Travelling waves for a nonlocal KdV-Burgersequation
Sabine Hittmeir
University of Vienna
joint work with:
Franz Achleitner, Carlota Cuesta, Christian Schmeiser
Anacapri, September 2015
Outline
Motivation
Nonlinear conservation laws with nonlocal diffusion
Travelling waves for the fractional KdV-Burgers equation
Motivation
The inviscid Burgers equation
∂tu + ∂xu2 = 0 (1)
has shock solutions u(x , t) = φ(x − ct) = φ(ξ) of the form
φ(ξ) =
{φ− ξ < 0φ+ ξ > 0
For {φ−, φ+, c} the Rankine-Hugoniot condition (RHC) has to hold
−c(φ+ − φ−) + φ2+ − φ2
− = 0 , i.e. c = φ+ + φ−
Both cases φ− > φ+ and φ− < φ+ provide solutions to (1).
How to obtain uniqueness?
Motivation
The inviscid Burgers equation
∂tu + ∂xu2 = 0 (1)
has shock solutions u(x , t) = φ(x − ct) = φ(ξ) of the form
φ(ξ) =
{φ− ξ < 0φ+ ξ > 0
For {φ−, φ+, c} the Rankine-Hugoniot condition (RHC) has to hold
−c(φ+ − φ−) + φ2+ − φ2
− = 0 , i.e. c = φ+ + φ−
Both cases φ− > φ+ and φ− < φ+ provide solutions to (1).
How to obtain uniqueness?
Motivation
The inviscid Burgers equation
∂tu + ∂xu2 = 0 (1)
has shock solutions u(x , t) = φ(x − ct) = φ(ξ) of the form
φ(ξ) =
{φ− ξ < 0φ+ ξ > 0
For {φ−, φ+, c} the Rankine-Hugoniot condition (RHC) has to hold
−c(φ+ − φ−) + φ2+ − φ2
− = 0 , i.e. c = φ+ + φ−
Both cases φ− > φ+ and φ− < φ+ provide solutions to (1).
How to obtain uniqueness?
Travelling waves for the viscous Burgers equation
∂tu + ∂xu2 = ∂2
xu ,
The travelling wave equation for u(t, x) = φ(ξ) with ξ = x − ct reads
h(φ(ξ)) := −c(φ(ξ)− φ−) + φ2(ξ)− φ2− = φ′(ξ)
The RHC is equivalent to h(φ+) = h(φ−) = 0.
Φ+ Φ-
Φ
hHΦL
Ξ
ΦHΞL
We obtain the entropy condition
φ− > φ+ .
Travelling waves for the viscous Burgers equation
∂tu + ∂xu2 = ∂2
xu ,
The travelling wave equation for u(t, x) = φ(ξ) with ξ = x − ct reads
h(φ(ξ)) := −c(φ(ξ)− φ−) + φ2(ξ)− φ2− = φ′(ξ)
The RHC is equivalent to h(φ+) = h(φ−) = 0.
Φ+ Φ-
Φ
hHΦL
Ξ
ΦHΞL
We obtain the entropy condition
φ− > φ+ .
Travelling waves for the viscous Burgers equation
∂tu + ∂xu2 = ∂2
xu ,
The travelling wave equation for u(t, x) = φ(ξ) with ξ = x − ct reads
h(φ(ξ)) := −c(φ(ξ)− φ−) + φ2(ξ)− φ2− = φ′(ξ)
The RHC is equivalent to h(φ+) = h(φ−) = 0.
Φ+ Φ-
Φ
hHΦL
Ξ
ΦHΞL
We obtain the entropy condition
φ− > φ+ .
Travelling waves for the KdV-Burgers equation
∂tu + ∂xu2 = ∂2
xu + τ∂3xu , where τ > 0 .
The travelling wave equation reads
h(φ) = φ′ + τφ′′
and as before we have the Rankine Hugoniot and entropy condition.
For phase plane analysis the system is linearised around φ±:(φ′
ψ′
)=
(0 1
2φ±−cτ − 1
τ
)(φψ
)
Travelling waves for the KdV-Burgers equation
∂tu + ∂xu2 = ∂2
xu + τ∂3xu , where τ > 0 .
The travelling wave equation reads
h(φ) = φ′ + τφ′′
and as before we have the Rankine Hugoniot and entropy condition.
For phase plane analysis the system is linearised around φ±:(φ′
ψ′
)=
(0 1
2φ±−cτ − 1
τ
)(φψ
)
Travelling waves for the KdV-Burgers equation
∂tu + ∂xu2 = ∂2
xu + τ∂3xu , where τ > 0 .
The travelling wave equation reads
h(φ) = φ′ + τφ′′
and as before we have the Rankine Hugoniot and entropy condition.
For phase plane analysis the system is linearised around φ±:(φ′
ψ′
)=
(0 1
2φ±−cτ − 1
τ
)(φψ
)
Eigenvalues for the linearised systems show:
φ− : saddle point
φ+:
{stable node for τ ≤ 1/(φ− − φ+) =: τ∗
stable spiral for τ > τ∗
Travelling wave solutions are
monotone for τ ≤ τ∗
oscillatory as ξ →∞ for τ > τ∗
for existence proof see Bona, Schonbeck 1985
Eigenvalues for the linearised systems show:
φ− : saddle point
φ+:
{stable node for τ ≤ 1/(φ− − φ+) =: τ∗
stable spiral for τ > τ∗
Travelling wave solutions are
monotone for τ ≤ τ∗
oscillatory as ξ →∞ for τ > τ∗
for existence proof see Bona, Schonbeck 1985
The fractional KdV-Burgers equation
Kluwick, Cox, Exner, Grinschgl (2010)
2d shallow water flow of an incompressible fluid with highReynolds-number
Interaction equation for the pressure p = p(t, x)
∂tp + ∂x(p − p2) = A∂xD1/3p + W ∂3xp
where
D1/3p(t, x) =1
Γ(2/3)
∫ x
−∞
∂yp(t, y)
(x − y)1/3dy
Nonlinear conservation laws with nonlocal diffusion
∂tu + ∂xu2 = ∂xDαu , (2)
where
Dαu = dα
∫ x
−∞
∂yu(t, y)
(x − y)αdy , 0 < α < 1 , dα =
1
Γ(1− α)
An alternative representation of ∂xDα:
F(∂xDαu)(k) = −Λ(k)u(t, k)
whereΛ(k) = (aα − ibαsgn(k))|k |α+1
with
aα = sin(απ/2) > 0 , bα = cos(απ/2) > 0.
Nonlinear conservation laws with nonlocal diffusion
∂tu + ∂xu2 = ∂xDαu , (2)
where
Dαu = dα
∫ x
−∞
∂yu(t, y)
(x − y)αdy , 0 < α < 1 , dα =
1
Γ(1− α)
An alternative representation of ∂xDα:
F(∂xDαu)(k) = −Λ(k)u(t, k)
whereΛ(k) = (aα − ibαsgn(k))|k |α+1
with
aα = sin(απ/2) > 0 , bα = cos(απ/2) > 0.
The Cauchy problem
∂tu + ∂xu2 = ∂xDαu, u(0, x) = u0(x) (3)
The semigroup generated by ∂xDα is given by the convolution with
K (t, x) = F−1e−Λ(k)t(x).
Mild formulation of (3)
u(t, x) = K (t, .) ∗ u0(x)−∫ t
0
K (t − τ, .) ∗ ∂xu2(τ, .)(x)dτ.
Theorem (Feller 1971): For 0 < α < 1, the kernel K is nonnegative:
K (t, x) ≥ 0 for all t > 0, x ∈ R.
The Cauchy problem
∂tu + ∂xu2 = ∂xDαu, u(0, x) = u0(x) (3)
The semigroup generated by ∂xDα is given by the convolution with
K (t, x) = F−1e−Λ(k)t(x).
Mild formulation of (3)
u(t, x) = K (t, .) ∗ u0(x)−∫ t
0
K (t − τ, .) ∗ ∂xu2(τ, .)(x)dτ.
Theorem (Feller 1971): For 0 < α < 1, the kernel K is nonnegative:
K (t, x) ≥ 0 for all t > 0, x ∈ R.
The Cauchy problem (II)
Theorem (Droniou, Gallouet, Vovelle 2003) If u0 ∈ L∞, then there existsa unique solution u ∈ L∞((0,∞)× R) of (3) satisfying the mildformulation (4) almost everywhere. In particular
‖u(t, .)‖∞ ≤ ‖u0‖∞, for t > 0.
Moreover, the solution satisfies u ∈ C∞((0,∞)× R).
Travelling wave solutions
Introducing ξ = x − ct we obtain the travelling wave problem
−cφ′ + (φ2)′ = (Dαφ)′ , φ(±∞) = φ± , ,
Integrating the equation from −∞ gives
h(φ) = Dαφ = dα
∫ ξ
−∞
φ′(y)
(ξ − y)αdy (4)
where as aboveh(φ) := −c(φ− φ−) + φ2 − φ2
−
and we have the Rankine-Hugoniot and entropy condition.
Travelling wave solutions
Introducing ξ = x − ct we obtain the travelling wave problem
−cφ′ + (φ2)′ = (Dαφ)′ , φ(±∞) = φ± , ,
Integrating the equation from −∞ gives
h(φ) = Dαφ = dα
∫ ξ
−∞
φ′(y)
(ξ − y)αdy (4)
where as aboveh(φ) := −c(φ− φ−) + φ2 − φ2
−
and we have the Rankine-Hugoniot and entropy condition.
Travelling wave solutions (II)
The equation is of Abel’s type. A well known transformation leads to
φ− φ− = Iα(h(φ)) := d1−α
∫ ξ
−∞
h(φ(y))
(ξ − y)1−α dy . (5)
Equivalence holds if φ ∈ C 1b (R) is monotone.
The linearizations
h′(φ−)v = Dαv , v = h′(φ−)Iαv ,
have solutionsv(ξ) = beλξ, b ∈ R,
where λ = h′(φ−)1/α.
Indeed these are the only solutions:
N (Dα − h′(u−)) = span{eλξ}
Travelling wave solutions (II)
The equation is of Abel’s type. A well known transformation leads to
φ− φ− = Iα(h(φ)) := d1−α
∫ ξ
−∞
h(φ(y))
(ξ − y)1−α dy . (5)
Equivalence holds if φ ∈ C 1b (R) is monotone.
The linearizations
h′(φ−)v = Dαv , v = h′(φ−)Iαv ,
have solutionsv(ξ) = beλξ, b ∈ R,
where λ = h′(φ−)1/α.
Indeed these are the only solutions:
N (Dα − h′(u−)) = span{eλξ}
Travelling wave solutions - Local existence
Lemma There exists a unique solution φ satisfyingφ− φ− ∈ H2((−∞, ξε]) with φ(ξε) = φ− − ε and ξε = log ε/λ.
Idea of the proof: Introduce the perturbation φ(ξ) = φ(ξ)− φ− + eλξ
and use fixed point argument involving Fourier transform. �
Travelling wave solutions - Local existence
Lemma There exists a unique solution φ satisfyingφ− φ− ∈ H2((−∞, ξε]) with φ(ξε) = φ− − ε and ξε = log ε/λ.
Idea of the proof: Introduce the perturbation φ(ξ) = φ(ξ)− φ− + eλξ
and use fixed point argument involving Fourier transform. �
Travelling wave solutions - Continuation principle
Lemma Let φ ∈ C 1b ((−∞, ξ0]) be a (continuation of a) solution of the
travelling wave equation (TWE) as constructed above. Then there existsa δ > 0, such that it can be extended uniquely to C 1
b ((−∞, ξ0 + δ)).
Proof. Writing the TWE as
φ(ξ) = f (ξ) + d1−α
∫ ξ
ξ0
h(φ(y))
(ξ − y)1−α dy ,
and considering
f (ξ) = φ− + d1−α
∫ ξ0
−∞
h(φ(y))
(ξ − y)1−α dy
as given inhomogenity, local existence of a smooth solution for ξ close toξ0 is a standard result for Volterra integral equation (see e.g. Linz 1985).�
Travelling wave solutions - Continuation principle
Lemma Let φ ∈ C 1b ((−∞, ξ0]) be a (continuation of a) solution of the
travelling wave equation (TWE) as constructed above. Then there existsa δ > 0, such that it can be extended uniquely to C 1
b ((−∞, ξ0 + δ)).
Proof. Writing the TWE as
φ(ξ) = f (ξ) + d1−α
∫ ξ
ξ0
h(φ(y))
(ξ − y)1−α dy ,
and considering
f (ξ) = φ− + d1−α
∫ ξ0
−∞
h(φ(y))
(ξ − y)1−α dy
as given inhomogenity, local existence of a smooth solution for ξ close toξ0 is a standard result for Volterra integral equation (see e.g. Linz 1985).�
Travelling wave solutions - Monotonicity
Lemma Let φ ∈ C 1b (−∞, ξ0] be (a continuation of) the solution
constructed above. Then φ is nonincreasing.
Proof. Let φm be the value, for which h′(φm) = 0 and
h′ < 0 in (φ+, φm) , h′ > 0 in (φm, φ−]
Φ+ Φm Φ-
hHΦL
Travelling wave solutions - Monotonicity
Lemma Let φ ∈ C 1b (−∞, ξ0] be (a continuation of) the solution
constructed above. Then φ is nonincreasing.
Proof. Let φm be the value, for which h′(φm) = 0 and
h′ < 0 in (φ+, φm) , h′ > 0 in (φm, φ−]
Φ+ Φm Φ-
hHΦL
(i) φ′ < 0 as long as φ ≥ φm: Assume to the contrary that
φ(ξ∗) ≥ φm , φ′(ξ∗) = 0 , φ′ < 0 in (−∞, ξ∗) .
This leads to a contradiction, since
0 = φ′(ξ∗) = d1−α
∫ ξ∗
−∞
h′(φ(y))φ′(y)
(ξ∗ − y)1−α dy < 0 .
Here we used∫ ξ−∞
h(φ(y))(ξ−y)1−α dy =
∫∞0
h(φ(ξ−y))y1−α dy
(ii) φ cannot become increasing for φ < φm. Assume the contrary
φ(ξ∗) < φm , φ′ > 0 in (ξ∗, ξ∗ + δ) , φ′ ≤ 0 in (−∞, ξ∗] ,
where δ is small enough s.t. φ(ξ∗ + δ) < φm. Then
Dαφ(ξ∗ + δ) = dα
∫ ξ∗+δ
−∞
φ′(y)
(ξ∗ + δ − y)αdy
= dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ + δ − y)αdy + dα
∫ ξ∗+δ
ξ∗
φ′(y)
(ξ∗ + δ − y)αdy
> dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ − y)αdy = Dαφ(ξ∗) .
But on the other hand we know
0 > h(φ(ξ∗ + δ))− h(φ(ξ∗)) = Dαφ(ξ∗ + δ)−Dαφ(ξ∗) > 0 ,
leading again to a contradiction. Therefore φ′ cannot get positive. �
(ii) φ cannot become increasing for φ < φm. Assume the contrary
φ(ξ∗) < φm , φ′ > 0 in (ξ∗, ξ∗ + δ) , φ′ ≤ 0 in (−∞, ξ∗] ,
where δ is small enough s.t. φ(ξ∗ + δ) < φm. Then
Dαφ(ξ∗ + δ) = dα
∫ ξ∗+δ
−∞
φ′(y)
(ξ∗ + δ − y)αdy
= dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ + δ − y)αdy + dα
∫ ξ∗+δ
ξ∗
φ′(y)
(ξ∗ + δ − y)αdy
> dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ − y)αdy = Dαφ(ξ∗) .
But on the other hand we know
0 > h(φ(ξ∗ + δ))− h(φ(ξ∗)) = Dαφ(ξ∗ + δ)−Dαφ(ξ∗) > 0 ,
leading again to a contradiction. Therefore φ′ cannot get positive. �
(ii) φ cannot become increasing for φ < φm. Assume the contrary
φ(ξ∗) < φm , φ′ > 0 in (ξ∗, ξ∗ + δ) , φ′ ≤ 0 in (−∞, ξ∗] ,
where δ is small enough s.t. φ(ξ∗ + δ) < φm. Then
Dαφ(ξ∗ + δ) = dα
∫ ξ∗+δ
−∞
φ′(y)
(ξ∗ + δ − y)αdy
= dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ + δ − y)αdy + dα
∫ ξ∗+δ
ξ∗
φ′(y)
(ξ∗ + δ − y)αdy
> dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ − y)αdy = Dαφ(ξ∗) .
But on the other hand we know
0 > h(φ(ξ∗ + δ))− h(φ(ξ∗)) = Dαφ(ξ∗ + δ)−Dαφ(ξ∗) > 0 ,
leading again to a contradiction. Therefore φ′ cannot get positive. �
Travelling wave solutions - Boundedness
Lemma Let φ ∈ C 1b (−∞, ξ0] be (a continuation of) the solution from
above. Thenφ+ < φ < φ−.
Proof. Suppose φ(ξ∗) = φ+ for some finite ξ∗. Then due to themonotonicity we obtain the contradiction
0 = h(φ+) = dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ − y)αdy < 0 .
�
Travelling wave solutions - Boundedness
Lemma Let φ ∈ C 1b (−∞, ξ0] be (a continuation of) the solution from
above. Thenφ+ < φ < φ−.
Proof. Suppose φ(ξ∗) = φ+ for some finite ξ∗. Then due to themonotonicity we obtain the contradiction
0 = h(φ+) = dα
∫ ξ∗
−∞
φ′(y)
(ξ∗ − y)αdy < 0 .
�
Travelling waves - Existence result
Theorem Then there exists a decreasing solution φ ∈ C 1b (R) of the
travelling wave problem (4). It is (up to a shift) unique among allφ ∈ φ− + H2((−∞, 0)) ∩ C 1
b (R).
Asymptotic stability of travelling waves
We change to the moving coordinates (t, ξ)
∂tu + ∂ξ(u2 − cu) = ∂ξDαu , u(0, ξ) = u0(ξ) (6)
We fix the shift in the travelling wave φ such that∫R
(u(t, ξ)− φ(ξ))dξ = 0
The perturbation U = u − φ satisfies
∂tU + ∂ξ((2φ− c)U) + ∂ξU2 = ∂ξDαU (7)
We test the equation with U and denote ‖U‖Hs = ‖|k |s U‖L2
1
2
d
dt‖U‖2
L2 +
∫Rφ′U2dξ ≤ −aα‖U‖2
H(1+α)/2
Asymptotic stability of travelling waves
We change to the moving coordinates (t, ξ)
∂tu + ∂ξ(u2 − cu) = ∂ξDαu , u(0, ξ) = u0(ξ) (6)
We fix the shift in the travelling wave φ such that∫R
(u(t, ξ)− φ(ξ))dξ = 0
The perturbation U = u − φ satisfies
∂tU + ∂ξ((2φ− c)U) + ∂ξU2 = ∂ξDαU (7)
We test the equation with U and denote ‖U‖Hs = ‖|k |s U‖L2
1
2
d
dt‖U‖2
L2 +
∫Rφ′U2dξ ≤ −aα‖U‖2
H(1+α)/2
Stability of travelling waves (II)
We introduce the primitive
W (t, ξ) =
∫ ξ
−∞U(t, η)dη
Integration of (7) gives,
∂tW + (2φ− c)∂ξW + (∂ξW )2 = ∂ξDαW (8)
We derive for
J(t) =1
2(‖W ‖2
L2 + γ‖U‖2L2 )
the estimate
d
dtJ + λ(‖W ‖H1 )
(‖W ‖2
H1+α
2+ γ‖U‖2
H1+α
2
)≤ 0
where
λ(‖W ‖H1 ) =aα2− L(‖W ‖H1 )
γ∗‖W ‖H1
Stability of travelling waves (II)
We introduce the primitive
W (t, ξ) =
∫ ξ
−∞U(t, η)dη
Integration of (7) gives,
∂tW + (2φ− c)∂ξW + (∂ξW )2 = ∂ξDαW (8)
We derive for
J(t) =1
2(‖W ‖2
L2 + γ‖U‖2L2 )
the estimate
d
dtJ + λ(‖W ‖H1 )
(‖W ‖2
H1+α
2+ γ‖U‖2
H1+α
2
)≤ 0
where
λ(‖W ‖H1 ) =aα2− L(‖W ‖H1 )
γ∗‖W ‖H1
Stability result
Theorem Let φ be a travelling wave solution as before. Let u0 be an
initial datum for (6) such that W0(ξ) =∫ ξ−∞(u0(η)− φ(η))dη satisfies
W0 ∈ H1 and let α > 1/2. If ‖W0‖H1 is small enough, then the Cauchyproblem for equation (6) with initial datum u0 has a unique globalsolution converging to the travelling wave in the sense that
limt→∞
∫ ∞t
‖u(τ, ·)− φ‖2L2dτ = 0 .
The fractional KdV-Burgers equation
∂tu + ∂xu2 = ∂xDαu + τ∂3
xu , x ∈ R , t ≥ 0 (9)
with τ > 0.
Travelling wave equation (TWE)
h(φ) = Dαφ+ τφ′′ , (10)
whereh(φ) := −c(φ− φ−) + φ2 − φ2
− .
Rankine-Hugoniot condition:
c = φ+ + φ−
Entropy conditionφ− > φ+
The fractional KdV-Burgers equation
∂tu + ∂xu2 = ∂xDαu + τ∂3
xu , x ∈ R , t ≥ 0 (9)
with τ > 0.
Travelling wave equation (TWE)
h(φ) = Dαφ+ τφ′′ , (10)
whereh(φ) := −c(φ− φ−) + φ2 − φ2
− .
Rankine-Hugoniot condition:
c = φ+ + φ−
Entropy conditionφ− > φ+
TWs for fKdV-Burgers - Local Existence
The linearisation about ξ = −∞ (or φ = φ−)
h′(φ−)v = Dαv + τv ′′ ,
has solutions of the form
v(ξ) = beλξ, b ∈ R,
where λ > 0 is the positive real root of
P(z) = τz2 + zα − h′(φ−) .
Assumption: N(τ∂2
ξ +Dα − h′(φ−)Id)
= span{eλξ} in H4(R)
Lemma There exists a unique solution φ satisfyingφ− φ− ∈ H4((−∞, ξε]) with φ(ξε) = φ− − ε and ξε = log ε/λ.
TWs for fKdV-Burgers - Local Existence
The linearisation about ξ = −∞ (or φ = φ−)
h′(φ−)v = Dαv + τv ′′ ,
has solutions of the form
v(ξ) = beλξ, b ∈ R,
where λ > 0 is the positive real root of
P(z) = τz2 + zα − h′(φ−) .
Assumption: N(τ∂2
ξ +Dα − h′(φ−)Id)
= span{eλξ} in H4(R)
Lemma There exists a unique solution φ satisfyingφ− φ− ∈ H4((−∞, ξε]) with φ(ξε) = φ− − ε and ξε = log ε/λ.
TWs for fKdV-Burgers - Local Existence
The linearisation about ξ = −∞ (or φ = φ−)
h′(φ−)v = Dαv + τv ′′ ,
has solutions of the form
v(ξ) = beλξ, b ∈ R,
where λ > 0 is the positive real root of
P(z) = τz2 + zα − h′(φ−) .
Assumption: N(τ∂2
ξ +Dα − h′(φ−)Id)
= span{eλξ} in H4(R)
Lemma There exists a unique solution φ satisfyingφ− φ− ∈ H4((−∞, ξε]) with φ(ξε) = φ− − ε and ξε = log ε/λ.
TWs for fKdV-Burgers - Continuation principle
Lemma Let φ ∈ C 3b ((−∞, ξ0]) be a solution of (10) as above. Then
∃δ > 0, s.t. φ can be extended uniquely to C 3b ((−∞, ξ0 + δ)).
Idea of Proof. Write the equation as a system of fractional differentialequations of orders α, 1− α and use local Lipschitz continuity as Jafariand Daftardar-Gejji 2006.
TWs for fKdV-Burgers - Continuation principle
Lemma Let φ ∈ C 3b ((−∞, ξ0]) be a solution of (10) as above. Then
∃δ > 0, s.t. φ can be extended uniquely to C 3b ((−∞, ξ0 + δ)).
Idea of Proof. Write the equation as a system of fractional differentialequations of orders α, 1− α and use local Lipschitz continuity as Jafariand Daftardar-Gejji 2006.
TWs for fKdV-Burgers - Boundedness
The key quantity for boundedness is the energy functional
H(φ) =
∫ φ
0
h(y)dy = −c φ2
2+φ3
3+ Aφ , with A = cφ− − φ2
− (11)
Lemma Let φ ∈ C 3b ((−∞, ξ0]) be a solution of the TWE. Then the
solution is bounded for ξ ∈ (−∞, ξ0) by
φ < φ(ξ) < φ− , where φ =3φ+ − φ−
2< φ+ (12)
is the second root ofH(φ)− H(φ−)
φ− φ−= 0 .
TWs for fKdV-Burgers - Boundedness
The key quantity for boundedness is the energy functional
H(φ) =
∫ φ
0
h(y)dy = −c φ2
2+φ3
3+ Aφ , with A = cφ− − φ2
− (11)
Lemma Let φ ∈ C 3b ((−∞, ξ0]) be a solution of the TWE. Then the
solution is bounded for ξ ∈ (−∞, ξ0) by
φ < φ(ξ) < φ− , where φ =3φ+ − φ−
2< φ+ (12)
is the second root ofH(φ)− H(φ−)
φ− φ−= 0 .
Proof of boundedness.
We first derive an energy type of estimate by multiplying the TWE by φ′
and integrating w.r.t. ξ:
H (φ(ξ))− H (φ−) =τ
2(φ′(ξ))
2+
∫ ξ
−∞φ′(y)Dαφ(y)dy . (13)
The first term on the RHS is clearly nonnegative.
Also the second term is nonnegative, since∫ ξ
−∞φ′(y)Dαφ(y)dy
!=
dα2
∫ ξ
−∞φ′(y)
∫ ξ
−∞
φ′(x)
|x − y |αdx dy
=dα2
∫ ξ
−∞φ′(y)
∫ y
−∞
φ′(x)
(y − x)αdx dy +
dα2
∫ ξ
−∞φ′(y)
∫ ξ
y
φ′(x)
|x − y |αdx dy
Proof of boundedness.
We first derive an energy type of estimate by multiplying the TWE by φ′
and integrating w.r.t. ξ:
H (φ(ξ))− H (φ−) =τ
2(φ′(ξ))
2+
∫ ξ
−∞φ′(y)Dαφ(y)dy . (13)
The first term on the RHS is clearly nonnegative.
Also the second term is nonnegative, since∫ ξ
−∞φ′(y)Dαφ(y)dy
!=
dα2
∫ ξ
−∞φ′(y)
∫ ξ
−∞
φ′(x)
|x − y |αdx dy
=dα2
∫ ξ
−∞φ′(y)
∫ y
−∞
φ′(x)
(y − x)αdx dy +
dα2
∫ ξ
−∞φ′(y)
∫ ξ
y
φ′(x)
|x − y |αdx dy
Proof of boundedness.
We first derive an energy type of estimate by multiplying the TWE by φ′
and integrating w.r.t. ξ:
H (φ(ξ))− H (φ−) =τ
2(φ′(ξ))
2+
∫ ξ
−∞φ′(y)Dαφ(y)dy . (13)
The first term on the RHS is clearly nonnegative.
Also the second term is nonnegative, since∫ ξ
−∞φ′(y)Dαφ(y)dy
!=
dα2
∫ ξ
−∞φ′(y)
∫ ξ
−∞
φ′(x)
|x − y |αdx dy
=dα2
∫ ξ
−∞φ′(y)
∫ y
−∞
φ′(x)
(y − x)αdx dy +
dα2
∫ ξ
−∞φ′(y)
∫ ξ
y
φ′(x)
|x − y |αdx dy
To see this, we observe that by changing the order of integration∫ ξ
−∞φ′(y)
∫ ξ
y
φ′(x)
(x − y)αdx dy =
∫ ξ
−∞φ′(x)
∫ x
−∞
φ′(y)
(x − y)αdy dx .
Ξx
Ξ
y
Ξx
y
Employing an extension φ′E ∈ L2(R) of φ′ to R so that φ′E (y) = 0 fory > ξ we can deduce∫ ξ
−∞φ′(y)Dαφ(y)dy =
dα2
∫Rφ′E (x)
∫R
φ′E (y)
|x − y |αdy dx ≥ 0 , (14)
where the last inequality was shown by Lieb and Loss 1997.
To see this, we observe that by changing the order of integration∫ ξ
−∞φ′(y)
∫ ξ
y
φ′(x)
(x − y)αdx dy =
∫ ξ
−∞φ′(x)
∫ x
−∞
φ′(y)
(x − y)αdy dx .
Ξx
Ξ
y
Ξx
y
Employing an extension φ′E ∈ L2(R) of φ′ to R so that φ′E (y) = 0 fory > ξ we can deduce∫ ξ
−∞φ′(y)Dαφ(y)dy =
dα2
∫Rφ′E (x)
∫R
φ′E (y)
|x − y |αdy dx ≥ 0 , (14)
where the last inequality was shown by Lieb and Loss 1997.
We have
H (φ(ξ))− H (φ−) =τ
2(φ′(ξ))
2+
∫ ξ
−∞φ′(y)Dαφ(y)dy ≥ 0
Upper bound φ ≤ φ−:
Suppose φ(ξ∗) = φ− for some ξ∗ <∞, then∫ ξ∗
−∞φ′(y)Dαφ(y)dy = 0,
implying φ′(ξ) = 0 for all ξ ∈ (−∞, ξ∗] (see Lieb, Loss).
Therefore non constant solution is always below φ−.
Lower bound:
We use the nonnegativity of
H (φ)− H (φ−) = −c
2(φ2 − (φ−)2) +
1
3(φ3 − (φ−)3) + A(φ− φ−) ≥ 0 .
Since φ− φ− < 0 in (−∞, ξ0], we obtain the condition
H(φ)− H(φ−)
φ− φ−= −c
2(φ+ φ−) +
1
3(φ2 + φφ− + (φ−)2) + A ≤ 0
and this implies exactly the lower bound. �
TWs for fKdV-Burgers - Far-field behaviour
Lemma Let φ be the TW solution from above. Suppose that
limξ→∞
φ = φ0 ∈ R .
Thenφ0 = φ+.
Proof. We argue by contradiction and assume that φ0 6= φ+, then
h(φ(ξ))→ h(φ0) 6= 0
and
Iαh(φ(ξ))→ ±∞ .
Use the integrated TWE to show contradiction...
TWs for fKdV-Burgers - Far-field behaviour
Lemma Let φ be the TW solution from above. Suppose that
limξ→∞
φ = φ0 ∈ R .
Thenφ0 = φ+.
Proof. We argue by contradiction and assume that φ0 6= φ+, then
h(φ(ξ))→ h(φ0) 6= 0
and
Iαh(φ(ξ))→ ±∞ .
Use the integrated TWE to show contradiction...
TWs for fKdV-Burgers - Far-field behaviour ctd.
Lemma Let φ be a solution as above. Then there exists a constantφ0 ∈ R such that
limξ→∞
φ(ξ) = φ0.
Idea of the proof. We rewrite
τφ′′ +Dαξ0φ+ φ = q(φ, ξ) (15)
for ξ ≥ ξ0, where
q(φ, ξ) = −dα∫ ξ0
−∞
φ′(y)
(ξ − y)αdy + h(φ(ξ)) + φ(ξ) .
TWs for fKdV-Burgers - Far-field behaviour ctd.
Lemma Let φ be a solution as above. Then there exists a constantφ0 ∈ R such that
limξ→∞
φ(ξ) = φ0.
Idea of the proof. We rewrite
τφ′′ +Dαξ0φ+ φ = q(φ, ξ) (15)
for ξ ≥ ξ0, where
q(φ, ξ) = −dα∫ ξ0
−∞
φ′(y)
(ξ − y)αdy + h(φ(ξ)) + φ(ξ) .
We can now write down the solution implicitly by applying Laplacetransform methods (see e.g. Gorenflo, Mainardi) to obtain a ’variationsof constants’ representation:
φ(ξ) = φ(ξ0) v(ξ)− φ′(ξ0) v ′(ξ)−∫ ξ
ξ0
q(φ(ξ − s), ξ − s)v ′(s)ds
where the function v and its derivatives are uniformly bounded and havepolynomial decay implying the integrability of the term with theinhomogeneity q as well as the decay of φ towards a constant. �
Travelling waves - Existence result
Theorem Assume that the exponential functions are the only solutions tothe linearised TWE. There exists a travelling wave solution φ ∈ C 3
b (R) ofthe travelling wave problem (4), which is (up to a shift) unique among allφ ∈ φ− + H4((−∞, 0)) ∩ C 3
b (R).
F. Achleitner, S. Hittmeir, C. Schmeiser: On nonlinear conservation lawswith a nonlocal diffusion term, J. Diff. Equ. 250, pp. 2177-2196 (2011)
F. Achleitner, C. Cuesta, S. Hittmeir: Travelling waves for a non-localKorteweg-de Vries-Burgers equation, J. Diff. Equ. 257, No. 3, pp.720-758 (2014)
Thank you for your attention!
F. Achleitner, S. Hittmeir, C. Schmeiser: On nonlinear conservation lawswith a nonlocal diffusion term, J. Diff. Equ. 250, pp. 2177-2196 (2011)
F. Achleitner, C. Cuesta, S. Hittmeir: Travelling waves for a non-localKorteweg-de Vries-Burgers equation, J. Diff. Equ. 257, No. 3, pp.720-758 (2014)
Thank you for your attention!