on the existence of global existence of certain uid models ... · on the existence of global...

On the existence of global existence of certain fluid models

Alexandru D. Ionescu

() 23 mai 2013 1 / 26

The Gravity Water Wave System in 2D

The evolution of a perfect fluid (with constant density equal to 1) withvelocity v and pressure p :

(vt + v · ∇v) = −∇p − gen x ∈ Ωt

∇ · v = 0 x ∈ Ωt

v(0, x) = v0(x) x ∈ Ω0 ,

where g = 1 is the gravitational constant. The free surface St := ∂Ωt

moves with the normal component of the velocity, and, in absence ofsurface tension, the pressure vanishes on the boundary :

∂t + v · ∇ is tangent to⋃

t St ⊂ Rn+1

p(t, x) = 0 , x ∈ St .(1)

() 23 mai 2013 2 / 26

In the case of irrotational flows, i.e. curl v = 0 one can reduce this to asystem on the boundary. Assume that Ωt ⊂ R2 is the region below thegraph of a function h : Rt × Rx → R and let Φ denote the velocitypotential : ∇Φ(t, x , y) = v(t, x , y), for (x , y) ∈ Ωt . Ifφ(t, x) := Φ(t, x , h(x , t)) is the restriction of Φ to the boundary St , theequations of motion reduce to the following system for the unknownsh, φ : Rt × Rx → R :

∂th = G (h)φ

∂tφ = −h − 12 |φx |

2 + 12(1+|hx |2)

(G (h)φ+ hxφx)2

G (h) :=

√1 + |hx |2N(h)

where N(h) is the Dirichlet-Neumann operator associated to Ωt .

() 23 mai 2013 3 / 26

Possible variants : 3D problem, periodic conditions, surface tension, finitebottom, nontrivial vorticity, two-fluid models.

Local wellposedness : Nalimov (1974), Yosihara (1982), Craig (1985),Wu (1997, 1999), Beyer–Gunther (1998), Christodoulou–Lindblad (2000),Ambrose–Masmoudi (2005), Lannes (2005), Lindblad (2005),Coutand–Shkoller (2007), Alazard–Burq–Zuily (2011), Shatah–Zeng(2012).

Global regularity : (”small” global solutions) 3D gravity water-wave (Wu,Germain–Masmoudi–Shatah), 3D capillary waves (surface tension, nogravity, Germain–Masmoudi–Shatah).

Blow-up solutions : ”splash” singularity(Castro–Cordoba–Fefferman–Gancedo–Gomez-Serrano).

Semiglobal (scattering) solutions : Amick–Kirchgassner (1989),Ming–Rousset–Tzvetkov (2013).

() 23 mai 2013 4 / 26

Almost global solutions 2D gravity water-waves : Wu (2009)

Energy (positive) conservation law : with Λ := |∂x |1/2,

E0(h, φ) :=1

∫φG (h)φ dx +

∫h2 dx ≈ ‖h + iΛφ‖2

Main Theorem (I.–Pusateri, 2012, 2013) Let N0 = 104,

N1 = N0/2 + 4, Λ := |∂x |1/2, and assume that

‖h0 + iΛφ0‖HN0+2 + ‖x∂x (h0 + iΛφ0)‖HN0/2+1 + ‖h0 + iΛφ0‖Z ≤ ε0.

(Global existence) Then the gravity WW initial-value problem admits aunique global solution with

(1 + t)−p0‖h(t) + iφx(t)‖HN0+ (1 + t)−p0‖S(h(t) + iφx(t))‖HN0/2

+ ‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t)‖WN1+4 . ε0,

where S := (1/2)t∂t + x∂x and p0 = 10−4.

() 23 mai 2013 5 / 26

E0(h, φ) :=1

∫φG (h)φ dx +

∫h2 dx ≈ ‖h + iΛφ‖2

+ ‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t)‖WN1+4 . ε0,

where S := (1/2)t∂t + x∂x and p0 = 10−4.

() 23 mai 2013 5 / 26

E0(h, φ) :=1

∫φG (h)φ dx +

∫h2 dx ≈ ‖h + iΛφ‖2

+ ‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t)‖WN1+4 . ε0,

where S := (1/2)t∂t + x∂x and p0 = 10−4.

() 23 mai 2013 5 / 26

(Modified scattering) Let u(t) := h(t) + iΛφ(t) and

G (ξ, t) :=|ξ|4

0|u(ξ, s)|2 ds

s + 1.

Then there is p1 > 0 such that

(1+t1)p1

∥∥∥(1+|ξ|)N1[e iG(ξ,t2)e it2Λ(ξ)u(ξ, t2)−e iG(ξ,t1)e it1Λ(ξ)u(ξ, t1)

]∥∥∥L2ξ

. ε0,

for any t1 ≤ t2 ∈ [0,T ]. In particular, there is w∞ ∈ L2 with such that

supt∈[0,∞)

(1 + t)p1

∥∥∥(1 + |ξ|)N1e iG(ξ,t)e itΛ(ξ)u(ξ, t)− w∞(ξ)∥∥∥L2ξ

. ε0.

Independent recent proof of a similar result by Alazard–Delort.

() 23 mai 2013 6 / 26

(Modified scattering) Let u(t) := h(t) + iΛφ(t) and

G (ξ, t) :=|ξ|4

0|u(ξ, s)|2 ds

s + 1.

Then there is p1 > 0 such that

(1+t1)p1

∥∥∥(1+|ξ|)N1[e iG(ξ,t2)e it2Λ(ξ)u(ξ, t2)−e iG(ξ,t1)e it1Λ(ξ)u(ξ, t1)

]∥∥∥L2ξ

. ε0,

for any t1 ≤ t2 ∈ [0,T ]. In particular, there is w∞ ∈ L2 with such that

supt∈[0,∞)

(1 + t)p1

∥∥∥(1 + |ξ|)N1e iG(ξ,t)e itΛ(ξ)u(ξ, t)− w∞(ξ)∥∥∥L2ξ

. ε0.

Independent recent proof of a similar result by Alazard–Delort.

() 23 mai 2013 6 / 26

Main ideas : The energy method (to recover the high Sobolev norm) +The Fourier transform method (to recover non-L2 norms).

We use both the Eulerian formulation and the Lagrangian formulation :

zt(t, α) = v(t, z(t, α)), z(0, α) = α + ih(α).

This satisfies the nonlinear system

ztt + i = iazα, z t = Hzz t ,

where a = −|zα|−1∂P/∂n is the Raileigh–Taylor coefficient and

(Hz)(t, α) :=1

f (t, β)

z(t, α)− z(t, β)zβ(t, β) dβ.

Wu’s modified Lagrangian variables : We use Wu’s change ofcoordinates k to obtain cubic equations amenable to energy estimates :

ζ := z k−1, u := zt k−1, w := ztt k−1,

where, with Kz = <Hz ,

k(t, α) := z(t, α) +1

2(I +Hz)(I +Kz)−1(z(t, α)− z(t, α)).

.() 23 mai 2013 7 / 26

(Hz)(t, α) :=1

f (t, β)

ζ := z k−1, u := zt k−1, w := ztt k−1,

k(t, α) := z(t, α) +1

2(I +Hz)(I +Kz)−1(z(t, α)− z(t, α)).

.() 23 mai 2013 7 / 26

(Hz)(t, α) :=1

f (t, β)

ζ := z k−1, u := zt k−1, w := ztt k−1,

k(t, α) := z(t, α) +1

2(I +Hz)(I +Kz)−1(z(t, α)− z(t, α)).

.() 23 mai 2013 7 / 26

Main bootstrap : letting

L(t, α) := (ζα(t, α)− 1, u(t, α),w(t, α),=ζ(t, α)) ,

we assume that, on some time interval [0,T ],

(1 + t)−p0‖(h(t), φx(t))‖XN0

+ ‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t))‖WN1+4,∞ ≤ ε1,

(1 + t)−p0‖L(t)‖XN0+ ‖L(t)‖HN1+5 +

√1 + t‖L(t)‖WN1,∞ ≤ ε1,

and‖kα(t)− 1‖WN0/2+3,∞ ≤ ε1 ,

where‖f (t)‖Xk

:= ‖f (t)‖Hk + ‖Sf (t)‖Hk/2 .

() 23 mai 2013 8 / 26

Proposition 1. (Control on the diffeomorphism k) We have

supt∈[0,T ]

‖kα(t)− 1‖WN0/2+3,∞ . ε0 + ε21.

It relies on a transport equation with a special null structure for thetransformation k :

(I −Hζ)(kt k−1) = [u,Hζ ]ζα − 1

Proposition 2. (Control on the high energy norm) We have

supt∈[0,T ]

(1 + t)−p0‖L(t)‖XN0. ε0 + ε2

It relies on Wu’s energy and vector-field estimates for the ”cubic”equations for the ”good unknowns”.

() 23 mai 2013 9 / 26

Proposition 1. (Control on the diffeomorphism k) We have

supt∈[0,T ]

‖kα(t)− 1‖WN0/2+3,∞ . ε0 + ε21.

It relies on a transport equation with a special null structure for thetransformation k :

(I −Hζ)(kt k−1) = [u,Hζ ]ζα − 1

Proposition 2. (Control on the high energy norm) We have

supt∈[0,T ]

(1 + t)−p0‖L(t)‖XN0. ε0 + ε2

It relies on Wu’s energy and vector-field estimates for the ”cubic”equations for the ”good unknowns”.

() 23 mai 2013 9 / 26

Proposition 3. (Elliptic transitions) For any t ∈ [0,T ] we have

‖(h(t), φx(t))‖XN0. ‖L(t)‖XN0

+ ε21,

‖L(t)‖HN1+5 . ‖h(t) + iΛφ(t)‖HN1+10 + ε21,

‖L(t)‖WN1,∞ . ‖h(t) + iΛφ(t))‖WN1+4,∞ + ε21.

Proposition 4. (Control of the Eulerian variables) For any t ∈ [0,T ] wehave

‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t)‖WN1+4,∞ . ε0 + ε21.

This is the main new ingredient.

() 23 mai 2013 10 / 26

Proposition 3. (Elliptic transitions) For any t ∈ [0,T ] we have

‖(h(t), φx(t))‖XN0. ‖L(t)‖XN0

+ ε21,

‖L(t)‖HN1+5 . ‖h(t) + iΛφ(t)‖HN1+10 + ε21,

‖L(t)‖WN1,∞ . ‖h(t) + iΛφ(t))‖WN1+4,∞ + ε21.

Proposition 4. (Control of the Eulerian variables) For any t ∈ [0,T ] wehave

‖h(t) + iΛφ(t)‖HN1+10 +√

1 + t‖h(t) + iΛφ(t)‖WN1+4,∞ . ε0 + ε21.

This is the main new ingredient.

() 23 mai 2013 10 / 26

Step 1. In Eulerian variables, the equations are, schematically, in the form

∂th = |∂x |φ− ∂x(h∂xφ)− |∂x |(h|∂x |φ) + Cubic(h, φx);

∂tφ = −h − (1/2)|φx |2 + (1/2)||∂x |φ|2 + Cubic(h, φx),

where the remainders are cubic expression of h, φx . We use a Shatahnormal form transformation H = h + A(h, h), Ψ = φ+ B(h, φ), forsuitable bilinear operators A and B to eliminate the quadraticnonlinearities and reduce this to an evolution equation of a complexvariable with a cubic nonlinearity. More precisely, letting V = H + iΛΨ, weshow that V satisfies an equation of the form

∂tV + iΛV = C (V , V ), Λ = |∂x |1/2,

where C is a nonlocal cubic quasilinear nonlinearity depending on allpossible combination of V and V , and some of their derivatives.

() 23 mai 2013 11 / 26

Step 2. As a model, consider the (semilinear) equation

∂tu + iΛu = −ic0u2u + c1u

3 + c2uu2 + c3u

where c0 ∈ R and c1, c2, c3 ∈ C. Let f (t) = e itΛu(t). The a prioriassumption gives, in this model,

(1 + t)−2p0[‖f (t)‖HN0−4 + ‖x∂x f (t)‖HN0/2−2

]. ε1,

and the Duhamel formula gives

(∂t f )(ξ, t) =−ic0

e it[Λ(ξ)−Λ(ξ−η)−Λ(η−σ)+Λ(σ)]

f (ξ − η, t)f (η − σ, t)f (σ, t) dηdσ + . . . .

By ”stationary phase” (η = 0, σ = −ξ), the main contribution in theintegral is

8π|ξ|3/2|f (ξ, t)|2

t + 1f (ξ, t).

() 23 mai 2013 12 / 26

H(ξ, t) :=2c0

π|ξ|3/2

0|f (ξ, s)|2 ds

s + 1, g(ξ, t) := e iH(ξ,t)f (ξ, t).

(∂t g)(ξ, t) =−ic0

4π2e iH(ξ,t)

[ ∫R2

f (ξ − η, t)f (η − σ, t)f (σ, t) dηdσ − 8π|ξ|3/2|f (ξ, t)|2

t + 1f (ξ, t)

]+ . . . .

We define, with β := 1/100,

‖h‖Z := ‖h(ξ)(|ξ|β + |ξ|N1+15)‖L∞ .

The identity above can be used to propagate control of the Z -norm of falong the flow (and g). Moreover, the Z -norm provides sharp L∞ decay,

‖e itΛh‖L∞ . (1+|t|)−1/2‖ |ξ|3/4h(ξ)‖L∞ξ +(1+|t|)−5/8[‖x · ∂xh‖L2+‖h‖H2

() 23 mai 2013 13 / 26

H(ξ, t) :=2c0

π|ξ|3/2

0|f (ξ, s)|2 ds

s + 1, g(ξ, t) := e iH(ξ,t)f (ξ, t).

(∂t g)(ξ, t) =−ic0

4π2e iH(ξ,t)

[ ∫R2

f (ξ − η, t)f (η − σ, t)f (σ, t) dηdσ − 8π|ξ|3/2|f (ξ, t)|2

t + 1f (ξ, t)

]+ . . . .

We define, with β := 1/100,

‖h‖Z := ‖h(ξ)(|ξ|β + |ξ|N1+15)‖L∞ .

The identity above can be used to propagate control of the Z -norm of falong the flow (and g). Moreover, the Z -norm provides sharp L∞ decay,

‖e itΛh‖L∞ . (1+|t|)−1/2‖ |ξ|3/4h(ξ)‖L∞ξ +(1+|t|)−5/8[‖x · ∂xh‖L2+‖h‖H2

() 23 mai 2013 13 / 26

The Euler–Maxwell two-fluid modelTwo compressible ion and electron fluids interact with their ownself-consistent electromagnetic field. The Euler-Maxwell system describesthe dynamical evolution of the functions ne , ni : R3 → R,ve , vi ,E ,B : R3 → R3, which evolve according to the quasi-linear coupledsystem,

∂tne + div(neve) = 0,

neme [∂tve + ve · ∇ve ] +∇pe = −nee[E +

vec× B

∂tni + div(nivi ) = 0,

niMi [∂tvi + vi · ∇vi ] +∇pi = Znie[E +

vic× B

∂tB + c∇× E = 0,

∂tE − c∇× B = 4πe [neve − Znivi ] ,

together with the elliptic equations

div(B) = 0, div(E ) = 4πe(Zni − ne)

and two equations of state expressing pe = pe(ne) and pi = pi (ni ).() 23 mai 2013 14 / 26

These equations describe a plasma composed of electrons and one speciesof ions. The electrons have charge −e, density ne , mass me , velocity ve ,and pressure pe , and the ions have charge Ze, density ni , mass Mi ,velocity vi , and pressure pi . In addition, c denotes the speed of light and Eand B denote the electric and magnetic field. The two elliptic equationsare propagated by the dynamic flow, provided that we assume that theyare satisfied at the initial time.

At the linear level, there are ion-acoustic waves, Langmuir waves, as wellas light waves. At the nonlinear level, the Euler-Maxwell system is theorigin of many well-known dispersive PDE, such as KdV, KP, Zakharov,Zakharov-Kuznetsov, and NLS, which can be derived from via differentscaling and asymptotic expansions. One can also derive the Euler–Poissonmodel, the cold-ion model, and quasi-neutral equations.

Variants : 2D model Euler–Maxwell, relativistic versions, periodicsolutions.

() 23 mai 2013 15 / 26

Main question : Are there any smooth nontrivial global solutions of theEuler–Maxwell system ? This is a system of nonlinear hyperbolic laws withno dissipation and no relaxation effects.

Constant solutions : (ne , ve , ni , vi ,E ,B) = (n0, 0, n0/Z , 0, 0, 0).

Adimensionalization : Assume quadratic pressure laws pe = Pen2e ,

pi = Pin2i . After rescaling, the system becomes becomes

∂tn + div((n + 1)v) = 0,

ε (∂tv + v · ∇v) + T∇n + E + v × B = 0,

∂tρ+ div((ρ+ 1)u) = 0,

(∂tu + u · ∇u) +∇ρ− E − u × B = 0,

∂tB +∇× E = 0,

∂t E −Cb

ε∇× B = [(n + 1)v − (ρ+ 1)u] ,

div(B) = 0, div(E ) = ρ− n,

where ε, T and Cb are parameters that satisfy

ε ≤ 10−3, T ∈ [1, 100], Cb ≥ 6T .() 23 mai 2013 16 / 26

∂tn + div((n + 1)v) = 0,

ε (∂tv + v · ∇v) + T∇n + E + v × B = 0,

∂tρ+ div((ρ+ 1)u) = 0,

(∂tu + u · ∇u) +∇ρ− E − u × B = 0,

∂tB +∇× E = 0,

∂t E −Cb

ε∇× B = [(n + 1)v − (ρ+ 1)u] ,

div(B) = 0, div(E ) = ρ− n,

ε ≤ 10−3, T ∈ [1, 100], Cb ≥ 6T .() 23 mai 2013 16 / 26

∂tn + div((n + 1)v) = 0,

ε (∂tv + v · ∇v) + T∇n + E + v × B = 0,

∂tρ+ div((ρ+ 1)u) = 0,

(∂tu + u · ∇u) +∇ρ− E − u × B = 0,

∂tB +∇× E = 0,

∂t E −Cb

ε∇× B = [(n + 1)v − (ρ+ 1)u] ,

div(B) = 0, div(E ) = ρ− n,

ε ≤ 10−3, T ∈ [1, 100], Cb ≥ 6T .() 23 mai 2013 16 / 26

Positive conserved energy :

[T |n|2 + ε(1 + n)|v |2 + |ρ|2 + (ρ+ 1)|u|2 + |E |2 +

ε|B|2

Main Theorem (Guo–I.–Pausader, 2012, 2013) Let N0 = 104 andassume that

‖(n0, v0, ρ0, u0, E 0, B0)‖HN0 + ‖(n0, v0, ρ0, u0, E 0, B0)‖Z ≤ δ0

div(E 0) + n0 − ρ0 = 0, B0 = ε∇× v0 = −∇× u0.

Then there exists a unique global solution(n, v , ρ, u, E , B) ∈ C ([0,∞) : HN0) of the Euler–Maxwell system withinitial data (n0, v0, ρ0, u0, E 0, B0). Moreover, for any t ∈ [0,∞),

div(E )(t) + n(t)− ρ(t) = 0, B(t) = ε∇× v(t) = −∇× u(t),

(generalized irrotationality) and, with β := 1/100 andf (t) ∈ (n(t), v(t), ρ(t), u(t), E (t), B(t)

‖f (t)‖HN0 + sup|α|≤4

(1 + t)1+β/2‖Dαx f (t)‖L∞ . δ0.

() 23 mai 2013 17 / 26

Positive conserved energy :

[T |n|2 + ε(1 + n)|v |2 + |ρ|2 + (ρ+ 1)|u|2 + |E |2 +

ε|B|2

Main Theorem (Guo–I.–Pausader, 2012, 2013) Let N0 = 104 andassume that

‖(n0, v0, ρ0, u0, E 0, B0)‖HN0 + ‖(n0, v0, ρ0, u0, E 0, B0)‖Z ≤ δ0

div(E 0) + n0 − ρ0 = 0, B0 = ε∇× v0 = −∇× u0.

Then there exists a unique global solution(n, v , ρ, u, E , B) ∈ C ([0,∞) : HN0) of the Euler–Maxwell system withinitial data (n0, v0, ρ0, u0, E 0, B0). Moreover, for any t ∈ [0,∞),

div(E )(t) + n(t)− ρ(t) = 0, B(t) = ε∇× v(t) = −∇× u(t),

(generalized irrotationality) and, with β := 1/100 andf (t) ∈ (n(t), v(t), ρ(t), u(t), E (t), B(t)

‖f (t)‖HN0 + sup|α|≤4

(1 + t)1+β/2‖Dαx f (t)‖L∞ . δ0.

() 23 mai 2013 17 / 26

Previous results :

Blow-up solutions for small irrotational initial data for the purecompressible Euler equation (Sideris 1985).

The Euler-Poisson model for the electrons :

∂tn + div((1 + n)v) = 0,

∂tv + v · ∇v +∇n = ∇φ,∆φ = n.

Here the magnetic field vanishes B ≡ 0, and the ions are treated asmotionless with a constant density.

Guo (1998) proved global stability of equilibrium solutions in the case ofinitial data (n0, v0) which are small, smooth, neutral and irrotational,∫

n0(x)dx = 0, ∇× v0 ≡ 0.

In this case the system can be reduced to a quasilinear Klein-Gordonequation with quadratic nonlinearity.

() 23 mai 2013 18 / 26

Previous results :

Blow-up solutions for small irrotational initial data for the purecompressible Euler equation (Sideris 1985).

The Euler-Poisson model for the electrons :

∂tn + div((1 + n)v) = 0,

∂tv + v · ∇v +∇n = ∇φ,∆φ = n.

Here the magnetic field vanishes B ≡ 0, and the ions are treated asmotionless with a constant density.

Guo (1998) proved global stability of equilibrium solutions in the case ofinitial data (n0, v0) which are small, smooth, neutral and irrotational,∫

n0(x)dx = 0, ∇× v0 ≡ 0.

In this case the system can be reduced to a quasilinear Klein-Gordonequation with quadratic nonlinearity.

() 23 mai 2013 18 / 26

The Euler-Poisson equation for the ions

∂tρ+ div((1 + ρ)u) = 0,

∂tu + u · ∇u +∇ρ = −∇φ,−∆φ = ρ− φ.

Here the electron dynamics with constant temperature is decoupled fromthe ion dynamics via the Boltzmann relation. The model equation is(

∂tt −∆ + (−∆)(1−∆)−1)α = |∇|Q(α,∇α).

This has intermediate behavior between wave and Klein–Gordon equations.There are strong degeneracies near the zero frequency, where thedispersion relation is similar to the wave dispersion up to third order.Guo–Pausader proved global regularity in the case of small, neutral, andirrotational perturbations by using a variation on the normal form method.A key property is the fact that the nonlinearity is an exact derivative (nullstructure), which helps compensate for the degeneracy at the 0 frequency.

() 23 mai 2013 19 / 26

The Euler-Maxwell equation for electrons :

∂tn + div((1 + n)v) = 0,

∂tv + v · ∇v +∇n = − [E + v × B] ,

∂tB +∇× E = 0,

∂tE − C∇× B = (1 + n)v

with constraints div(B) = 0 and div(E ) = n.

In the irrotational case this can reduced to a system of 2 quasilinearKlein–Gordon equations, with different speeds :

(−∂2t +c2

1 ∆+1)u = Q1(∇2u,∇2v), (−∂2t +c2

2 ∆+1)u = Q2(∇2u,∇2v).

The vector-field method does not apply in this case, and there are largesets of resonances. Germain (semilinear problem) and Germain–Masmoudi(quasilinear problem) proved global existence (under generic conditions onthe parameters c1, c2 and special structure on the quasilinear terms), withweak L∞ decay like t−1/2, using the spacetime resonances method.I.–Pausader proved a robust theorem, with strong t−1−β decay, for allparameters.

() 23 mai 2013 20 / 26

The Euler-Maxwell equation for electrons :

∂tn + div((1 + n)v) = 0,

∂tv + v · ∇v +∇n = − [E + v × B] ,

∂tB +∇× E = 0,

∂tE − C∇× B = (1 + n)v

with constraints div(B) = 0 and div(E ) = n.

In the irrotational case this can reduced to a system of 2 quasilinearKlein–Gordon equations, with different speeds :

(−∂2t +c2

1 ∆+1)u = Q1(∇2u,∇2v), (−∂2t +c2

2 ∆+1)u = Q2(∇2u,∇2v).

The vector-field method does not apply in this case, and there are largesets of resonances. Germain (semilinear problem) and Germain–Masmoudi(quasilinear problem) proved global existence (under generic conditions onthe parameters c1, c2 and special structure on the quasilinear terms), withweak L∞ decay like t−1/2, using the spacetime resonances method.I.–Pausader proved a robust theorem, with strong t−1−β decay, for allparameters.

() 23 mai 2013 20 / 26

Main idea in our case : The energy method (to recover the high Sobolevnorm) + The Fourier transform method (to recover non-L2 norms).

Energy estimate (on the physical variables) :

EN(t ′)− EN(t) .∫ t′

tA(s)EN(s) ds,

A(s) :=∑

f ∈n,v ,ρ,u,B,E

‖∇f (s)‖L∞ + ‖u‖L∞ + ‖v‖L∞ + ‖B‖L∞ .

Dispersive model : first order pseudo-differential equations

(∂t + iΛi )Ui = Ni , (∂t + iΛe)Ue = Ne , (∂t + iΛb)Ub = Nb.

() 23 mai 2013 21 / 26

EN(t ′)− EN(t) .∫ t′

tA(s)EN(s) ds,

A(s) :=∑

f ∈n,v ,ρ,u,B,E

‖∇f (s)‖L∞ + ‖u‖L∞ + ‖v‖L∞ + ‖B‖L∞ .

() 23 mai 2013 21 / 26

EN(t ′)− EN(t) .∫ t′

tA(s)EN(s) ds,

A(s) :=∑

f ∈n,v ,ρ,u,B,E

‖∇f (s)‖L∞ + ‖u‖L∞ + ‖v‖L∞ + ‖B‖L∞ .

() 23 mai 2013 21 / 26

Λi (ξ) := ε−1/2

√√√√(1 + ε) + (T + ε)|ξ|2 −√

((1− ε) + (T − ε)|ξ|2)2 + 4ε

Λe(ξ) := ε−1/2

√√√√(1 + ε) + (T + ε)|ξ|2 +√

((1− ε) + (T − ε)|ξ|2)2 + 4ε

Λb(ξ) := ε−1/2√

1 + ε+ Cb|ξ|2.

Model phases

Λi (ξ) ≈ |ξ|

√2 + |ξ|21 + |ξ|2

Λe(ξ) ≈ C√

1 + A|ξ|2,

Λb(ξ) ≈ C√

1 + B|ξ|2,

where C 1, A,B ∈ [1,∞), B ≥ 2A.() 23 mai 2013 22 / 26

Let Vσ(t) = e itΛσUσ(t) and use Duhamel’s formula

Vσ(ξ, t)− Vσ(ξ, 0) = c∑µ,ν

e is[Λσ(ξ)±Λµ(ξ−η)±Λν(η)]

×mσ;µ,ν(ξ, η)Vµ(ξ − η, s)Vν(η, s) dηds,

for σ ∈ i , e, b.

We need a space Z such that

T : Z ∩ HN0 × Z ∩ HN0 → Z .

and‖e itΛσ f ‖L∞ . (1 + t)−1−β‖f ‖Z .

() 23 mai 2013 23 / 26

Let Vσ(t) = e itΛσUσ(t) and use Duhamel’s formula

Vσ(ξ, t)− Vσ(ξ, 0) = c∑µ,ν

e is[Λσ(ξ)±Λµ(ξ−η)±Λν(η)]

×mσ;µ,ν(ξ, η)Vµ(ξ − η, s)Vν(η, s) dηds,

for σ ∈ i , e, b.

We need a space Z such that

T : Z ∩ HN0 × Z ∩ HN0 → Z .

and‖e itΛσ f ‖L∞ . (1 + t)−1−β‖f ‖Z .

() 23 mai 2013 23 / 26

Critical points (spacetime resonances) :

(ξ, η) : Φσ;µ,ν(ξ, η) = 0 and ∇ηΦσ;µ,ν(ξ, η) = 0.

In our case, all possible types of resonances

(ξ, η) = (rω,Rω), (ξ, η) = (rω, 0), (ξ, η) = (0,Rω),

where ω ∈ S2.

We need to study the contributions of sub-level sets

(ξ, η) : |Φσ;µ,ν(ξ, η)| ≤ δ1 and |∇ηΦσ;µ,ν(ξ, η)| ≤ δ2.

We use localization and L2orthogonality arguments to bound thesecontributions.

() 23 mai 2013 24 / 26

Critical points (spacetime resonances) :

(ξ, η) : Φσ;µ,ν(ξ, η) = 0 and ∇ηΦσ;µ,ν(ξ, η) = 0.

In our case, all possible types of resonances

(ξ, η) = (rω,Rω), (ξ, η) = (rω, 0), (ξ, η) = (0,Rω),

where ω ∈ S2.

We need to study the contributions of sub-level sets

(ξ, η) : |Φσ;µ,ν(ξ, η)| ≤ δ1 and |∇ηΦσ;µ,ν(ξ, η)| ≤ δ2.

We use localization and L2orthogonality arguments to bound thesecontributions.

() 23 mai 2013 24 / 26

The Z -norm : We define

Z := f ∈ L2(R3) : ‖f ‖Z := sup(k,j)‖φj(x) · Pk f (x)‖Bk,j

Here k := min(k , 0), k+ := max(k , 0),

β := 1/100, α := β/2, γ := 3/2− 4β,

‖g‖Bk,j:= inf

g=g1+g2

[‖g1‖B1

k,j+ ‖g2‖B2

‖h‖B1k,j

:= (2αk + 210k)[2(1+β)j‖h‖L2 + 2(1/2−β)k‖h‖L∞

‖h‖B2k,j

:= 220|k|[2(1−β)j‖h‖L2 + ‖h‖L∞

+ 2γj supR∈[2−j ,2k ], ξ0∈R3

R−2‖h‖L1(B(ξ0,R))

() 23 mai 2013 25 / 26

The Z -norm : We define

Z := f ∈ L2(R3) : ‖f ‖Z := sup(k,j)‖φj(x) · Pk f (x)‖Bk,j

Here k := min(k , 0), k+ := max(k , 0),

β := 1/100, α := β/2, γ := 3/2− 4β,

‖g‖Bk,j:= inf

g=g1+g2

[‖g1‖B1

k,j+ ‖g2‖B2

‖h‖B1k,j

:= (2αk + 210k)[2(1+β)j‖h‖L2 + 2(1/2−β)k‖h‖L∞

‖h‖B2k,j

:= 220|k|[2(1−β)j‖h‖L2 + ‖h‖L∞

+ 2γj supR∈[2−j ,2k ], ξ0∈R3

R−2‖h‖L1(B(ξ0,R))

() 23 mai 2013 25 / 26

Extensions : The Euler–Maxwell global regularity result extends to

• general smooth barotropic pressure laws pe = pe(ne), pi = pi (ni ) ;

• other models in 3D : the Euler–Poisson 2-fluid system, the 1-fluidrelativistic Euler–Maxwell system, the 2-fluid relativistic Euler–Maxwellsystem, which have Galilean/Lorentz symmetries.

Natural open problems :

• global solutions in 2-fluid models in 2D (partial result by I.–Pausader forthe Euler–Poisson electron model) ;

• ”stable” water-wave models : surface tension + gravity in 2D or 3D,two-fluid models in 2D or 3D ;

• global solutions with nontrivial vorticity. Or, at least, conditional resultssaying that solutions stay smooth as long as the vorticity is controlled.

• global solutions in periodic settings.

() 23 mai 2013 26 / 26

Extensions : The Euler–Maxwell global regularity result extends to

• general smooth barotropic pressure laws pe = pe(ne), pi = pi (ni ) ;

• other models in 3D : the Euler–Poisson 2-fluid system, the 1-fluidrelativistic Euler–Maxwell system, the 2-fluid relativistic Euler–Maxwellsystem, which have Galilean/Lorentz symmetries.

Natural open problems :

• global solutions in 2-fluid models in 2D (partial result by I.–Pausader forthe Euler–Poisson electron model) ;

• ”stable” water-wave models : surface tension + gravity in 2D or 3D,two-fluid models in 2D or 3D ;

• global solutions with nontrivial vorticity. Or, at least, conditional resultssaying that solutions stay smooth as long as the vorticity is controlled.

• global solutions in periodic settings.

() 23 mai 2013 26 / 26

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