigenfunctions of 1− D2
x
with the boundary conditions for a full analysis.
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Scaling properties
Let βλ(x) = λ−nβ(xλ−1).
As λ→ 0, the ”λ” problem tends to the equation of classicalelasticity.
What happens to the solution?
Same question (with different scaling) for the nonlinearperidynamic problem.
Limiting properies
More generally suppose the kernels βj tend to a certain β0 in somesense. What happens to the solutions?
Partial result in ”good” cases.
Peridynamic problem?
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Scaling properties
Let βλ(x) = λ−nβ(xλ−1).
As λ→ 0, the ”λ” problem tends to the equation of classicalelasticity.
What happens to the solution?
Same question (with different scaling) for the nonlinearperidynamic problem.
Limiting properies
More generally suppose the kernels βj tend to a certain β0 in somesense. What happens to the solutions?
Partial result in ”good” cases.
Peridynamic problem?
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Numerical results
Some initial experiments.
May shed light to the previous questions.
Asymptotic behavior of solutions
The double dispersive problem is promising.
Decay stimates for the semigroup S(t).
May lead to dealing with less smooth data.
How to deal with more complicated problems
The state based peridynamic problem.
Non ODE cases: Bad kernels; The case r < 2 for the locallynonlinear nonlocal problem.
Thank you
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Numerical results
Some initial experiments.
May shed light to the previous questions.
Asymptotic behavior of solutions
The double dispersive problem is promising.
Decay stimates for the semigroup S(t).
May lead to dealing with less smooth data.
How to deal with more complicated problems
The state based peridynamic problem.
Non ODE cases: Bad kernels; The case r < 2 for the locallynonlinear nonlocal problem.
Thank you
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Numerical results
Some initial experiments.
May shed light to the previous questions.
Asymptotic behavior of solutions
The double dispersive problem is promising.
Decay stimates for the semigroup S(t).
May lead to dealing with less smooth data.
How to deal with more complicated problems
The state based peridynamic problem.
Non ODE cases: Bad kernels; The case r < 2 for the locallynonlinear nonlocal problem.
Thank you
A. Erkip, Nonlocal Models and Peridynamics
Further Questions
Numerical results
Some initial experiments.
May shed light to the previous questions.
Asymptotic behavior of solutions
The double dispersive problem is promising.
Decay stimates for the semigroup S(t).
May lead to dealing with less smooth data.
How to deal with more complicated problems
The state based peridynamic problem.
Non ODE cases: Bad kernels; The case r < 2 for the locallynonlinear nonlocal problem.
Thank you
A. Erkip, Nonlocal Models and Peridynamics
References
C. Babaoglu, H. A. Erbay, A. Erkip, ”Global exitence and blow-up of solutionsfor a general class of doublly dispersive nonlocal nonlineer wave equations”Nonlinear Anal. TMA 73 2013 82-93.
Q. Du, K. Zhou, ”Mathematical analysis for the peridynamic nonlocal continuumtheory” M2AN Math. Model. Numer. Anal. 45 2011, 217-234.
N. Duruk, H.A.Erbay, A. Erkip, ”Global existence and blow-up for a class ofnonlocal nonlinear Cauchy problems arising in elasticity”, Nonlinearity, Vol.23,2010, 107-118.
N. Duruk Mutlubas, H.A. Erbay, A. Erkip, ” Blow-up and global existence for ageneral class of nonlocal nonlinear coupled wave equations” Journal ofDifferential Equations, Vol.250, 2011, 1448-1459.
E. Emmrich, O. Weckner, ”On the well-posedness of the linear peridynamicsmodel and its convergence towards the Navier equation of linear elasticity”,Communications in Mathematical Sciences, Vol. 5, 2007, 851-864 .
E. Emmrich, D. Puhst, ”Well-posedness of the peridynamic model with Lipschitzcontinuous pairwise force function”, preprint 2012
A. Erkip, Nonlocal Models and Peridynamics
References
H.A. Erbay, S. Erbay, A. Erkip, ”The Cauchy problem for a class oftwo-dimensional nonlocal nonlinear wave equations governing anti-plane shearmotions in elastic materials”, Nonlinearity, Vol.24, 2011, 1347-1359.
H. A. Erbay, A. Erkip and G. M. Muslu, ”The Cauchy problem for the onedimensional nonlinear peridynamic model”, Journal of Differential Equations,Vol.252, 2012, 4392-4409.
S. A. Silling, ”Reformulation of elasticity theory for discontinuities and long-rangeforces”, Journal of the Mechanics and Physics of Solids, Vol. 48, 2000, 175-209.
J. Stubbe, ”Existence and stability of solitary waves of Boussinesq-typeequations”, Portugaliae Mathematica, Vol. 46, 1989, 501-516.
A. Erkip, Nonlocal Models and Peridynamics