Asymptotic regimes for the Euler–Korteweg system
S. Benzoni-Gavage1
D. Chiron2
P. Noble3
L.M. Rodrigues1
Institut Camille Jordan
Université Claude Bernard Lyon 1
2 Laboratoire J.A. Dieudonné
Université Nice Sophia Antipolis
3 Institut
de Mathématiques de Toulouse
INSA Toulouse
1 / 25
EK at a glance
Outline
1
EK at a glance
2
Long waves asymptotics
3
Modulated wave trains
2 / 25
EK at a glance
Euler equations
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
0.
−∇ · (K (ρ)∇ρ ⊗ ∇ρ) .
With ρ = density, u = velocity,
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
∇(ρ∇ · (K (ρ)∇ρ))
+ 12 (K (ρ) − ρK 0 (ρ))|∇ρ|2 ) + 12 (K (ρ) −
−∇ · (K (ρ)∇ρ ⊗ ∇ρ) .
With ρ = density, u = velocity, K (ρ) = capillarity,
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
ρ∇(K (ρ)∆ρ + 12 K 0 (ρ)|∇ρ|2 ) .
+ 21 (K (ρ) − ρK 0 (ρ))|∇ρ|2 )
With ρ = density, u = velocity, K (ρ) = capillarity,
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
ρK ∇∆ρ .
+ 21 (K (ρ) − ρK 0 (ρ))|∇ρ|2 )
With ρ = density, u = velocity, K = constant capillarity,
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
ρ∇
√ ∆ ρ
√
2 ρ .
−∇ · (K (ρ)∇ρ ⊗ ∇ρ) .
With ρ = density, u = velocity,
1
= quantum capillarity,
4ρ
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
∇(ρ∇ · (K (ρ)∇ρ))
+ 12 (K (ρ) − ρK 0 (ρ))|∇ρ|2 )
−∇ · (K (ρ)∇ρ ⊗ ∇ρ) . −∇ · (K (ρ)∇ρ ⊗
With ρ = density, u = velocity, K (ρ) = capillarity,
p(ρ) = pressure.
3 / 25
EK at a glance
Euler equations with capillarity
Conservation of mass
∂t ρ + ∇ · (ρu) = 0 .
Conservation of momentum
∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p(ρ) =
∇(ρ∇ · (K (ρ)∇ρ))
+ 12 (K (ρ) − ρK 0 (ρ))|∇ρ|2 )
−∇ · (K (ρ)∇ρ ⊗ ∇ρ) . −∇ · (K (ρ)∇ρ ⊗
With ρ = density, u = velocity, K (ρ) = capillarity,
p(ρ) = pressure = ρF 0 (ρ) − F (ρ).
Conservation of energy
∂t
2
1
2 ρ|u|
+ 12 K (ρ)|∇ρ|2 + F (ρ) + ∇ · 12 ρ|u|2 + F (ρ) + p(ρ) u =
∇ · ρK (ρ)∆ρ + 21 ρK 0 (ρ)|∇ρ|2 u − (∇ · (ρu))K (ρ)∇ρ .
3 / 25
EK at a glance
Dispersive fluids
4 / 25
EK at a glance
Dispersive fluids
4 / 25
EK at a glance
Dispersive fluids
4 / 25
EK at a glance
Dispersive fluids
4 / 25
EK at a glance
Dispersive fluids
4 / 25
EK at a glance
Dispersive fluids
c Wan et al
Nature Physics 2007
4 / 25
EK at a glance
What’s known on general EK
Local well-posedness for initial data
s+1 (Rd ) × H s (Rd )d , s > 1 + d/2
(ρ0 , u0 ) ∈ (ρ
− , u) + H
[SBG–Danchin–Descombes’07]
+ dispersive smoothing ‘à la Kato’ [Audiard’12]
5 / 25
EK at a glance
What’s known on general EK
Local well-posedness for initial data
s+1 (Rd ) × H s (Rd )d , s > 1 + d/2
(ρ0 , u0 ) ∈ (ρ
− , u) + H
[SBG–Danchin–Descombes’07]
+ dispersive smoothing ‘à la Kato’ [Audiard’12]
Existence of global, travelling wave solutions
planar ones (kinks, solitons, periodic) (ρ, u) = (ρ, u)(x · n − σt)
merely follow from phase portrait analysis ;
5 / 25
EK at a glance
What’s known on general EK
Local well-posedness for initial data
s+1 (Rd ) × H s (Rd )d , s > 1 + d/2
(ρ0 , u0 ) ∈ (ρ
− , u) + H
[SBG–Danchin–Descombes’07]
+ dispersive smoothing ‘à la Kato’ [Audiard’12]
Existence of global, travelling wave solutions
planar ones (kinks, solitons, periodic) (ρ, u) = (ρ, u)(x · n − σt)
merely follow from phase portrait analysis ;
Conditional, orbital stability of travelling wave solutions :
all kinks, 1D solitary waves under Boussinesq/Grillakis–Shatah–Strauss
stability criterion [SBG–Danchin–Descombes–Jamet’05]
5 / 25
EK at a glance
What’s known on ‘general’ EK
Local well-posedness for initial data
s+1 (Rd ) × H s (Rd )d , s > 1 + d/2
(ρ0 , u0 ) ∈ (ρ
− , u) + H
[SBG–Danchin–Descombes’07]
+ dispersive smoothing ‘à la Kato’ [Audiard’12]
Existence of global, travelling wave solutions
planar ones (kinks, solitons, periodic) (ρ, u) = (ρ, u)(x · n − σt)
merely follow from phase portrait analysis ;
more general ones (Roberts’ programme for NLS)
(ρ, u) = (ρ, u)(x · n − σt, x − (x · n)n) by variational methods
[Béthuel–Saut’99], ... [Béthuel–Gravejat–Saut’09], [Mariş’13],
[Chiron-Mariş’12-14].
Conditional, orbital stability of travelling wave solutions :
all kinks, 1D solitary waves under Boussinesq/Grillakis–Shatah–Strauss
stability criterion [SBG–Danchin–Descombes–Jamet’05]
[Béthuel–Gravejat–Saut–Smets’08] [Chiron’13]
soliton-like ‘bubbles’ [Chiron-Mariş’12-14]
5 / 25
Long waves asymptotics
Outline
1
EK at a glance
2
Long waves asymptotics
3
Modulated wave trains
6 / 25
Long waves asymptotics
Linear regime
Nonlinear system
∂t ρ + u · ∇ρ + ρ (∇ · u) = 0 ,
∂t u + (u · ∇)u + ∇(g (ρ)) = ∇(K (ρ)∆ρ + 12 K 0 (ρ)|∇ρ|2 ) ,
with g := F 0 .
7 / 25
Long waves asymptotics
Linear regime
Nonlinear system
∂t ρ + u · ∇ρ + ρ (∇ · u) = 0 ,
∂t u + (u · ∇)u + ∇(g (ρ)) = ∇(K (ρ)∆ρ + 12 K 0 (ρ)|∇ρ|2 ) ,
with g := F 0 .
Linearized system
Formal limit when (ρ, u) → (%, 0), ρ̂ := ρ − %,
∂t ρ̂ + % (∇ · u) = 0 ,
∂t u + g 0 (%)∇ρ̂ = 0 .
7 / 25
Long waves asymptotics
Linear regime
Nonlinear system
∂t ρ + u · ∇ρ + ρ (∇ · u) = 0 ,
∂t u + (u · ∇)u + ∇(g (ρ)) = ∇(K (ρ)∆ρ + 12 K 0 (ρ)|∇ρ|2 ) ,
with g := F 0 .
Linearized system
Formal limit when (ρ, u) → (%, 0), ρ̂ := ρ − %,
∂t ρ̂ + % (∇ · u) = 0 ,
∂t u + g 0 (%)∇ρ̂ = 0 .
p
Well-posed if g 0 (%) > 0. Sound speed c := %g 0 (%).
7 / 25
Long waves asymptotics
Linear regime
Nonlinear system
∂t ρ + u · ∇ρ + ρ (∇ · u) = 0 ,
∂t u + (u · ∇)u + ∇(g (ρ)) = ∇(K (ρ)∆ρ + 12 K 0 (ρ)|∇ρ|2 ) ,
with g := F 0 .
Linearized system
Formal limit when (ρ, u) → (%, 0), ρ̂ := ρ − %,
∂t ρ̂ + % (∇ · u) = 0 ,
∂t u + g 0 (%)∇ρ̂ = 0 .
p
Well-posed if g 0 (%) > 0. Sound speed c := %g 0 (%).
Fully justified limit for GP [Béthuel–Danchin–Smets’10].
7 / 25
Long waves asymptotics
Toward weakly nonlinear regimes
0 < ε 1, 0 < η 1.
8 / 25
Long waves asymptotics
Toward weakly nonlinear regimes
0 < ε 1, 0 < η 1.
(ρ, u)(x, t) = (%, 0) + η (ρ̂, û)(X, T ),
X := εx, T := εt.
8 / 25
Long waves asymptotics
Toward weakly nonlinear regimes
0 < ε 1, 0 < η 1.
(ρ, u)(x, t) = (%, 0) + η (ρ̂, û)(X, T ),
X := εx, T := εt.
Rescaled, nonlinear system
∂T ρ̂ + η û · ∇X ρ̂ + (% + η ρ̂) (∇X · û) = 0 ,
∂t û + η (û · ∇X )û + g 0 (% + η ρ̂)∇X ρ̂
= ε2 ∇X (K (% + η ρ̂)∆X ρ̂)
+ η2 K 0 (% + η ρ̂)|∇X ρ|2 ) .
8 / 25
Long waves asymptotics
1D heuristics
Linearized system
∂T ρ̂ + % ∂X û
= 0,
∂T û + g 0 (%)∂X ρ̂ = 0 .
9 / 25
Long waves asymptotics
1D heuristics
Linearized system
∂T ρ̂ + % ∂X û
= 0,
∂T û + g 0 (%)∂X ρ̂ = 0 .
2w := ρ̂ + c% û
⇐⇒
2v := ρ̂ − c% û
∂T w + c ∂X w
∂T v − c ∂X v
= 0,
= 0.
9 / 25
Long waves asymptotics
1D heuristics
Linearized system
∂T ρ̂ + % ∂X û
= 0,
∂T û + g 0 (%)∂X ρ̂ = 0 .
2w := ρ̂ + c% û
⇐⇒
2v := ρ̂ − c% û
∂T w + c ∂X w
∂T v − c ∂X v
= 0,
= 0.
Nonlinear system
∂T ρ̂ + η û ∂X ρ̂ + (% + η ρ̂) ∂X û
= 0,
∂T û + η û ∂X û + g 0 (% + η ρ̂)∂X ρ̂ = ε2 ∂X (K (% + η ρ̂)∂X2 ρ̂)
+ η2 K 0 (% + η ρ̂)(∂X ρ̂)2 ) .
9 / 25
Long waves asymptotics
1D heuristics
Linearized system
∂T ρ̂ + % ∂X û
= 0,
∂T û + g 0 (%)∂X ρ̂ = 0 .
2w := ρ̂ + c% û
⇐⇒
2v := ρ̂ − c% û
∂T w + c ∂X w
∂T v − c ∂X v
= 0,
= 0.
Nonlinear system
∂T ρ̂ + η û ∂X ρ̂ + (% + η ρ̂) ∂X û
= 0,
∂T û + η û ∂X û + g 0 (% + η ρ̂)∂X ρ̂ = ε2 ∂X (K (% + η ρ̂)∂X2 ρ̂)
+ η2 K 0 (% + η ρ̂)(∂X ρ̂)2 ) .
⇐⇒
∂T w + c ∂X w + η Γ w ∂X w − ε2 κ ∂X3 w
∂T v − c ∂X v − η Γ v ∂X v + ε2 κ ∂X3 v
= ε2 κ ∂X3 v + i.t + l.o.t ,
= −ε2 κ ∂X3 w + i.t + l.o.t .
9 / 25
Long waves asymptotics
1D heuristics
Linearized system
∂T ρ̂ + % ∂X û
= 0,
∂T û + g 0 (%)∂X ρ̂ = 0 .
2w := ρ̂ + c% û
⇐⇒
2v := ρ̂ − c% û
∂T w + c ∂X w
∂T v − c ∂X v
= 0,
= 0.
Nonlinear system
∂T ρ̂ + η û ∂X ρ̂ + (% + η ρ̂) ∂X û
= 0,
∂T û + η û ∂X û + g 0 (% + η ρ̂)∂X ρ̂ = ε2 ∂X (K (% + η ρ̂)∂X2 ρ̂)
+ η2 K 0 (% + η ρ̂)(∂X ρ̂)2 ) .
⇐⇒
∂T w + c ∂X w + η Γ w ∂X w − ε2 κ ∂X3 w
∂T v − c ∂X v − η Γ v ∂X v + ε2 κ ∂X3 v
Γ :=
3c
%g 00 (%)
+
,
2%
2c
= ε2 κ ∂X3 v + i.t + l.o.t ,
= −ε2 κ ∂X3 w + i.t + l.o.t .
κ :=
%
K (%),
2c
9 / 25
Long waves asymptotics
1D result
Theorem ([SBG–Chiron’14])
If s > 5 and T 7→ (ρ̂, û)(T ) ∈ H s+1 × H s solves the rescaled EK,
10 / 25
Long waves asymptotics
1D result
Theorem ([SBG–Chiron’14])
If s > 5 and T 7→ (ρ̂, û)(T ) ∈ H s+1 × H s solves the rescaled EK, define
w := (ρ̂ + c% û)/2, v := (ρ̂ − c% û)/2, solve in H s
∂θ W + ΓW ∂X W = 0 , W (0) = w (0) ,
∂θ V − ΓV ∂X V = 0 , V (0) = v (0) ,
10 / 25
Long waves asymptotics
1D result
Theorem ([SBG–Chiron’14])
If s > 5 and T 7→ (ρ̂, û)(T ) ∈ H s+1 × H s solves the rescaled EK, define
w := (ρ̂ + c% û)/2, v := (ρ̂ − c% û)/2, solve in H s
∂θ W + ΓW ∂X W = 0 , W (0) = w (0) ,
∂θ V − ΓV ∂X V = 0 , V (0) = v (0) ,
Then there exist Θ and C , depending only on s,
R b
k(ρ̂, û)(0)kH s + εkρ̂(0)kH s+1 , and supa,b∈R a (ρ̂, û)(X , 0)dX 10 / 25
Long waves asymptotics
1D result
Theorem ([SBG–Chiron’14])
If s > 5 and T 7→ (ρ̂, û)(T ) ∈ H s+1 × H s solves the rescaled EK, define
w := (ρ̂ + c% û)/2, v := (ρ̂ − c% û)/2, solve in H s
∂θ W + ΓW ∂X W = 0 , W (0) = w (0) ,
∂θ V − ΓV ∂X V = 0 , V (0) = v (0) ,
Then there exist Θ and C , depending only on s,
R b
k(ρ̂, û)(0)kH s + εkρ̂(0)kH s+1 , and supa,b∈R a (ρ̂, û)(X , 0)dX so that for 0 6 T 6 Θ/η, 0 6 σ 6 s − 5,
kw (T ) − W (ηT , · − cT )kH σ + kv (T ) − V (ηT , · + cT )kH σ 6
4
2
C η + εη + εη ,
10 / 25
Long waves asymptotics
1D result
Theorem ([SBG–Chiron’14])
If s > 5 and T 7→ (ρ̂, û)(T ) ∈ H s+1 × H s solves the rescaled EK, define
w := (ρ̂ + c% û)/2, v := (ρ̂ − c% û)/2, solve in H s
∂θ W + ΓW ∂X W = 0 , W (0) = w (0) ,
∂θ V − ΓV ∂X V = 0 , V (0) = v (0) ,
2
∂θ W + Γ W ∂X W = εη κ ∂X3 W , W (0) = w (0) ,
2
∂θ V − Γ V ∂X V = − εη κ ∂X3 V , V (0) = v (0) .
Then there exist Θ and C , depending only on s,
R b
k(ρ̂, û)(0)kH s + εkρ̂(0)kH s+1 , and supa,b∈R a (ρ̂, û)(X , 0)dX so that for 0 6 T 6 Θ/η, 0 6 σ 6 s − 5,
kw (T ) − W (ηT , · − cT )kH σ + kv (T ) − V (ηT , · + cT )kH σ 6
4
2
C η + εη + εη ,
kW (T ) − W (ηT , · − cT )kH σ + kv (T ) − V (ηT , · + cT )kH σ 6
4
C η + ε2 + εη .
10 / 25
Long waves asymptotics
Various regimes
Parameters’
range
Error bound
for Burgers
Error bound
for KdV
Time of
validity
ε≈η
η
η
1/ε2
ε2 η ε
ε2 /η
η
1/ε2
ε2 ≈ η
1
η
1/ε3
11 / 25
Long waves asymptotics
Various regimes
Parameters’
range
Error bound
for Burgers
Error bound
for KdV
Time of
validity
ε≈η
η
η
1/ε2
ε2 η ε
ε2 /η
η
1/ε2
ε2 ≈ η
1
η
1/ε3
Extends [Béthuel–Gravejat–Saut–Smets’10], [Chiron’14].
11 / 25
Long waves asymptotics
Crucial ingredient
For any s > 1 +
d
, Λ > 0, if (ρ̂0 , û0 ) ∈ Bε (Λ),
2
Bε (Λ) := {(ρ̂, û) ∈ H s+1 (Rd ) × (H s (Rd ))d ; k(ρ̂, û)kH s + εkρ̂kH s+1 6 Λ},
there exists Θ > 0, depending only on Λ, s and d, such that the maximal
solution to rescaled EK such that (ρ̂, û)(0) = (ρ̂0 , û0 ) exists at least on
[0, Θ/η], and (ρ̂, û)(T ) ∈ Bε (2Λ) for all T ∈ [0, Θ/η].
12 / 25
Long waves asymptotics
Proof based on extended system...
∂T ρ̂ + η û · ∇X ρ̂ + (% + η ρ̂) (∇X · û) = 0 ,
∂t û + η (û · ∇X )û + g 0 (% + η ρ̂)∇X ρ̂
= ε2 ∇X (K (% + η ρ̂)∆X ρ̂)
+ η2 K 0 (% + η ρ̂)|∇X ρ|2 ) .
13 / 25
Long waves asymptotics
Proof based on extended system...
∂T ρ̂ + η û · ∇X ρ̂ + (% + η ρ̂) (∇X · û) = 0 ,
∂t û + η (û · ∇X )û + g 0 (% + η ρ̂)∇X ρ̂
= ε2 ∇X (K (% + η ρ̂)∆X ρ̂)
+ η2 K 0 (% + η ρ̂)|∇X ρ|2 ) .
q
K (%+η ρ̂)
m ẑ := û + i ŵ, ŵ := ε
%+η ρ̂ ∇X ρ̂
13 / 25
Long waves asymptotics
Proof based on extended system...
∂T ρ̂ + η û · ∇X ρ̂ + (% + η ρ̂) (∇X · û) = 0 ,
∂t û + η (û · ∇X )û + g 0 (% + η ρ̂)∇X ρ̂
= ε2 ∇X (K (% + η ρ̂)∆X ρ̂)
+ η2 K 0 (% + η ρ̂)|∇X ρ|2 ) .
q
K (%+η ρ̂)
m ẑ := û + i ŵ, ŵ := ε
%+η ρ̂ ∇X ρ̂
∂T ρ̂ +η û · ∇X ρ̂
+(% + η ρ̂) (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(% + η ρ̂)ŵ
+iε∇X (a(% + η ρ̂)∇X · ẑ) = 0 .
13 / 25
Long waves asymptotics
Proof based on extended system...
∂T ρ̂ + η û · ∇X ρ̂ + (% + η ρ̂) (∇X · û) = 0 ,
∂t û + η (û · ∇X )û + g 0 (% + η ρ̂)∇X ρ̂
= ε2 ∇X (K (% + η ρ̂)∆X ρ̂)
+ η2 K 0 (% + η ρ̂)|∇X ρ|2 ) .
q
K (%+η ρ̂)
m ẑ := û + i ŵ, ŵ := ε
%+η ρ̂ ∇X ρ̂
∂T ρ̂ +η û · ∇X ρ̂
+(% + η ρ̂) (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(% + η ρ̂)ŵ
+iε∇X (a(% + η ρ̂)∇X · ẑ) = 0 .
a(ρ) :=
p
ρg 0 (ρ)
ρK (ρ), b(ρ) :=
.
a(ρ)
13 / 25
Long waves asymptotics
... and energy estimates
Physical energy
η2
E := d
2ε
Z
Rd
2
2
(% + η ρ̂)|ẑ| + 2 (F (% + η ρ̂) − F (%)) dX.
η
14 / 25
Long waves asymptotics
... and energy estimates
Physical energy
η2
E := d
2ε
Z
Rd
2
2
(% + η ρ̂)|ẑ| + 2 (F (% + η ρ̂) − F (%)) dX.
η
Mathematical energy
E0 (ρ̂, ẑ) :=
1
2
Z
(% + η ρ̂)|ẑ|2 + g 0 (% + η ρ̂)ρ̂2 dX.
Rd
14 / 25
Long waves asymptotics
... and energy estimates
Physical energy
η2
E := d
2ε
Z
Rd
2
2
(% + η ρ̂)|ẑ| + 2 (F (% + η ρ̂) − F (%)) dX.
η
Mathematical energy
E0 (ρ̂, ẑ) :=
1
2
Z
(% + η ρ̂)|ẑ|2 + g 0 (% + η ρ̂)ρ̂2 dX.
Rd
E0 (ρ̂, ẑ) ≈ kûk2L2 + kρ̂k2L2 + ε2 kρ̂k2H 1 .
14 / 25
Long waves asymptotics
... and energy estimates
Physical energy
η2
E := d
2ε
Z
Rd
2
2
(% + η ρ̂)|ẑ| + 2 (F (% + η ρ̂) − F (%)) dX.
η
Mathematical energy
E0 (ρ̂, ẑ) :=
1
2
Z
(% + η ρ̂)|ẑ|2 + g 0 (% + η ρ̂)ρ̂2 dX.
Rd
E0 (ρ̂, ẑ) ≈ kûk2L2 + kρ̂k2L2 + ε2 kρ̂k2H 1 .
d
E0 (ρ̂, ẑ) . ηk(∇X ρ̂, ∇X û)kL∞ E0 (ρ̂, ẑ).
dT
14 / 25
Long waves asymptotics
Higher order estimates
Recall the extended system
∂T ρ̂ +η û · ∇X ρ̂
+ρ (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(ρ)ŵ + iε∇X (a(ρ)∇X · ẑ) = 0 .
15 / 25
Long waves asymptotics
Higher order estimates
Recall the extended system
∂T ρ̂ +η û · ∇X ρ̂
+ρ (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(ρ)ŵ + iε∇X (a(ρ)∇X · ẑ) = 0 .
Z
1
Energy E0 (ρ̂, ẑ) =
ρ|ẑ|2 + g 0 (ρ)ρ̂2 dX , a(ρ)b(ρ) = ρg 0 (ρ) .
2 Rd
Z
s
X
X
σ!
a(ρ)σ ρ|∂ α ẑ|2 + g 0 (ρ)(∂ α ρ̂)2 dX .
Es (ρ̂, ẑ) :=
2α! Rd
d
σ=0 α ∈ N ,
0
|α| = σ
15 / 25
Long waves asymptotics
Higher order estimates
Recall the extended system
∂T ρ̂ +η û · ∇X ρ̂
+ρ (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(ρ)ŵ + iε∇X (a(ρ)∇X · ẑ) = 0 .
Z
1
Energy E0 (ρ̂, ẑ) =
ρ|ẑ|2 + g 0 (ρ)ρ̂2 dX , a(ρ)b(ρ) = ρg 0 (ρ) .
2 Rd
Z
s
X
X
σ!
a(ρ)σ ρ|∂ α ẑ|2 + g 0 (ρ)(∂ α ρ̂)2 dX .
Es (ρ̂, ẑ) :=
2α! Rd
d
σ=0 α ∈ N ,
0
|α| = σ
Es (ρ̂, ẑ) ≈ (kûk2H s + kρ̂k2H s + ε2 kρ̂k2H s+1 ) .
15 / 25
Long waves asymptotics
Higher order estimates
Recall the extended system
∂T ρ̂ +η û · ∇X ρ̂
+ρ (∇X · û) = 0 ,
∂T ẑ +η(û · ∇X )ẑ +iη(∇X ẑ)ŵ + 1ε b(ρ)ŵ + iε∇X (a(ρ)∇X · ẑ) = 0 .
Z
1
Energy E0 (ρ̂, ẑ) =
ρ|ẑ|2 + g 0 (ρ)ρ̂2 dX , a(ρ)b(ρ) = ρg 0 (ρ) .
2 Rd
Z
s
X
X
σ!
a(ρ)σ ρ|∂ α ẑ|2 + g 0 (ρ)(∂ α ρ̂)2 dX .
Es (ρ̂, ẑ) :=
2α! Rd
d
σ=0 α ∈ N ,
0
|α| = σ
Es (ρ̂, ẑ) ≈ (kûk2H s + kρ̂k2H s + ε2 kρ̂k2H s+1 ) .
d
Es (ρ̂, ẑ) . η k(∇X ρ̂, ∇X û)kL∞ +εkDX2 ρ̂kL∞ (1+ηεk∇X ρ̂kL∞ )Es (ρ̂, ẑ).
dT
15 / 25
Modulated wave trains
Outline
1
EK at a glance
2
Long waves asymptotics
3
Modulated wave trains
16 / 25
Modulated wave trains
Principles of modulation
Evolution equation ∂t U = ∂x N [U]
with a N-parameter family of periodic
waves (N > 2).
17 / 25
Modulated wave trains
Principles of modulation
Evolution equation ∂t U = ∂x N [U]
with a N-parameter family of periodic
waves (N > 2).
‘Two-timing’ method [Whitham’70]
Look for
U(t, x) = U0 (|{z}
εt , |{z}
εx , φ(εt, εx)/ε) +
| {z }
T
X
θ
ε U1 (εt, εx, φ(εt, εx)/ε, ε) + o(ε) ,
with U0 and U1 one-periodic in θ.
17 / 25
Modulated wave trains
Principles of modulation
Evolution equation ∂t U = ∂x N [U]
with a N-parameter family of periodic
waves (N > 2).
‘Two-timing’ method [Whitham’70]
Look for
U(t, x) = U0 (|{z}
εt , |{z}
εx , φ(εt, εx)/ε) +
| {z }
T
X
θ
ε U1 (εt, εx, φ(εt, εx)/ε, ε) + o(ε) ,
with U0 and U1 one-periodic in θ.
Exact examples are travelling waves, of spatial period 1/k and speed σ,
U(t, x) = U(k(x − σt)), for which U0 (T , X , θ) = U(θ),
φ(T , X ) = k(X − σT ).
17 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws
∂t U = ∂x N [U]
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
∂t Q(U) = ∂x S [U] ⇒ ∂T hQ(U0 )i = ∂X hS [U0 ]i.
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
∂t Q(U) = ∂x S [U] ⇒ ∂T hQ(U0 )i = ∂X hS [U0 ]i.
Remark 1 Does not mean that asymptotic expansion is justified.
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
∂t Q(U) = ∂x S [U] ⇒ ∂T hQ(U0 )i = ∂X hS [U0 ]i.
Remark 1 Does not mean that asymptotic expansion is justified.
[Doelman–Sandstede–Scheel–Schneider’09],
[Johnson–Noble–Rodrigues–Zumbrun’14], [Düll–Schneider’09].
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
∂t Q(U) = ∂x S [U] ⇒ ∂T hQ(U0 )i = ∂X hS [U0 ]i.
Remark 2 ‘the relation of the stability of the periodic wave with the type
of the [modulated equations is] given in the previous papers’ in
[Whitham’70].
18 / 25
Modulated wave trains
Principles of modulation, cont’d
Conservation of waves
k := φX , ω := φT , σ := −ω/k
⇒
∂T k + ∂X (σk) = 0.
Other conservation laws Using that ∂t = ε∂T + ω∂θ , ∂x = ε∂X + k∂θ ,
∂t U = ∂x N [U]
formally
=⇒
ω∂θ U0 = k∂θ N [U0 ] ,
∂T U0 = ∂X N [U0 ] + k∂θ (dN [U0 ]U1 ) ,
⇓
(averaging)
∂T hU0 i = ∂X hN [U0 ]i .
∂t Q(U) = ∂x S [U] ⇒ ∂T hQ(U0 )i = ∂X hS [U0 ]i.
Remark 2 ‘the relation of the stability of the periodic wave with the type
of the [modulated equations is] given in the previous papers’ in
[Whitham’70].
[Oh–Zumbrun’03], [Serre’05], [Johnson–Zumbrun–Bronski’10],
[Noble–Rodrigues’10-11], [SBG–Noble–Rodrigues’13].
18 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
− σ∂x −
ρ + ∂x (ρ
− u) = 0
2
− σ∂x u + u∂x u + ∂x (g (ρ
ρ + 21 K 0 (ρ
ρ )2 )
− )) = ∂x (K (ρ
− )∂x −
− )(∂x −
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
2
− σ∂x u + u∂x u + ∂x (g (ρ
ρ + 21 K 0 (ρ
ρ )2 )
− )) = ∂x (K (ρ
− )∂x −
− )(∂x −
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
2
2
− 12 j 2 /ρ
− g (ρ
ρ + 12 K 0 (ρ
ρ )2 =: λ
− ) + K (ρ
− )∂x −
− )(∂x −
−
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
1 2
−
2 j /ρ
1
− F (ρ
ρ )2 − λρ
− ) + 2 K (ρ
− )(∂x −
− =: µ
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
1 2
−
2 j /ρ
1
− F (ρ
ρ )2 − λρ
− ) + 2 K (ρ
− )(∂x −
− =: µ
of the form
1 2
2 v̇
+ W (v ; j, λ) = µ
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
1 2
−
2 j /ρ
1
− F (ρ
ρ )2 − λρ
− ) + 2 K (ρ
− )(∂x −
− =: µ
of the form
1 2
2 v̇
+ W (v ; j, λ) = µ
19 / 25
Modulated wave trains
Periodic waves of EK
(ρ, u) = (ρ
− , u)(x − σt) solves
∂t ρ + ∂x (ρu) = 0
∂t u + u∂x u + ∂x (g (ρ)) = ∂x (K (ρ)∂x2 ρ + 12 K 0 (ρ)(∂x ρ)2 )
iff
ρ (u − σ) =: j
−
1 2
−
2 j /ρ
1
− F (ρ
ρ )2 − λρ
− ) + 2 K (ρ
− )(∂x −
− =: µ
of the form
1 2
2 v̇
+ W (v ; j, λ) = µ
19 / 25
Modulated wave trains
Modulated EK
∂T k + ∂X (σk) = 0 ,
∂T hρi + ∂X hρui = 0 ,
∂T hui + ∂X h 21 u 2 + g (ρ) − k 2 (K (ρ)∂θ2 ρ + 12 K 0 (ρ)(∂θ ρ)2 )i = 0 ,
∂T hρui + ∂X hρu 2 + p(ρ)i
− ∂X hk 2 (ρK (ρ)∂θ2 ρ − 12 (K (ρ) − ρK 0 (ρ))(∂θ ρ)2 )i = 0 .
20 / 25
Modulated wave trains
Modulated EK
∂T k + ∂X (σk) = 0 ,
∂T hρi + ∂X hρui = 0 ,
∂T hui + ∂X h 21 u 2 + g (ρ) − k 2 (K (ρ)∂θ2 ρ + 12 K 0 (ρ)(∂θ ρ)2 )i = 0 ,
∂T hρui + ∂X hρu 2 + p(ρ)i
− ∂X hk 2 (ρK (ρ)∂θ2 ρ − 12 (K (ρ) − ρK 0 (ρ))(∂θ ρ)2 )i = 0 .
Evolutionary ?
20 / 25
Modulated wave trains
Modulated EK
∂T k + ∂X (σk) = 0 ,
∂T hρi + ∂X hρui = 0 ,
∂T hui + ∂X h 21 u 2 + g (ρ) − k 2 (K (ρ)∂θ2 ρ + 12 K 0 (ρ)(∂θ ρ)2 )i = 0 ,
∂T hρui + ∂X hρu 2 + p(ρ)i
− ∂X hk 2 (ρK (ρ)∂θ2 ρ − 12 (K (ρ) − ρK 0 (ρ))(∂θ ρ)2 )i = 0 .
Evolutionary ?
Hyperbolic ?
20 / 25
Modulated wave trains
Modulated EK
∂T k + ∂X (σk) = 0 ,
∂T hρi + ∂X hρui = 0 ,
∂T hui + ∂X h 21 u 2 + g (ρ) − k 2 (K (ρ)∂θ2 ρ + 12 K 0 (ρ)(∂θ ρ)2 )i = 0 ,
∂T hρui + ∂X hρu 2 + p(ρ)i
− ∂X hk 2 (ρK (ρ)∂θ2 ρ − 12 (K (ρ) − ρK 0 (ρ))(∂θ ρ)2 )i = 0 .
Evolutionary ?
Hyperbolic ?
Nature of characteristic fields ? ?
20 / 25
Modulated wave trains
Action
Travelling waves ODE
1
ρ )2
− )(∂x −
2 K (ρ
− 21 −
ρ u 2 − F (ρ
− ) + σρ
− u − λρ
− + ju = µ
21 / 25
Modulated wave trains
Action
Travelling waves ODE
1
ρ )2 − 21 −
ρ u 2 − F (ρ
− )(∂x −
− ) + σρ
− u − λρ
− + ju = µ
2 K (ρ
⇒ h 21 K (ρ
ρ )2 + 12 −
ρ u 2 + F (ρ
− ) − σρ
− u + λρ
− − ju + µi
− )(∂x −
= hK (ρ
ρ )2 i
− )(∂x −
21 / 25
Modulated wave trains
Action
Travelling waves ODE
1
ρ )2 − 21 −
ρ u 2 − F (ρ
− )(∂x −
− ) + σρ
− u − λρ
− + ju = µ
2 K (ρ
⇒ h 21 K (ρ
ρ )2 + 12 −
ρ u 2 + F (ρ
− ) − σρ
− u + λρ
− − ju + µi
− )(∂x −
= hK (ρ
ρ )2 i
− )(∂x −
Definition of the action
A := k1 h 12 K (ρ
ρ )2 + 12 −
ρ u 2 + F (ρ
− )(∂x −
− ) − σρ
− u + λρ
− − ju + µi
21 / 25
Modulated wave trains
Action
Travelling waves ODE
1
ρ )2 − 21 −
ρ u 2 − F (ρ
− )(∂x −
− ) + σρ
− u − λρ
− + ju = µ
2 K (ρ
⇒ h 21 K (ρ
ρ )2 + 12 −
ρ u 2 + F (ρ
− ) − σρ
− u + λρ
− − ju + µi
− )(∂x −
= hK (ρ
ρ )2 i
− )(∂x −
Definition of the action
A := k1 h 12 K (ρ
ρ )2 + 12 −
ρ u 2 + F (ρ
− )(∂x −
− ) − σρ
− u + λρ
− − ju + µi
Proposition
When A is viewed as a function of (µ, λ, j, σ), its partial derivatives are
1
1
A µ = k1 , A λ = k1 hρ
− i , A j = − k hui , A σ = − k hρ
− ui .
21 / 25
Modulated wave trains
Action and modulation
Modulated equations
equivalently read


∂T A µ + σ ∂X A µ − A µ ∂X σ = 0 ,






 ∂T A λ + σ ∂X A λ + A µ ∂X j = 0 ,


∂T A j + σ ∂X A j + A µ ∂X λ = 0 ,





 ∂ A
σ + σ ∂X A σ − A µ ∂X µ = 0 .
T
22 / 25
Modulated wave trains
Action and modulation
Modulated equations
equivalently read


∂T A µ + σ ∂X A µ − A µ ∂X σ = 0 ,






 ∂T A λ + σ ∂X A λ + A µ ∂X j = 0 ,


∂T A j + σ ∂X A j + A µ ∂X λ = 0 ,





 ∂ A
σ + σ ∂X A σ − A µ ∂X µ = 0 .
T
Quasilinear form (∂T + σ ∂X ) Σ W + A µ S ∂X W = 0 ,



µ
0 0



λ 
 0 0
Σ := d2 A , W := 
 j  , S :=  0 1
σ
−1 0

0 −1
1 0 
.
0 0 
0 0
22 / 25
Modulated wave trains
Action and modulation
Modulated equations
equivalently read


∂T A µ + σ ∂X A µ − A µ ∂X σ = 0 ,






 ∂T A λ + σ ∂X A λ + A µ ∂X j = 0 ,


∂T A j + σ ∂X A j + A µ ∂X λ = 0 ,





 ∂ A
σ + σ ∂X A σ − A µ ∂X µ = 0 .
T
Quasilinear form (∂T + σ ∂X ) Σ W + A µ S ∂X W = 0 ,



µ
0 0



λ 
 0 0
Σ := d2 A , W := 
 j  , S :=  0 1
σ
−1 0

0 −1
1 0 
.
0 0 
0 0
Evolutionarity and hyperbolicity fully encoded by d A , nature of
2
characteristic fields depends also on
d3 A
.
22 / 25
Modulated wave trains
Action and stability of periodic waves
Failure of hyperbolicity of modulated equations (modulational
instability) ⇒ side-band instability of underlying periodic wave.
23 / 25
Modulated wave trains
Action and stability of periodic waves
Failure of hyperbolicity of modulated equations (modulational
instability) ⇒ side-band instability of underlying periodic wave.
Action encodes (through d2 A ) a necessary condition (det d2 A ≥ 0),
for the stability of underlying periodic wave with respect to
co-periodic perturbations.
23 / 25
Modulated wave trains
Action and stability of periodic waves
Failure of hyperbolicity of modulated equations (modulational
instability) ⇒ side-band instability of underlying periodic wave.
Action encodes (through d2 A ) a necessary condition (det d2 A ≥ 0),
and also a sufficient condition (...)
for the stability of underlying periodic wave with respect to
co-periodic perturbations.
23 / 25
Modulated wave trains
Action and stability of periodic waves
Failure of hyperbolicity of modulated equations (modulational
instability) ⇒ side-band instability of underlying periodic wave.
Action encodes (through d2 A ) a necessary condition (det d2 A ≥ 0),
and also a sufficient condition (...)
for the stability of underlying periodic wave with respect to
co-periodic perturbations.
similar to [Johnson’09], [Bronski–Johnson’10] for KdV.
23 / 25
Modulated wave trains
Dispersive shocks
c Jacques Dassié
[Wan–Jia–Fleischer’Nature Physics’07]
[El’Chaos’05]
[El–Grimshaw–Kamchatnov’JFM’07]
[El–Grimshaw–Smyth’Phys. Fluids’06]
[Hoefer–Ablowitz–Engels’PRL’08]
[Hoefer’arXiv’13].
24 / 25
Modulated wave trains
Dispersive shocks
c Jacques Dassié
[Wan–Jia–Fleischer’Nature Physics’07]
[El’Chaos’05]
[El–Grimshaw–Kamchatnov’JFM’07]
[El–Grimshaw–Smyth’Phys. Fluids’06]
[Hoefer–Ablowitz–Engels’PRL’08]
[Hoefer’arXiv’13].
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Modulated wave trains
Further reading
S. Benzoni-Gavage and D. Chiron
Long wave asymptotics for the Euler–Korteweg system.
Forthcoming.
S. Benzoni-Gavage, P. Noble, and L.M. Rodrigues Slow modulations of
periodic waves in Hamiltonian PDEs, with application to capillary fluids,
J. Nonlinear Sci. 2014
S. Benzoni-Gavage, P. Noble, and L.M. Rodrigues, Stability of periodic waves
in Hamiltonian PDEs,
GDR Analyse des EDP 2013.
S. Benzoni-Gavage, C. Mietka and L. M. Rodrigues.
Co-periodic stability of periodic waves in some Hamiltonian PDEs.
Forthcoming.
25 / 25