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!