On Whithams modulated equations for the EulerKorteweg system S. - PowerPoint PPT Presentation

On Whitham’s modulated equations for the Euler–Korteweg system S. Benzoni-Gavage P. Noble L.M. Rodrigues Institut Camille Jordan Universit´ e Claude Bernard Lyon 1 June 26, 2012 1 / 21

The Euler–Korteweg system and travelling waves Outline The Euler–Korteweg system and travelling waves 1 Modulated equations and periodic waves 2 2 / 21

② The Euler–Korteweg system and travelling waves General model for isothermal capillary fluids Euler–Lagrange equations for Lagrangian 1 2 ρ � u � 2 − E ( ρ, ∇ ρ ) − ρ∂ t ϕ − ρ u · ∇ ϕ in coordinates ( ρ, ϕ, p ) � ∂ t ρ + div ( ρ u ) = 0 , ✇ ∂ t u + ( u · ∇ ) u + ∇ (E ρ E ) = 0 , ρ = density, u = velocity, E = free energy density, E ρ E = variational derivative of E . 3 / 21

The Euler–Korteweg system and travelling waves General model for isothermal capillary fluids Euler–Lagrange equations for Lagrangian 1 2 ρ � u � 2 − E ( ρ, ∇ ρ ) − ρ∂ t ϕ − ρ u · ∇ ϕ in coordinates ( ρ, ϕ, p ) � ∂ t ρ + div ( ρ u ) = 0 , ✇ ∂ t u + ( u · ∇ ) u + ∇ (E ρ E ) = 0 , ρ = density, u = velocity, E = free energy density, E ρ E = variational derivative of E . Standard compressible fluids: E = E ( ρ ), E ρ E = d E d ρ . ② ‘compressible’ Euler equations. 3 / 21

The Euler–Korteweg system and travelling waves General model for isothermal capillary fluids Euler–Lagrange equations for Lagrangian 1 2 ρ � u � 2 − E ( ρ, ∇ ρ ) − ρ∂ t ϕ − ρ u · ∇ ϕ in coordinates ( ρ, ϕ, p ) � ∂ t ρ + div ( ρ u ) = 0 , ✇ ∂ t u + ( u · ∇ ) u + ∇ (E ρ E ) = 0 , ρ = density, u = velocity, E = free energy density, E ρ E = variational derivative of E . Standard compressible fluids: E = E ( ρ ), E ρ E = d E d ρ . ② ‘compressible’ Euler equations. � ∂ E � Capillary fluids: E = E ( ρ, ∇ ρ ), E ρ E := ∂ E ∂ρ − � d . j =1 D x j ∂ρ x j 3 / 21

② ② The Euler–Korteweg system and travelling waves Special cases Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E ( ρ, ∇ ρ ) = F ( ρ ) + 1 2 K ( ρ ) �∇ ρ � 2 . 4 / 21

② The Euler–Korteweg system and travelling waves Special cases Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E ( ρ, ∇ ρ ) = F ( ρ ) + 1 2 K ( ρ ) �∇ ρ � 2 . Quantum fluids (Schr¨ odinger, Madelung, Gross–Pitaevskii) F ′ ( ρ ) − ∆ √ ρ � � ∂ t ρ + div ( ρ u ) = 0 , ∂ t u + ( u · ∇ ) u + ∇ 2 √ ρ = 0 . ② ρ K ( ρ ) ≡ 1 4 . 4 / 21

The Euler–Korteweg system and travelling waves Special cases Korteweg capillarity theory (after Rayleigh, van der Waals,...), see [Rowlinson & Widom’82] E ( ρ, ∇ ρ ) = F ( ρ ) + 1 2 K ( ρ ) �∇ ρ � 2 . Quantum fluids (Schr¨ odinger, Madelung, Gross–Pitaevskii) F ′ ( ρ ) − ∆ √ ρ � � ∂ t ρ + div ( ρ u ) = 0 , ∂ t u + ( u · ∇ ) u + ∇ 2 √ ρ = 0 . ② ρ K ( ρ ) ≡ 1 4 . Vortex-filaments (Levi-Civita & Da Rios, Hasimoto), see [Arnold & Khesin’98] √ ρ � ρ � 4 + ∂ 2 x ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t u + u ∂ x u = ∂ x . 2 √ ρ 8 ρ 2 , d = 1. ② ρ K ( ρ ) ≡ 1 4 , F ( ρ ) = − 1 4 / 21

⑨❡ ⑨❡ ⑨❡ ⑨❡ ⑨❡ ⑨❡ The Euler–Korteweg system and travelling waves Two formulations of 1D model Eulerian coordinates � ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , ρ = density, u = velocity, E = E ( ρ, ρ x ) energy density, � ∂ E � E ρ E := ∂ E ∂ρ − D x ∂ρ x 5 / 21

The Euler–Korteweg system and travelling waves Two formulations of 1D model Mass Lagrangian coordinates Eulerian coordinates � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , ρ = density, u = velocity, v = specific volume, u = velocity, E = E ( ρ, ρ x ) energy density, ⑨❡ = ⑨❡ ( v , v y ) specific energy, � ∂ E � ∂ ⑨❡ � � E ρ E := ∂ E E v ⑨❡ := ∂ ⑨❡ ∂ρ − D x ∂ v − D y ∂ρ x ∂ v y 5 / 21

The Euler–Korteweg system and travelling waves Two formulations of 1D model Mass Lagrangian coordinates Eulerian coordinates � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ( ∂ t + u ∂ x ) u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , ρ = density, u = velocity, v = specific volume, u = velocity, E = E ( ρ, ρ x ) energy density, ⑨❡ = ⑨❡ ( v , v y ) specific energy, � ∂ E � ∂ ⑨❡ � � E ρ E := ∂ E E v ⑨❡ := ∂ ⑨❡ ∂ρ − D x ∂ v − D y ∂ρ x ∂ v y d y = ρ d x − ρ u d t ⇔ d x = v d y + u d t 5 / 21

The Euler–Korteweg system and travelling waves Two formulations of 1D model Mass Lagrangian coordinates Eulerian coordinates � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , ρ = density, u = velocity, v = specific volume, u = velocity, E = E ( ρ, ρ x ) energy density, ⑨❡ = ⑨❡ ( v , v y ) specific energy, � ∂ E � ∂ ⑨❡ � � E ρ E := ∂ E E v ⑨❡ := ∂ ⑨❡ ∂ρ − D x ∂ v − D y ∂ρ x ∂ v y d y = ρ d x − ρ u d t ⇔ d x = v d y + u d t ∂ x (E ρ E ) = − ∂ y (E v ⑨❡ ) 5 / 21

The Euler–Korteweg system and travelling waves More special cases Korteweg / Cahn–Hilliard energy again E ( ρ, ρ x ) = F ( ρ ) + 1 ⑨❡ ( v , v y ) = f ( v ) + 1 2 K ( ρ ) ρ 2 2 κ ( v ) v 2 ⇔ y , x κ ( v ) := ρ 5 K ( ρ ) . with F ( ρ ) = ρ f ( v ) , Water waves [Boussinesq’72], [Bona & Sachs’88]: v 3 gH 3 ∂ 3 2 H v 2 ) = 1 3 ∂ t v = ∂ y u , ∂ t u − gH ∂ y ( v + y v , g H 3 gH 3 , f ( v ) = 1 2 gH v 2 (1 + v ② κ = − 1 H ). y 6 / 21

The Euler–Korteweg system and travelling waves More special cases Korteweg / Cahn–Hilliard energy again E ( ρ, ρ x ) = F ( ρ ) + 1 ⑨❡ ( v , v y ) = f ( v ) + 1 2 K ( ρ ) ρ 2 2 κ ( v ) v 2 ⇔ y , x κ ( v ) := ρ 5 K ( ρ ) . with F ( ρ ) = ρ f ( v ) , Water waves [Boussinesq’72], [Bona & Sachs’88]: v 3 gH 3 ∂ 3 2 H v 2 ) = 1 3 ∂ t v = ∂ y u , ∂ t u − gH ∂ y ( v + y v , g H 3 gH 3 , f ( v ) = 1 2 gH v 2 (1 + v ② κ = − 1 H ). y van der Waals fluids 6 / 21

The Euler–Korteweg system and travelling waves Travelling wave profiles Eulerian coordinates Lagrangian coordinates ( ρ, u ) = ( R , U )( x − σ t ) solution of ( v , u ) = ( V , W )( y + jt ) solution of � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , � � ∂ ξ ( R ( U − σ )) = 0 , ∂ ζ ( W − j V ) = 0 , iff iff ( U − σ ) ∂ ξ U + ∂ ξ (E ρ E ) = 0 . ∂ ζ (E v ⑨❡ − j W ) = 0 . 7 / 21

The Euler–Korteweg system and travelling waves Travelling wave profiles Eulerian coordinates Lagrangian coordinates ( ρ, u ) = ( R , U )( x − σ t ) solution of ( v , u ) = ( V , W )( y + jt ) solution of � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , � � R ( U − σ ) ≡ j , W − j V ≡ σ , iff iff ( U − σ ) ∂ ξ U + ∂ ξ (E ρ E ) = 0 . ∂ ζ (E v ⑨❡ − j W ) = 0 . 7 / 21

The Euler–Korteweg system and travelling waves Travelling wave profiles Eulerian coordinates Lagrangian coordinates ( ρ, u ) = ( R , U )( x − σ t ) solution of ( v , u ) = ( V , W )( y + jt ) solution of � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , � � R ( U − σ ) ≡ j , W − j V ≡ σ , iff iff ( U − σ ) ∂ ξ U + ∂ ξ (E ρ E ) = 0 . ∂ ζ (E v ⑨❡ − j W ) = 0 . R ( ξ ) V ( Z ( ξ )) = 1 , U ( ξ ) = W ( Z ( ξ )) , d Z 1 d ξ = R = V ( Z ) . 7 / 21

The Euler–Korteweg system and travelling waves Travelling wave profiles Eulerian coordinates Lagrangian coordinates ( ρ, u ) = ( R , U )( x − σ t ) solution of ( v , u ) = ( V , W )( y + jt ) solution of � � ∂ t ρ + ∂ x ( ρ u ) = 0 , d t v = ∂ y u , ∂ t u + u ∂ x u + ∂ x (E ρ E ) = 0 , d t u = ∂ y (E v ⑨❡ ) , � � R ( U − σ ) ≡ j , W − j V ≡ σ , iff iff ( U − σ ) ∂ ξ U + ∂ ξ (E ρ E ) = 0 . ∂ ζ (E v ⑨❡ − j W ) = 0 . R ( ξ ) V ( Z ( ξ )) = 1 , U ( ξ ) = W ( Z ( ξ )) , d Z 1 d ξ = R = V ( Z ) . Proposition Relationships above yield a one-to-one correspondence between Eulerian travelling waves s.t. R ( R ) is compact in R + ∗ , and Lagrangian travelling waves s.t. V ( R ) is compact in R + ∗ . 7 / 21

The Euler–Korteweg system and travelling waves Key to one-to-one correspondence • PDEs solutions, Eulerian vs mass Lagrangian coordinates: x ↔ ξ ↔ y . ρ ( χ ( ξ, t ) , t ) v ( y ( χ ( ξ, t ) , t ) , t ) = 1 , u ( χ ( ξ, t ) , t ) = w ( y ( χ ( ξ, t ) , t ) , t ) , � ∂ t χ = u ( χ, t ) , � y ( χ ( ξ, t ) , t ) = y 0 ( ξ ) , χ ( ξ, 0) = ξ , y 0 ′ ( ξ ) = ρ ( ξ, 0) . 8 / 21