Travelling waves for a nonlocal KdV-Burgers equation Sabine - PowerPoint PPT Presentation

Travelling waves for a nonlocal KdV-Burgers equation 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 ∂ t u + ∂ x u 2 = 0 (1) has shock solutions u ( x , t ) = φ ( x − ct ) = φ ( ξ ) of the form � φ − ξ < 0 φ ( ξ ) = φ + ξ > 0

Motivation The inviscid Burgers equation ∂ t u + ∂ x u 2 = 0 (1) has shock solutions u ( x , t ) = φ ( x − ct ) = φ ( ξ ) of the form � φ − ξ < 0 φ ( ξ ) = φ + ξ > 0 For { φ − , φ + , c } the Rankine-Hugoniot condition (RHC) has to hold − c ( φ + − φ − ) + φ 2 + − φ 2 − = 0 , i.e. c = φ + + φ −

Motivation The inviscid Burgers equation ∂ t u + ∂ x u 2 = 0 (1) has shock solutions u ( x , t ) = φ ( x − ct ) = φ ( ξ ) of the form � φ − ξ < 0 φ ( ξ ) = φ + ξ > 0 For { φ − , φ + , c } the Rankine-Hugoniot condition (RHC) has to hold − c ( φ + − φ − ) + φ 2 + − φ 2 − = 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 ∂ t u + ∂ x u 2 = ∂ 2 x u , The travelling wave equation for u ( t , x ) = φ ( ξ ) with ξ = x − ct reads h ( φ ( ξ )) := − c ( φ ( ξ ) − φ − ) + φ 2 ( ξ ) − φ 2 − = φ ′ ( ξ ) The RHC is equivalent to h ( φ + ) = h ( φ − ) = 0.

Travelling waves for the viscous Burgers equation ∂ t u + ∂ x u 2 = ∂ 2 x u , The travelling wave equation for u ( t , x ) = φ ( ξ ) with ξ = x − ct reads h ( φ ( ξ )) := − c ( φ ( ξ ) − φ − ) + φ 2 ( ξ ) − φ 2 − = φ ′ ( ξ ) The RHC is equivalent to h ( φ + ) = h ( φ − ) = 0.

Travelling waves for the viscous Burgers equation ∂ t u + ∂ x u 2 = ∂ 2 x u , The travelling wave equation for u ( t , x ) = φ ( ξ ) with ξ = x − ct reads h ( φ ( ξ )) := − c ( φ ( ξ ) − φ − ) + φ 2 ( ξ ) − φ 2 − = φ ′ ( ξ ) The RHC is equivalent to h ( φ + ) = h ( φ − ) = 0. h � Φ � Φ � Ξ � Φ Φ � Φ � Ξ We obtain the entropy condition φ − > φ + .

Travelling waves for the KdV-Burgers equation ∂ t u + ∂ x u 2 = ∂ 2 x u + τ∂ 3 x u , 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 ′ � � � � φ � 0 1 = 2 φ ± − c ψ ′ − 1 ψ τ τ

Travelling waves for the KdV-Burgers equation ∂ t u + ∂ x u 2 = ∂ 2 x u + τ∂ 3 x u , 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 ′ � � � � φ � 0 1 = 2 φ ± − c ψ ′ − 1 ψ τ τ

Travelling waves for the KdV-Burgers equation ∂ t u + ∂ x u 2 = ∂ 2 x u + τ∂ 3 x u , 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 ′ � � � � φ � 0 1 = 2 φ ± − c ψ ′ − 1 ψ τ τ

Eigenvalues for the linearised systems show: φ − : saddle point � stable node for τ ≤ 1 / ( φ − − φ + ) =: τ ∗ φ + : 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 � stable node for τ ≤ 1 / ( φ − − φ + ) =: τ ∗ φ + : 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 high Reynolds-number

Interaction equation for the pressure p = p ( t , x ) ∂ t p + ∂ x ( p − p 2 ) = A ∂ x D 1 / 3 p + W ∂ 3 x p where � x 1 ∂ y p ( t , y ) D 1 / 3 p ( t , x ) = ( x − y ) 1 / 3 dy Γ(2 / 3) −∞

Nonlinear conservation laws with nonlocal diffusion ∂ t u + ∂ x u 2 = ∂ x D α u , (2) where � x ∂ y u ( t , y ) 1 D α u = d α ( x − y ) α dy , 0 < α < 1 , d α = Γ(1 − α ) −∞ An alternative representation of ∂ x D α : F ( ∂ x D α u )( k ) = − Λ( k ) � u ( t , k ) where Λ( k ) = ( a α − ib α sgn ( k )) | k | α +1 with a α = sin( απ/ 2) > 0 , b α = cos( απ/ 2) > 0 .

Nonlinear conservation laws with nonlocal diffusion ∂ t u + ∂ x u 2 = ∂ x D α u , (2) where � x ∂ y u ( t , y ) 1 D α u = d α ( x − y ) α dy , 0 < α < 1 , d α = Γ(1 − α ) −∞ An alternative representation of ∂ x D α : F ( ∂ x D α u )( k ) = − Λ( k ) � u ( t , k ) where Λ( k ) = ( a α − ib α sgn ( k )) | k | α +1 with a α = sin( απ/ 2) > 0 , b α = cos( απ/ 2) > 0 .

The Cauchy problem ∂ t u + ∂ x u 2 = ∂ x D α u , u (0 , x ) = u 0 ( x ) (3) The semigroup generated by ∂ x D α is given by the convolution with K ( t , x ) = F − 1 e − Λ( k ) t ( x ) . Mild formulation of (3) � t K ( t − τ, . ) ∗ ∂ x u 2 ( τ, . )( x ) d τ. u ( t , x ) = K ( t , . ) ∗ u 0 ( x ) − 0 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 ∂ t u + ∂ x u 2 = ∂ x D α u , u (0 , x ) = u 0 ( x ) (3) The semigroup generated by ∂ x D α is given by the convolution with K ( t , x ) = F − 1 e − Λ( k ) t ( x ) . Mild formulation of (3) � t K ( t − τ, . ) ∗ ∂ x u 2 ( τ, . )( x ) d τ. u ( t , x ) = K ( t , . ) ∗ u 0 ( x ) − 0 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 u 0 ∈ L ∞ , then there exists a unique solution u ∈ L ∞ ((0 , ∞ ) × R ) of (3) satisfying the mild formulation (4) almost everywhere. In particular � u ( t , . ) � ∞ ≤ � u 0 � ∞ , for t > 0 . Moreover, the solution satisfies u ∈ C ∞ ((0 , ∞ ) × R ) .

Travelling wave solutions Introducing ξ = x − ct we obtain the travelling wave problem − c φ ′ + ( φ 2 ) ′ = ( D α φ ) ′ , φ ( ±∞ ) = φ ± , , Integrating the equation from −∞ gives � ξ φ ′ ( y ) h ( φ ) = D α φ = d α ( ξ − y ) α dy (4) −∞ where as above h ( φ ) := − c ( φ − φ − ) + φ 2 − φ 2 − and we have the Rankine-Hugoniot and entropy condition.

Travelling wave solutions Introducing ξ = x − ct we obtain the travelling wave problem − c φ ′ + ( φ 2 ) ′ = ( D α φ ) ′ , φ ( ±∞ ) = φ ± , , Integrating the equation from −∞ gives � ξ φ ′ ( y ) h ( φ ) = D α φ = d α ( ξ − y ) α dy (4) −∞ where as above h ( φ ) := − c ( φ − φ − ) + φ 2 − φ 2 − 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 � ξ h ( φ ( y )) φ − φ − = I α ( h ( φ )) := d 1 − α ( ξ − y ) 1 − α dy . (5) −∞ Equivalence holds if φ ∈ C 1 b ( R ) is monotone. The linearizations h ′ ( φ − ) v = D α v , v = h ′ ( φ − ) I α v , have solutions v ( ξ ) = be λξ , b ∈ R , where λ = h ′ ( φ − ) 1 /α . Indeed these are the only solutions: � e λξ � N ( D α − h ′ ( u − )) = span

Travelling wave solutions (II) The equation is of Abel’s type. A well known transformation leads to � ξ h ( φ ( y )) φ − φ − = I α ( h ( φ )) := d 1 − α ( ξ − y ) 1 − α dy . (5) −∞ Equivalence holds if φ ∈ C 1 b ( R ) is monotone. The linearizations h ′ ( φ − ) v = D α v , v = h ′ ( φ − ) I α v , have solutions v ( ξ ) = be λξ , b ∈ R , where λ = h ′ ( φ − ) 1 /α . Indeed these are the only solutions: � e λξ � N ( D α − h ′ ( u − )) = span

Travelling wave solutions - Local existence Lemma There exists a unique solution φ satisfying φ − φ − ∈ H 2 (( −∞ , ξ ε ]) with φ ( ξ ε ) = φ − − ε and ξ ε = log ε/λ . Idea of the proof: Introduce the perturbation ¯ φ ( ξ ) = φ ( ξ ) − φ − + e λξ and use fixed point argument involving Fourier transform. �

Travelling wave solutions - Local existence Lemma There exists a unique solution φ satisfying φ − φ − ∈ H 2 (( −∞ , ξ ε ]) with φ ( ξ ε ) = φ − − ε and ξ ε = log ε/λ . Idea of the proof: Introduce the perturbation ¯ φ ( ξ ) = φ ( ξ ) − φ − + e λξ and use fixed point argument involving Fourier transform. �