Water Waves with vorticity David Lannes Joint work with Angel - PowerPoint PPT Presentation

The equations The free surface Bernoulli equations (H1)’ ∂ t U + ( U · ∇ X , z ) U = − 1 ρ ∇ X , z P − g e z in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ U · n = 0 on { z = − H 0 + b ( X ) } � 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)’ lim | ( X , z ) |→∞ | ζ ( t , X ) | + | U ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ U · n = 0 on { z = − H 0 + b ( X ) } � 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)’ lim | ( X , z ) |→∞ | ζ ( t , X ) | + | U ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ ∂ n Φ = 0 on { z = − H 0 + b ( X ) } . � 1 + |∇ ζ | 2 U · n = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)’ lim | ( X , z ) |→∞ | ζ ( t , X ) | + | U ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ ∂ n Φ = 0 on { z = − H 0 + b ( X ) } . � 1 + |∇ ζ | 2 ∂ n Φ = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)’ lim | ( X , z ) |→∞ | ζ ( t , X ) | + | U ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ ∂ n Φ = 0 on { z = − H 0 + b ( X ) } . � 1 + |∇ ζ | 2 ∂ n Φ = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)” lim | ( X , z ) |→∞ | ζ ( t , X ) | + |∇ X , z Φ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ ∂ n Φ = 0 on { z = − H 0 + b ( X ) } . � 1 + |∇ ζ | 2 ∂ n Φ = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)” lim | ( X , z ) |→∞ | ζ ( t , X ) | + |∇ X , z Φ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . Definition Equations (H1)’-(H9)’ are called free surface Bernoulli equations. David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The free surface Bernoulli equations 2 |∇ X , z Φ | 2 + gz = − 1 (H1)’ ∂ t Φ + 1 ρ ( P − P atm ) in Ω t (H2)’ ∆ X , z Φ = 0 (H3)’ U = ∇ X , z Φ (H4)’ Ω t = { ( X , z ) ∈ R d +1 , − H 0 + b ( X ) < z < ζ ( t , X ) } . (H5)’ ∂ n Φ = 0 on { z = − H 0 + b ( X ) } . � 1 + |∇ ζ | 2 ∂ n Φ = 0 on { z = ζ ( t , X ) } . (H6)’ ∂ t ζ − (H7)’ P = P atm on { z = ζ ( t , X ) } . (H8)” lim | ( X , z ) |→∞ | ζ ( t , X ) | + |∇ X , z Φ( t , X , z ) | = 0 (H9)’ ∃ H min > 0 , H 0 + ζ ( t , X ) − b ( X ) ≥ H min . Definition Equations (H1)’-(H9)’ are called free surface Bernoulli equations. � ONE unknown function ζ on a fixed domain R d � ONE unknown function Φ on a moving, unknown domain Ω t David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 4 / 33

The equations The Zakharov/Craig-Sulem formulation Zakharov 68 : 1 Define ψ ( t , X ) = Φ( t , X , ζ ( t , X )) . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 5 / 33

The equations The Zakharov/Craig-Sulem formulation Zakharov 68 : 1 Define ψ ( t , X ) = Φ( t , X , ζ ( t , X )) . 2 ζ and ψ fully determine Φ: indeed, the equation � ∆ X , z Φ = 0 in Ω t , Φ | z = ζ = ψ, ∂ n Φ | z = − H 0+ b = 0 . has a unique solution Φ. David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 5 / 33

The equations The Zakharov/Craig-Sulem formulation Zakharov 68 : 1 Define ψ ( t , X ) = Φ( t , X , ζ ( t , X )) . 2 ζ and ψ fully determine Φ: indeed, the equation � ∆ X , z Φ = 0 in Ω t , Φ | z = ζ = ψ, ∂ n Φ | z = − H 0+ b = 0 . has a unique solution Φ. 3 The equations can be put under the canonical Hamiltonian form � ζ � � � 0 1 ∂ t = grad ζ,ψ H ψ − 1 0 with the Hamiltonian � � H = 1 R d g ζ 2 + | U | 2 2 Ω David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 5 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 Craig-Sulem 93: Definition (Dirichlet-Neumann operator) � 1 + |∇ ζ | 2 ∂ n Φ | z = ζ . ψ G [ ζ, b ] : �→ G [ ζ, b ] ψ = David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 Craig-Sulem 93: Definition (Dirichlet-Neumann operator) � 1 + |∇ ζ | 2 ∂ n Φ | z = ζ . ψ G [ ζ, b ] : �→ G [ ζ, b ] ψ =  ∆ X , z Φ = 0 ,  ∂ n Φ | z = − H 0+ b = 0 ,  Φ | z = ζ = ψ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 Craig-Sulem 93: Definition (Dirichlet-Neumann operator) � 1 + |∇ ζ | 2 ∂ n Φ | z = ζ . ψ G [ ζ, b ] : �→ G [ ζ, b ] ψ =  ∆ X , z Φ = 0 ,  ∂ n Φ | z = − H 0+ b = 0 ,  Φ | z = ζ = ψ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 Craig-Sulem 93: Definition (Dirichlet-Neumann operator) � 1 + |∇ ζ | 2 ∂ n Φ | z = ζ . ψ G [ ζ, b ] : �→ G [ ζ, b ] ψ = � The equation on ζ can be written ∂ t ζ − G [ ζ, b ] ψ = 0 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation Question What are the equations on ζ and ψ ??? • Equation on ζ . It is given by the kinematic equation � ∂ t ζ − 1 + |∇ ζ | 2 ∂ n Φ | z = ζ = 0 Craig-Sulem 93: Definition (Dirichlet-Neumann operator) � 1 + |∇ ζ | 2 ∂ n Φ | z = ζ . ψ G [ ζ, b ] : �→ G [ ζ, b ] ψ = � The equation on ζ can be written ∂ t ζ − G [ ζ, b ] ψ = 0 Remark. One has the exact relation � ζ with h = H 0 + ζ − b and V = 1 G [ ζ, b ] ψ = −∇· ( hV ) V ( X , z ) dz h − H 0 + b David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 6 / 33

The equations The Zakharov/Craig-Sulem formulation • Equation on ψ . We use (H1)” and (H7)” ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) AND P | z = ζ = P atm David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 7 / 33

The equations The Zakharov/Craig-Sulem formulation • Equation on ψ . We use (H1)” and (H7)” ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) AND P | z = ζ = P atm � � ∂ t Φ | z = ζ + 1 2 |∇ X , z Φ | 2 | z = ζ + g ζ = 0 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 7 / 33

The equations The Zakharov/Craig-Sulem formulation • Equation on ψ . We use (H1)” and (H7)” ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) AND P | z = ζ = P atm � � ∂ t Φ | z = ζ + 1 2 |∇ X , z Φ | 2 | z = ζ + g ζ = 0 � The equation on ψ can be written 2 |∇ ψ | 2 − ( G [ ζ, b ] ψ + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 = 0 . 2(1 + |∇ ζ | 2 ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 7 / 33

The equations The Zakharov/Craig-Sulem formulation • Equation on ψ . We use (H1)” and (H7)” ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) AND P | z = ζ = P atm � � ∂ t Φ | z = ζ + 1 2 |∇ X , z Φ | 2 | z = ζ + g ζ = 0 � The equation on ψ can be written 2 |∇ ψ | 2 − ( G [ ζ, b ] ψ + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 = 0 . 2(1 + |∇ ζ | 2 ) The Zakharov-Craig-Sulem equations  ∂ t ζ − G [ ζ, b ] ψ = 0 ,  2 |∇ ψ | 2 − ( G [ ζ, b ] ψ + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1  = 0 . 2(1 + |∇ ζ | 2 ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 7 / 33

Asymptotic expansions Nondimensionalization Goal Derive simpler asymptotic models describing the solutions to the water waves equations in shallow water. David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 8 / 33

Asymptotic expansions Nondimensionalization Goal Derive simpler asymptotic models describing the solutions to the water waves equations in shallow water. For the sake of simplicity, we consider here a flat bottom ( b = 0). David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 8 / 33

Asymptotic expansions Nondimensionalization Goal Derive simpler asymptotic models describing the solutions to the water waves equations in shallow water. For the sake of simplicity, we consider here a flat bottom ( b = 0). We introduce three characteristic scales The characteristic water depth H 0 1 The characteristic horizontal scale L 2 The order of the free surface amplitude a 3 Two independent dimensionless parameters can be formed from these three scales. We choose: a = ε (amplitude parameter ) , H 0 H 2 0 L 2 = µ (shallowness parameter ) . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 8 / 33

Asymptotic expansions Nondimensionalization We proceed to the simple nondimensionalizations X ′ = X z ′ = z ζ ′ = ζ L , , a , etc. H 0 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 9 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 Shallow water asymptotics ( µ ≪ 1) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V This is obtained through an asymtotic description of V in the fluid. David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V This is obtained through an asymtotic description of V in the fluid. This is obtained through an asympotic description of Φ in the fluid, Φ ∼ Φ 0 + µ Φ 1 + µ 2 Φ 2 + . . . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations  ∂ t ζ + ∇ · ( hV ) = 0 ,  2 ∇|∇ ψ | 2 − εµ ∇ ( −∇ · ( hV ) + ∇ ( εζ ) · ∇ ψ ) 2 ∂ t ∇ ψ + ∇ ζ + ε  = 0 , 2(1 + ε 2 µ |∇ ζ | 2 ) where in dimensionless form � εζ V = 1 h = 1 + εζ and V ( x , z ) dz . h − 1 Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V This is obtained through an asymtotic description of V in the fluid. This is obtained through an asympotic description of Φ in the fluid, Φ ∼ Φ 0 + µ Φ 1 + µ 2 Φ 2 + . . . At first order, we have a columnar motion and therefore ∇ ψ = V + O ( µ ). David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations � ∂ t ζ + ∇ · ( hV ) = 0 , Saint-Venant ∂ t V + ∇ ζ + ε V · ∇ V = 0 . where we dropped all O ( µ ) terms. Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V This is obtained through an asymtotic description of V in the fluid. This is obtained through an asympotic description of Φ in the fluid, Φ ∼ Φ 0 + µ Φ 1 + µ 2 Φ 2 + . . . At first order, we have a columnar motion and therefore ∇ ψ = V + O ( µ ). David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Asymptotic expansions Nondimensionalized equations � ∂ t ζ + ∇ · ( hV ) = 0 , Saint-Venant ∂ t V + ∇ ζ + ε V · ∇ V = 0 . where we dropped all O ( µ ) terms. Shallow water asymptotics ( µ ≪ 1) We look for an asymptotic description with respect to µ of ∇ ψ in terms of ζ and V This is obtained through an asymtotic description of V in the fluid. This is obtained through an asympotic description of Φ in the fluid, Φ ∼ Φ 0 + µ Φ 1 + µ 2 Φ 2 + . . . At first order, we have a columnar motion and therefore ∇ ψ = V + O ( µ ). All this procedure can be fully justified (cf Walter Craig for KdV ! ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 10 / 33

Numerical computations 1D simulations “ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 11 / 33

Numerical computations 1D simulations “ Bonneton, Chazel, L. , Marche, Tissier 2011-2012 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 12 / 33

Numerical computations 2D computations 2D configurations can also be handled (D.L. & F. Marche, 2014): • Tsunami island David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 13 / 33

Numerical computations 2D computations 2D configurations can also be handled (D.L. & F. Marche, 2014): • Beach David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 13 / 33

Numerical computations 2D computations 2D configurations can also be handled (D.L. & F. Marche, 2014): • Overtopping David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 13 / 33

Numerical computations 2D computations David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 14 / 33

Water waves with vorticity Basic assumptions (H1) The fluid is homogeneous and inviscid (H2) The fluid is incompressible (H3) ///// The///// flow /// is////////////// irrotational (H4) The surface and the bottom can be parametrized as graphs above the still water level (H5) The fluid particles do not cross the bottom (H6) The fluid particles do not cross the surface (H7) There is no surface tension and the external pressure is constant. (H8) The fluid is at rest at infinity (H9) The water depth is always bounded from below by a nonnegative constant Refs: Lindblad, Coutand-Shkoller, Shatah-Zeng, Zhang-Zhang, Masmoudi-Rousset, ... David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 15 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 16 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Irrotational case We get from curl U = 0 that U = ∇ X , z Φ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 16 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Irrotational case We get from curl U = 0 that U = ∇ X , z Φ We replace Euler’s equation on U by Bernoulli’s equation on Φ ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 16 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Irrotational case We get from curl U = 0 that U = ∇ X , z Φ We replace Euler’s equation on U by Bernoulli’s equation on Φ ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) We eliminate the pressure by taking the trace on the interface David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 16 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Irrotational case We get from curl U = 0 that U = ∇ X , z Φ We replace Euler’s equation on U by Bernoulli’s equation on Φ ∂ t Φ + 1 2 |∇ X , z Φ | 2 + gz = − 1 ρ ( P − P atm ) We eliminate the pressure by taking the trace on the interface We reduce the problem to an equation on ζ and ψ ( t , X ) = Φ( t , X , ζ ( t , x )). David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 16 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 17 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Rotational case One has curl U = ω � = 0 and ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 17 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Rotational case One has curl U = ω � = 0 and ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . One cannot work with the Benouilli equation � How can we use the boundary condition on the pressure P ? David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 17 / 33

Water waves with vorticity Euler’s equations − 1 ∂ t U + U · ∇ X , z U = ρ ∇ X , z P − g e z , ∇ X , z · U = 0 , P | z = ζ = P atm Rotational case One has curl U = ω � = 0 and ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . One cannot work with the Benouilli equation � How can we use the boundary condition on the pressure P ? One can remark that � ∇ ( P | z = ζ ) � ( ∇ X , z P ) | z = ζ = + N ∂ z P | z = ζ 0 = 0 + N ∂ z P | z = ζ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 17 / 33

Water waves with vorticity New formulation One has ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z and � −∇ ζ � ( ∇ X , z P ) | z = ζ = N ∂ z P | z = ζ , with N = . 1 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 18 / 33

Water waves with vorticity New formulation One has ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z and � −∇ ζ � ( ∇ X , z P ) | z = ζ = N ∂ z P | z = ζ , with N = . 1 � One can eliminate the pressure by 1 Taking the trace of Euler’s equation at the surface David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 18 / 33

Water waves with vorticity New formulation One has ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z and � −∇ ζ � ( ∇ X , z P ) | z = ζ = N ∂ z P | z = ζ , with N = . 1 � One can eliminate the pressure by 1 Taking the trace of Euler’s equation at the surface 2 Take the vectorial product of the resulting equation with N . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 18 / 33

Water waves with vorticity New formulation One has ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z and � −∇ ζ � ( ∇ X , z P ) | z = ζ = N ∂ z P | z = ζ , with N = . 1 � One can eliminate the pressure by 1 Taking the trace of Euler’s equation at the surface 2 Take the vectorial product of the resulting equation with N . � This leads to an equation on the tangential part of the velocity at the surface David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 18 / 33

Water waves with vorticity New formulation One has ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z and � −∇ ζ � ( ∇ X , z P ) | z = ζ = N ∂ z P | z = ζ , with N = . 1 � One can eliminate the pressure by 1 Taking the trace of Euler’s equation at the surface 2 Take the vectorial product of the resulting equation with N . � This leads to an equation on the tangential part of the velocity at the surface Notation With U = ( V , w ) = U | z = ζ , we write � � − U ⊥ � U � = V + w ∇ ζ so that U × N = − U ⊥ � · ∇ ζ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 18 / 33

Water waves with vorticity New formulation � � ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z × N | z = ζ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 19 / 33

Water waves with vorticity New formulation � � ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z × N | z = ζ � � ( with some computations) � � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 19 / 33

Water waves with vorticity New formulation � � ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z × N | z = ζ � � ( with some computations) � � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ What does it give in the irrotational case? David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 19 / 33

Water waves with vorticity New formulation � � ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z × N | z = ζ � � ( with some computations) � � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ What does it give in the irrotational case? In the irrotational case, one has U � = ∇ ψ. David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 19 / 33

Water waves with vorticity New formulation � � ∂ t U + U · ∇ X , z U = − 1 ρ ∇ X , z P − g e z × N | z = ζ � � ( with some computations) � � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ What does it give in the irrotational case? In the irrotational case, one has U � = ∇ ψ. How do we generalize to the rotational case? We decompose U � into U � = ∇ ψ + ∇ ⊥ � ψ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 19 / 33

Water waves with vorticity New formulation We have found � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ and decomposed U � = ∇ ψ + ∇ ⊥ � ψ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 20 / 33

Water waves with vorticity New formulation We have found � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ and decomposed U � = ∇ ψ + ∇ ⊥ � ψ The question is now to find equations on ψ and � ψ . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 20 / 33

Water waves with vorticity New formulation We have found � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ and decomposed U � = ∇ ψ + ∇ ⊥ � ψ The question is now to find equations on ψ and � ψ . This is done by projecting the equation onto its “gradient” and “orthogonal gradient” components David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 20 / 33

Water waves with vorticity New formulation We have found � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ and decomposed U � = ∇ ψ + ∇ ⊥ � ψ The question is now to find equations on ψ and � ψ . This is done by projecting the equation onto its “gradient” and “orthogonal gradient” components ⊥ This is done by applying div and div to the equation ∆ ∆ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 20 / 33

Water waves with vorticity New formulation We have found � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ and decomposed U � = ∇ ψ + ∇ ⊥ � ψ The question is now to find equations on ψ and � ψ . This is done by projecting the equation onto its “gradient” and “orthogonal gradient” components ⊥ This is done by applying div and div to the equation ∆ ∆ The “orthogonal gradient” component yields ∂ t ( ω · N − ∇ ⊥ · U � ) = 0 , which is trivially true and does not bring any information David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 20 / 33

Water waves with vorticity New formulation � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 21 / 33

Water waves with vorticity New formulation � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ The “orthogonal gradient “component of the equation does not bring any information David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 21 / 33

Water waves with vorticity New formulation � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ The “orthogonal gradient “component of the equation does not bring any information The “gradient” component of the equation is obtained by applying div ∆ . After remarking that div div ∆ ( ∇ ψ + ∇ ⊥ � ∆ U � = ψ ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 21 / 33

Water waves with vorticity New formulation � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ The “orthogonal gradient “component of the equation does not bring any information The “gradient” component of the equation is obtained by applying div ∆ . After remarking that div div ∆ ( ∇ ψ + ∇ ⊥ � ∆ U � = ψ ) = ψ, we get David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 21 / 33

Water waves with vorticity New formulation � (1 + |∇ ζ | 2 ) w 2 � ∂ t U � + g ∇ ζ + 1 2 ∇| U � | 2 − 1 + ω · NV ⊥ = 0 . 2 ∇ The “orthogonal gradient “component of the equation does not bring any information The “gradient” component of the equation is obtained by applying div ∆ . After remarking that div div ∆ ( ∇ ψ + ∇ ⊥ � ∆ U � = ψ ) = ψ, we get � (1 + |∇ ζ | 2 ) w 2 � � ω · NV ⊥ � ∂ t ψ + g ζ + 1 2 | U � | 2 − 1 + ∇ ∆ · = 0 2 David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 21 / 33

Water waves with vorticity New formulation Irrotational case  ∂ t ζ − U · N = 0 ,    2 |∇ ψ | 2 − ( U · N + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 (ZCS) = 0  2(1 + |∇ ζ | 2 )   ω = 0 . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 22 / 33

Water waves with vorticity New formulation Irrotational case  ∂ t ζ − U · N = 0 ,    2 |∇ ψ | 2 − ( U · N + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 (ZCS) = 0  2(1 + |∇ ζ | 2 )   ω = 0 . Moreover, U · N = G [ ζ ] ψ David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 22 / 33

Water waves with vorticity New formulation Irrotational case  ∂ t ζ − U · N = 0 ,    2 |∇ ψ | 2 − ( U · N + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 (ZCS) = 0  2(1 + |∇ ζ | 2 )   ω = 0 . Moreover, U · N = G [ ζ ] ψ ⇒ (ZCS) is a closed system of equations in ( ζ, ψ ) . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 22 / 33

Water waves with vorticity New formulation Irrotational case  ∂ t ζ − U · N = 0 ,    2 |∇ ψ | 2 − ( U · N + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 (ZCS) = 0  2(1 + |∇ ζ | 2 )   ω = 0 . Moreover, U · N = G [ ζ ] ψ ⇒ (ZCS) is a closed system of equations in ( ζ, ψ ) . Rotational case   ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )    ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 22 / 33

Water waves with vorticity New formulation Irrotational case  ∂ t ζ − U · N = 0 ,    2 |∇ ψ | 2 − ( U · N + ∇ ζ · ∇ ψ ) 2 ∂ t ψ + g ζ + 1 (ZCS) = 0  2(1 + |∇ ζ | 2 )   ω = 0 . Moreover, U · N = G [ ζ ] ψ ⇒ (ZCS) is a closed system of equations in ( ζ, ψ ) . Rotational case   ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )    ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . � is this a closed system of equations in ( ζ, ψ, ω ) ? David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 22 / 33

Water waves with vorticity New formulation  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . We want to prove that this is a closed system of equations in ( ζ, ψ, ω ): David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 23 / 33

Water waves with vorticity New formulation  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . We want to prove that this is a closed system of equations in ( ζ, ψ, ω ): It is enough to prove that U is fully determined by ( ζ, ψ, ω ) David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 23 / 33

Water waves with vorticity New formulation  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . We want to prove that this is a closed system of equations in ( ζ, ψ, ω ): It is enough to prove that U is fully determined by ( ζ, ψ, ω ) We recall that by definition of ψ and � ψ , U � = ∇ ψ + ∇ ⊥ � ψ, and we have already used the fact that ω · N = ∇ ⊥ · U � ; therefore David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 23 / 33

Water waves with vorticity New formulation  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . We want to prove that this is a closed system of equations in ( ζ, ψ, ω ): It is enough to prove that U is fully determined by ( ζ, ψ, ω ) We recall that by definition of ψ and � ψ , U � = ∇ ψ + ∇ ⊥ � ψ, and we have already used the fact that ω · N = ∇ ⊥ · U � ; therefore U � = ∇ ψ + ∇ ⊥ ∆ ω · N . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 23 / 33

Water waves with vorticity New formulation  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . We want to prove that this is a closed system of equations in ( ζ, ψ, ω ): It is enough to prove that U is fully determined by ( ζ, ψ, ω ) We recall that by definition of ψ and � ψ , U � = ∇ ψ + ∇ ⊥ � ψ, and we have already used the fact that ω · N = ∇ ⊥ · U � ; therefore U � = ∇ ψ + ∇ ⊥ ∆ ω · N . We are therefore led to solve  curl U = in Ω ω    div U = 0 in Ω ∇ ψ + ∇ ⊥ ∆ − 1 ( ω · N ) U � = at the surface    U | z = − H 0 · N b = 0 at the bottom . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 23 / 33

Water waves with vorticity The div-curl problem  curl U = ω in Ω    div U = 0 in Ω ∇ ψ + ∇ ⊥ ∆ − 1 ( ω · N ) U � = at the surface    U | z = − H 0 · N b = 0 at the bottom . Proposition For all ω ∈ H b (div 0 , Ω) and all ψ ∈ ˙ H 3 / 2 ( R d ) , (1) There is a unique solution U ∈ H 1 (Ω) d +1 to the div-curl problem, and � � � U � 2 + �∇ X , z U � 2 ≤ C ( 1 , | ζ | W 2 , ∞ ) � ω � 2 , b + |∇ ψ | H 1 / 2 . h min David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 24 / 33

Water waves with vorticity The div-curl problem  curl U = ω in Ω    div U = 0 in Ω ∇ ψ + ∇ ⊥ ∆ − 1 ( ω · N ) U � = at the surface    U | z = − H 0 · N b = 0 at the bottom . Proposition For all ω ∈ H b (div 0 , Ω) and all ψ ∈ ˙ H 3 / 2 ( R d ) , (2) The solution U can be written U = curl A + ∇ X , z Φ with  curl curl A = ω in Ω ,     div A = 0 in Ω ,     N b × A | bott = 0 N · A | surf = 0     = ∇ ⊥ ∆ − 1 ω · N ,  (curl A ) �    N · curl A | bott = 0 , [...] David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 24 / 33

Water waves with vorticity The div-curl problem  curl U = ω in Ω    div U = 0 in Ω ∇ ψ + ∇ ⊥ ∆ − 1 ( ω · N ) U � = at the surface    U | z = − H 0 · N b = 0 at the bottom . Proposition For all ω ∈ H b (div 0 , Ω) and all ψ ∈ ˙ H 3 / 2 ( R d ) , (2) [...] while Φ ∈ ˙ H 1 (Ω) solves � ∆ X , z Φ = 0 in Ω , Φ | z = εζ = ψ, ∂ n Φ | z = − 1+ β b = 0 . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 24 / 33

Water waves with vorticity The div-curl problem Proof.  curl curl A = ω     N b × A | z = − H 0 = 0 N · A | z = ζ = 0   � �   = ∇ ⊥ � curl A | z = ζ ψ. � Step 4. Solving ∆ ˜ ψ = ω · N in ˙ H 1 / 2 ( R d ). Use Lax-Milgram in ˙ H 1 ( R d ) to solve the variational formulation of the equation: for all v ∈ ˙ H 1 ( R d ) � � R d ∇ v · ∇ ˜ ψ = R d ω · Nv � � R d ω b · Nv ext ω · ∇ X , z v ext = | z = − 1+ β b − Ω � � � | D | − 1 ω b · N b � ≤ ( H 1 / 2 + � ω � 2 ) |∇ v | 2 � �� � := � ω � 2 , b David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 25 / 33

Water waves with vorticity Well-posedness  ∂ t ζ − U · N = 0 ,    2 | U � | 2 − ( U · N + ∇ ζ · U � ) 2 = ∇ ⊥ ∂ t ψ + g ζ + 1 (ZCS) gen ∆ · ( ω · NV )  2(1 + |∇ ζ | 2 )   ∂ t ω + U · ∇ X , z ω = ω · ∇ X , z U . Corollary This is a closed system of equations in ( ζ, ψ, ω ) . David Lannes (DMA, ENS et CNRS) Water Waves with vorticity Toronto, January 10th, 2014 26 / 33