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Mod elisation math ematique des vagues David Lannes Institut de - - PowerPoint PPT Presentation

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Mod elisation math ematique des vagues David Lannes Institut de Math ematiques de Bordeaux et CNRS UMR 5251 Journ ee des doctorants David Lannes (IMB) Mod elisation math ematique des vagues Valenciennes, 10/09/2015 1 / 30

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Mod´ elisation math´ ematique des vagues

David Lannes

Institut de Math´ ematiques de Bordeaux et CNRS UMR 5251

Journ´ ee des doctorants

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 1 / 30

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Goal

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

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Goal

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

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Goal

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 2 / 30

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Where do waves come from? How are they created? Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 3 / 30

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Where do waves come from? What is their speed?

Sir Isaac Newton (1642-1727) Principia Mathematica, 1687

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 4 / 30

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Where do waves come from? What is their speed? David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 4 / 30

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Where do waves come from? What is their speed?

Leonhard Euler (1707-1783) M´ emoires de l’Acad´ emie royale des sciences et des belles lettres de Berlin, 1757 Equations of fluid mechanics ρ(∂tU + U · ∇X,zU) = − ∇X,zP + ρg div U =0

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 5 / 30

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Where do waves come from? What is their speed?

Leonhard Euler (1707-1783) M´ emoires de l’Acad´ emie royale des sciences et des belles lettres de Berlin, 1757 Equations of fluid mechanics ρ(∂tU + U · ∇X,zU) = − ∇X,zP + ρg div U =0 This equations are very general What do they tell us about waves?

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 5 / 30

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Where do waves come from? What is their speed?

Giuseppe Lodovico Lagrangia (Joseph Louis Lagrange) (1736-1813) M´ emoire sur la th´ eorie du mouvement des fluides, 1781

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 6 / 30

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Where do waves come from? What is their speed? David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 6 / 30

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Where do waves come from? What is their speed? Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 7 / 30

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Where do waves come from? What is their speed?

Comparison of Newton and Lagrange’s formulas Lagrange: c = √gH. All waves have same speed

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

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Where do waves come from? What is their speed?

Comparison of Newton and Lagrange’s formulas Lagrange: c = √gH. All waves have same speed Newton: c =

1 √ 2π

√gL where L is the wave length of the wave Waves of different wavelength propagate differently

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

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Where do waves come from? What is their speed?

Comparison of Newton and Lagrange’s formulas Lagrange: c = √gH. All waves have same speed Newton: c =

1 √ 2π

√gL where L is the wave length of the wave Waves of different wavelength propagate differently This is dispersion:

Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

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Where do waves come from? What is their speed?

Comparison of Newton and Lagrange’s formulas Lagrange: c = √gH. All waves have same speed Newton: c =

1 √ 2π

√gL where L is the wave length of the wave Waves of different wavelength propagate differently Comparison for a wave a0(x) = sin(x) + 0.5 sin(2x).

Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 8 / 30

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Where do waves come from? What is their speed?

Recall how waves are created

Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011

So the good formula should be

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 9 / 30

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Where do waves come from? What is their speed?

Recall how waves are created

Source: Les vagues en ´ equations, Pour la Science, no 409, novembre 2011

So the good formula should be Newton: c =

1 √ 2π

√gL

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 9 / 30

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Closer to the shore Another formula!

Closer to the shore we observe: And the relevant formula is

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 10 / 30

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Closer to the shore Another formula!

Closer to the shore we observe: And the relevant formula is Lagrange: c = √gH

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 10 / 30

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Closer to the shore What happens?

Sim´ eon Denis Poisson (1780–1840) Augustin Louis Cauchy (1789–1857) Sir George Biddell Airy (1801–1892) Sir George Gabriel Stokes (1819–1903)

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 11 / 30

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Closer to the shore What happens?

Sim´ eon Denis Poisson (1780–1840) Augustin Louis Cauchy (1789–1857) Sir George Biddell Airy (1801–1892) Sir George Gabriel Stokes (1819–1903) A single formula with two different asymptotic regimes Lagrange’s formula in shallow water (H/L → 0), Newton’s formula in deep water (H/L → ∞).

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 11 / 30

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Modern mathematical approaches Notations David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 12 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}.

Definition Equations (H1)-(H9) are called free surface Euler equations.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}.

Definition Equations (H1)-(H9) are called free surface Euler equations. ONE unknown function ζ on a fixed domain Rd THREE unknown functions U on a moving, unknown domain Ωt

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 13 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 curl U = 0 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)} 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 div U = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)} 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tU + (U · ∇X,z)U = − 1

ρ∇X,zP − gez in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)} 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tΦ + 1

2|∇X,zΦ|2 + gz = − 1 ρ(P − Patm) in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 U · n = 0 on {z = −H0 + b(X)} 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tΦ + 1

2|∇X,zΦ|2 + gz = − 1 ρ(P − Patm) in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 ∂nΦ = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2U · n = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tΦ + 1

2|∇X,zΦ|2 + gz = − 1 ρ(P − Patm) in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 ∂nΦ = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2∂nΦ = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}. David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tΦ + 1

2|∇X,zΦ|2 + gz = − 1 ρ(P − Patm) in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 ∂nΦ = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2∂nΦ = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}.

Definition Equations (H1)’-(H9)’ are called free surface Bernoulli equations.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations

1 ∂tΦ + 1

2|∇X,zΦ|2 + gz = − 1 ρ(P − Patm) in Ωt

2 ∆X,zΦ = 0 3 U = ∇X,zΦ 4 Ωt = {(X, z) ∈ Rd+1, −H0 + b(X) < z < ζ(t, X)}. 5 ∂nΦ = 0 on {z = −H0 + b(X)}. 6 ∂tζ −

  • 1 + |∇ζ|2∂nΦ = 0 on {z = ζ(t, X)}.

7 P = Patm on {z = ζ(t, X)}.

Definition Equations (H1)’-(H9)’ are called free surface Bernoulli equations. ONE unknown function ζ on a fixed domain Rd ONE unknown function Φ on a moving, unknown domain Ωt

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 14 / 30

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Modern mathematical approaches Working with a fix domain

The Lagrangian approach

One parametrizes any fluid particle of Ωt by its initial position through the diffeomorphism Σ

  • ∂tΣ(t, X, z) = U(t, Σ(t, X, z)),

Σ(0, X, z) = (X, z). Writing ˜ U(t, X, z) =U(t, Σ(t, X, z)), A(t, X, z) =|∇X,zΣ|−1 we get

  • ∂tU + U · ∇X,zU =

−∇X,zP + g div (U) = in Ωt

  • ∂t ˜

U = −A∇X,z ˜ P + g Tr (A∇X,z ˜ U) = in Ω0

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 15 / 30

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Modern mathematical approaches Working with a fix domain

The geometric approach (I)

The Lagrangian diffeomorphism

  • ∂tΣ(t, X, z) = U(t, Σ(t, X, z)),

Σ(0, X, z) = (X, z). is volume preserving since U is divergence free Σ ∈ H = {Σ : Ω0 → Rd+1, Σ volume preserving. Moreover the energy is conserved H = 1 2

  • Ωt

|U|2 + g 2

  • Rd ζ2

= 1 2

  • Ω0

|∂tΣ|2 + gG(Σ)

  • :=L(Σ,∂tΣ)

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 16 / 30

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Modern mathematical approaches Working with a fix domain

The geometric approach (II)

Defining TΣH TΣH = {Σ′ Ω0 → Rd+1, div (Σ′ ◦ Σ) = 0}, the free surface Euler equations can be viewed as a critical point of the action L(Σ, ∂tΣ) = = 1 2

  • Ω0

|∂tΣ|2 − gG(Σ). Remark Arnold (1966): the Euler equation for an incompressible inviscid fluid can be viewed as the geodesic equation on the group of volume-preserving diffeomorphisms.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 17 / 30

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Modern mathematical approaches Working with a fix domain

Lagrangian interface formulation (I)

We consider a Lagrangian parametrization of the surface Γt = {M(t, α), α ∈ R}, with

  • ∂tM(t, α) = U(t, M(t, α)),

M(0, α) = (α, ζ0(α)). One then has ∂2

t M =∂tU + ∂tM · ∇X,zU

=∂tU + U · ∇X,zU = − gez − 1 ρ∇X,zP And since P = Patm is constant at the surface ∂2

t M + gez = 1

ρ(−∂nP)n

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 18 / 30

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Modern mathematical approaches Working with a fix domain

Lagrangian interface formulation (II)

∂2

t M + gez = 1

ρ(−∂nP)n ∂αM1∂2

t M1 + (g + ∂2 t M2)∂αM2 = 0

We still need a relation between ∂tM1 and ∂tM2 !!!! Complex analysis (x, z) ∈ R2 x + iz ∈ C Incompressibility+Irrotationality=Cauchy Riemann for U U is holomorphic in Ωt ∂tM = U(t, M(t, α)) is the boundary of a holomorphic function, therefore ∂tM = H(Γt)∂tM with H(Γt)f (t, α) = 1 iπp.v.

  • f (t, α′)∂αM(t, α′)

M(t, α) − M(t, α′)dα′.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 19 / 30

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Modern mathematical approaches Working with a fix domain

An Eulerian approach: The Zakharov-Craig-Sulem formulation

Zakharov 68:

1 Define ψ(t, X) = Φ(t, X, ζ(t, X)) . David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 20 / 30

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Modern mathematical approaches Working with a fix domain

An Eulerian approach: 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,zΦ = 0 in Ωt, Φ|z=ζ = ψ, ∂nΦ|z=−H0+b = 0. has a unique solution Φ.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 20 / 30

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Modern mathematical approaches Working with a fix domain

An Eulerian approach: 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,zΦ = 0 in Ωt, Φ|z=ζ = ψ, ∂nΦ|z=−H0+b = 0. has a unique solution Φ.

3 The equations can be put under the canonical Hamiltonian form

∂t ζ ψ

  • =
  • 1

−1

  • gradζ,ψH

with the Hamiltonian H = 1 2

  • Rd gζ2 +
  • Ω

|U|2

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 20 / 30

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Modern mathematical approaches Working with a fix domain

Question What are the equations on ζ and ψ???

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 21 / 30

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Modern mathematical approaches Working with a fix domain

Question What are the equations on ζ and ψ???

  • Equation on ζ. It is given by the kinematic equation

∂tζ −

  • 1 + |∇ζ|2∂nΦ|z=ζ = 0

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 21 / 30

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Modern mathematical approaches Working with a fix domain

Question What are the equations on ζ and ψ???

  • Equation on ζ. It is given by the kinematic equation

∂tζ −

  • 1 + |∇ζ|2∂nΦ|z=ζ = 0

Craig-Sulem 93: Definition (Dirichlet-Neumann operator) G[ζ] : ψ → G[ζ]ψ =

  • 1 + |∇ζ|2∂nΦ|z=ζ

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 21 / 30

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Modern mathematical approaches Working with a fix domain

Question What are the equations on ζ and ψ???

  • Equation on ζ. It is given by the kinematic equation

∂tζ −

  • 1 + |∇ζ|2∂nΦ|z=ζ = 0

Craig-Sulem 93: Definition (Dirichlet-Neumann operator) G[ζ] : ψ → G[ζ]ψ =

  • 1 + |∇ζ|2∂nΦ|z=ζ

The equation on ζ can be written ∂tζ − G[ζ]ψ = 0

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 21 / 30

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Modern mathematical approaches Working with a fix domain

  • Equation on ψ. We use (H1)” and (H7)”

∂tΦ + 1 2|∇X,zΦ|2 + gz = −1 ρ(P − Patm) AND P|z=ζ = Patm

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 22 / 30

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Modern mathematical approaches Working with a fix domain

  • Equation on ψ. We use (H1)” and (H7)”

∂tΦ + 1 2|∇X,zΦ|2 + gz = −1 ρ(P − Patm) AND P|z=ζ = Patm

  • ∂tΦ|z=ζ + 1

2|∇X,zΦ|2

|z=ζ + gζ = 0

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 22 / 30

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SLIDE 55

Modern mathematical approaches Working with a fix domain

  • Equation on ψ. We use (H1)” and (H7)”

∂tΦ + 1 2|∇X,zΦ|2 + gz = −1 ρ(P − Patm) AND P|z=ζ = Patm

  • ∂tΦ|z=ζ + 1

2|∇X,zΦ|2

|z=ζ + gζ = 0

The equation on ψ can be written ∂tψ + gζ + 1 2|∇ψ|2 − (G[ζ]ψ + ∇ζ · ∇ψ)2 2(1 + |∇ζ|2) = 0.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 22 / 30

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SLIDE 56

Modern mathematical approaches Working with a fix domain

  • Equation on ψ. We use (H1)” and (H7)”

∂tΦ + 1 2|∇X,zΦ|2 + gz = −1 ρ(P − Patm) AND P|z=ζ = Patm

  • ∂tΦ|z=ζ + 1

2|∇X,zΦ|2

|z=ζ + gζ = 0

The equation on ψ can be written ∂tψ + gζ + 1 2|∇ψ|2 − (G[ζ]ψ + ∇ζ · ∇ψ)2 2(1 + |∇ζ|2) = 0. The Zakharov-Craig-Sulem equations    ∂tζ − G[ζ]ψ = 0, ∂tψ + gζ + 1 2|∇ψ|2 − (G[ζ]ψ + ∇ζ · ∇ψ)2 2(1 + |∇ζ|2) = 0. Two scalar equations on the fix d-dimensional domain Rd!

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 22 / 30

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SLIDE 57

Local well posedness Linearized equations around the rest state

Linearized equations

  • ∂tζ − G[0]ψ = 0,

∂tψ + gζ = 0. and G[0] = |D| tanh(H|D|) and therefore ∂2

t ζ + g|D| tanh(H|D|)ζ = 0

Newton and Lagrange’s formulas: ∂2

t ζ − gH∂2 xζ = 0 in shallow water

∂2

t ζ + g|D|ζ = 0 in deep water.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 23 / 30

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SLIDE 58

Local well posedness Linearized equations around the rest state

Quasilinearized equations

After differentiation and change of unknowns, the structure is (∂t + V · ∇) ˜ ζ ˜ ψ

  • +

−G[ζ] a ˜ ζ ˜ ψ

  • = l.o.t.

Symbolic approximation G[ζ] = |D| + order 0 Jordan block a > 0 This is the Rayleigh-Taylor criterion (−∂zP)|z=ζ > 0. Theorem The (ZCS) is locally well posed.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 24 / 30

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SLIDE 59

Asymptotic expansions Nondimensionalization

Asymptotic models

Goal Derive simpler asymptotic models describing the solutions to the water waves equations in shallow water.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 25 / 30

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SLIDE 60

Asymptotic expansions Nondimensionalization

Asymptotic models

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 (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 25 / 30

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SLIDE 61

Asymptotic expansions Nondimensionalization

Asymptotic models

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

1

The characteristic water depth H0

2

The characteristic horizontal scale L

3

The order of the free surface amplitude a

Two independent dimensionless parameters can be formed from these three scales. We choose: a H0 = ε (amplitude parameter ), H2 L2 = µ (shallowness parameter ).

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 25 / 30

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SLIDE 62

Asymptotic expansions Nondimensionalization

We proceed to the simple nondimensionalizations X ′ = X L , z′ = z H0 , ζ′ = ζ a, etc.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 26 / 30

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SLIDE 63

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz.

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 64

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz. Shallow water asymptotics (µ ≪ 1)

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 65

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz. 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 (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 66

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz. 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 (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 67

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz. 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 ∼ Φ0 + µΦ1 + µ2Φ2 + . . .

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 68

Asymptotic expansions Nondimensionalized equations

   ∂tζ + ∇ · (hV ) = 0, ∂t∇ψ + ∇ζ + ε 2∇|∇ψ|2 − εµ∇(−∇ · (hV ) + ∇(εζ) · ∇ψ)2 2(1 + ε2µ|∇ζ|2) = 0, where in dimensionless form h = 1 + εζ and V = 1 h εζ

−1

V (x, z)dz. 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 ∼ Φ0 + µΦ1 + µ2Φ2 + . . . At first order, we have a columnar motion and therefore ∇ψ = V + O(µ).

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 69

Asymptotic expansions Nondimensionalized equations

Saint-Venant ∂tζ + ∇ · (hV ) = 0, ∂tV + εV · ∇V + ∇ζ = 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 ∼ Φ0 + µΦ1 + µ2Φ2 + . . . At first order, we have a columnar motion and therefore ∇ψ = V + O(µ).

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 70

Asymptotic expansions Nondimensionalized equations

Green-Nadghi ∂tζ + ∇ · (hV ) = 0, (I + µT )

  • ∂tV + εV · ∇V
  • + ∇ζ + µQ(V ) = 0.

where we dropped all O(µ2) 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 ∼ Φ0 + µΦ1 + µ2Φ2 + . . . At first order, we have a columnar motion and therefore ∇ψ = V + O(µ). Next order approximation: Green-Naghdi

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 27 / 30

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SLIDE 71

Asymptotic expansions Nondimensionalized equations

Justification

One needs to prove that the solution exists on a time interval [0, T/ε] with T independent of µ One needs bounds on the solution on this time scale

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 28 / 30

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SLIDE 72

Asymptotic expansions Nondimensionalized equations

Justification

One needs to prove that the solution exists on a time interval [0, T/ε] with T independent of µ One needs bounds on the solution on this time scale The previous proof does not work! Beware the W 1,∞\C 1(Rd) waves!!!

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 28 / 30

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SLIDE 73

Asymptotic expansions Nondimensionalized equations

Justification

One needs to prove that the solution exists on a time interval [0, T/ε] with T independent of µ One needs bounds on the solution on this time scale The previous proof does not work! Beware the big waves!!!

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 28 / 30

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SLIDE 74

Asymptotic expansions Nondimensionalized equations

Justification

One needs to prove that the solution exists on a time interval [0, T/ε] with T independent of µ One needs bounds on the solution on this time scale The previous proof does not work! G[ζ]ψ ∼ |D|ψ + order 0 (symbolic analysis) G[ζ]ψ ∼ µ∇((1 + εζ)∇ψ) + O(µ2) (shallow water expansion) Symbolic analysis and shallow water expansions are not compatible Justification OK away from wave breaking and shoreline

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 28 / 30

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SLIDE 75

Open problems in coastal oceanography

Numerically, we can handle Shoreline Wavebreaking

David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 29 / 30

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SLIDE 76

Open problems in coastal oceanography David Lannes (IMB) Mod´ elisation math´ ematique des vagues Valenciennes, 10/09/2015 30 / 30