Global well-posedness of the quasi-linear Euler-Korteweg system for - - PowerPoint PPT Presentation

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Global well-posedness of the quasi-linear Euler-Korteweg system for small irrotational data

Corentin Audiard (LJLL, Paris 6) and Boris Haspot (Ceremade, Paris Dauphine)

1 Presentation of the results 2 Idea of the Proof 3 Perspectives Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Let us recall the compressible Navier-Stokes equations: Mass equation : ∂tρ + divρu = 0, Momentum equation : ∂t(ρu) + div(ρu ⊗ u) + ∇P(ρ) = divK, , Initial data : (ρ, u)/t=0 = (ρ0, u0). Here u = u(t, x) ∈ RN stands for the velocity field, ρ = ρ(t, x) ∈ R+ is the density, P(ρ) the pressure (in the sequel we will only consider P(ρ) = ρ2 ) and the general Korteweg tensor reads as follows: divK = div

• ρκ(ρ)∆ρ + 1

2 (κ(ρ) + ρκ

′(ρ))|∇ρ|2

Id − κ(ρ)∇ρ ⊗ ∇ρ

• .

(1) Here κ is the capillary coefficient and is regular far away of 0 (we can think to κ(ρ) = ρα with α ∈ R)

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Dispersive Equations: Following an idea of F. Coquel, setting: a(ρ) =

• ρ κ(ρ), ω = −
• κ(ρ)

ρ ∇ρ and z = u + iω we get the following extended formulation:

• ∂tρ + div(ρu) = 0,

∂tz − i∇(a(ρ)divu) + u · z + i∇z · ω + ∇f(ρ) = 0. with f′(ρ) = P ′(ρ)

ρ

. We are in presence of a quasilinear degenerate Schr¨

• dinger

equation which is degenerate because we have only ∇(a(ρ)divz) and not div(a(ρ)∇z). Remark It is generally delicate to work with such equation since it requires non trapping conditions on the bicharacteristics, Kato smoothing effects. Due to the Mizohata condition we also have to work in weight space in order to ensure ”energy” estimates. When κ(ρ) = κ1

ρ

and with irrotationnal velocity we have ∇(a(ρ)divz) = √κ1∆z. In particular we have in this case a semilinear Schr¨

• dinger equation. This case

corresponds to the so called quantum pressure.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

The Madelung transform When the velocity u = ∇θ is irrotational, the Madelung transform ψ = √ρe

i

θ 2√κ1 allows formally to rewrite the Euler

Korteweg system as the Gross-Pitaevski equation (GP):

• 2i√κ1∂tψ = −2κ1∆ψ + (|ψ|2 − 1)ψ,

ψ(0, ·) = ψ0. (2) with the boundary condition lim|x|→+∞ ψ = 1. The Gross-Pitaevskii equation is the Hamiltonian evolution associated to the Ginzburg-Landau energy: E(ψ)=

• RN

√κ1 2 |∇ψ(t, x)|2 + 1 4 (|ψ|2 − 1)2 dx =

• RN

√κ1 2 |∇ϕ(t, x)|2 + 1 4 (2Reϕ + |ϕ|2)2 dx. (3) with ψ = 1 + ϕ. Up to a change of variable we may take κ1 = 1 and we consider the equation on ϕ = ψ − 1:

• i∂tϕ + ∆ϕ − 2Reϕ = F(ϕ),

F(ϕ) = (ϕ + 2 ¯ ϕ + |ϕ|2)ϕ. (4) Remark The Gross-Pitaevskii equation is close from a defocusing cubic nonlinear Schr¨

• dinger equation, one of the main difference is that there exists traveling

waves.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Some results of ”weak-strong” solutions for Euler Korteweg

• S. Benzoni, R. Danchin and S. Descombes [03,04], Existence of strong

solution in finite time for (ρ − 1, u0) ∈ H

N 2 +2+ε × H N 2 +1+ε . They use an

energy method via the introduction of a suitable gauge.

• P. Antonelli and P. Marcati [09], Existence of global weak solution when

κ(ρ) = 1

ρ and u0 = ∇θ0 for N = 2, 3 (Madelung transformation).

• C. Audiard and BH [14], Existence of global strong solution for N = 3 with

small initial data u0 = ∇θ0 when κ(ρ) = 1

ρ .

The crucial part of the proof is relied to dispersive estimate in weight space. What about the Gross-Pitaevskii equation? N ≥ 2 [Gallo, G´ erard] , Existence of global strong solution in the energy space. N ≥ 2 [Bethuel, Chiron, Gravejat, Maris, Saut, Smets · · · ] Existence of traveling waves when N ≥ 2 via variational technics with 0 < |c| < √ 2 of finite energy, it means solutions of the form: uc(x1 − ct, x2, · · · , xN). N ≥ 3 [Gustafson,Nakanishi,Tsai] Scattering results for small initial data (Normal form, Space-time resonance).

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Strategy to get global strong solution for the Euler Korteweg system Let us look at the simpler case of the quantum pressure κ(ρ) = 1

ρ . It suffices to

solves the Gross-Pitaevskii equation and to verify that the solution ψ does not vanish in order to use the Madelung transform. Indeed in this case we can transfer the regularity of ψ on the density ρ and the velocity u = ∇θ . To do this (since ψ = 1 + ϕ) we shall prove that ϕL∞ remains small all along the time. Remark This is generally not right, indeed for the linear Schr¨

• dinger equation we can

choose arbitrary small L∞ initial data which blow up in L∞ norm for arbitrary small time. To prove such result, the natural tool is to use global dispersive Strichartz estimate (however in the case N = 3, it is not sufficient because we have quadratic nonlinearities such that their exponent corresponds to the Strauss exponent). Why it is difficult to deal with a quadratic nonlinearity? We have: ei(t−s)∆ϕ2(s, ·)L3 ϕ(s)2

L3

(t − s) is not in L1(0, t). In particular we shall use the theory of space-time resonance developed by S. Gustafson, K. Nakanishi , T-P. Tsai and also P. Germain, N. Masmoudi, J. Shatah.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Theorem (C. Audiard, BH) Let N = 3 and P ′(1) > 0, if u0 = ∇φ0 is irrotational, there exists δ > 0 and N0 large enough such that if u0HN0−1 + ρ0 − 1HN0 + xu0L2 + x(ρ0 − 1)L2 ≤ δ, then there exists a global strong solution for the Euler Korteweg system with ρ − 1L∞(R+×Rd) ≤ 1/2. Remark Let us mention that our solution scatters. We are going to combine space-time resonance theory and energy estimates, these type of technics have allowed to prove for the first time the existence of global strong solution with small initial data for the water wave equation (see P. Germain, N. Masmoudi and J Shatah). Remark We have similar results when N ≥ 4. The problem remains open when N ≥ 2. This case is much tricky since we are in a situation where the lower exponent of the nonlinearities is below the Strauss exponent. Furthermore for κ(ρ) = 1

ρ , it exists traveling waves for the Gross Pitaevskii of

arbitrary small energy when N = 2. It seems more complicated to exhibit dispersion properties.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Sketch of the Proof: Setting w =

• κ(ρ)

ρ ∇ρ, L the primitive of

• κ(ρ)

ρ

such that L(1) = 1, L = L(ρ), z = u + iw the Euler-Korteweg system rewrites ∂tl + u · ∇l + a(1 + l)divu = 0, ∂tz + u · ∇z + iw · ∇z + i∇(a(1 + l)divz) = g′

0(1 + l)w.

(5) with l = L − 1. In the potential case u = ∇φ, the system on φ, l then reads

• ∂tφ − ∆l + 2l = (a(1 + l) − 1)∆l − 1

2

• |∇φ|2 − |∇l|2

+ (2l − g(1 + l)), ∂tl + ∆φ = −∇φ · ∇l + (1 − a(1 + l))∆φ. (6) with g(1) = 0 since we look for integrable functions. Up to a change of variable we can assume that g′(1) = 2. The linear part precisely corresponds to the linear part of the Gross-Pitaevskii equation. Diagonalization of the system: In order to diagonalize the system we set U =

• −∆

2 − ∆ , H =

• −∆(2 − ∆), φ1 = Uφ, l1 = l.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

The equation writes in the new variables    ∂tφ1 + Hl1 = U

• (a(1 + l1) − 1)∆l1 − 1

2

• |∇U−1φ1|2 − |∇l1|2

+ (2l1 − g(1 + l1))

• ,

∂tl1 − Hφ1 = −∇U−1φ1 · ∇l1 − (1 − a(1 + l1))Hφ. (7) More compactly, if we set ψ = φ1 + il1, ψ0 = (Uφ + il)|t=0 , the Duhamel formula gives ψ(t) = eitHψ0 + t ei(t−s)HN(ψ(s))ds, (8) N(ψ) = U

• (a(1 + l1) − 1)∆l1 − 1

2

• |∇U−1φ1|2 − |∇l1|2

+ (2l1 − g(1 + l1))

• +i
• − ∇U−1φ1 · ∇l1 −
• 1 − a(1 + l1)
• Hφ
• .

(9) What are the difficulties for getting solution of (8)-(16)? How to deal with the quadratic terms? How to avoid the possible loss of derivatives?

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

In order to overcome these difficulties, we must combine space-time resonance theory and energy estimates. The idea to combine energy estimates and dispersion estimates is independently due to G. Ponce, S. Klainerman and J. Shatah for the quasilinear wave equation. Energy estimate: Proposition Under the following assumptions (∇φ0, l) ∈ H2n × H2n+1 L(x, t) = 1 + l(x, t) ≥ m > 0 for (x, t) ∈ Rd × [0, T], then for n > N/4 + 1/2, there exists C > 0 such that the solution of the Euler Korteweg system satisfies the following estimate ∇φH2n + lH2n+1 ≤

• ∇φ0H2n + l0H2n+1
• × exp
• C

t ∇φ(s)W 1,∞ + l(s)W 2,∞ds

• .

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Fixed point theorem By Duhamel formula we have: ψ(t) = eitHψ0 + t ei(t−s)HN(ψ(s))ds, (10) We recall that for 2 ≤ p ≤ +∞, 0 ≤ θ ≤ 1, s ∈ R, and σ = 1

2 − 1 p we have (S.

Gustafson et al): e−itHvBs

p,2 |t|−(N−θ)σU(N−2+3θ)σ∇2θσvBs p′,2,

(11) For some ε > 0, 1/p = 1/6 − ε, k sufficiently large and N0 sufficiently large compared to k we set

• ψXT

= ψL∞

T HN0 + xe−itHψL∞ T L2 + s1+3εψL∞ T W k,p,

ψYT = xe−itHψL∞

T L2 + s1+3εψL∞ T W k,p.

(12) We would like to prove a fixed point theorem in X∞ . How to prove the stability of the space X∞ We set Jψ = eitHxe−itHψ (a type of ”pseudo conform transform”). Let us apply J to the bilinear terms and we have in Fourier space some terms of the form: e−itH(ξ) t

• RN ∇ξ
• eis(
• H(ξ)±H(η)±H(ξ−η)
• B(η, ξ − η)

f(s, η) g(s, ξ − η)

• dηds,

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

where f, g can be either e−isHψ or eisHψ and B is the symbol of a Coifman-Meyer operator. Non space-time resonance When we derive in ξ the previous estimate, we have three terms and one is particularly delicate. It is: I = e−itH(ξ) t

• RN s∇ξΩ eisΩ(ξ)B(η, ξ − η)

f(s, η) g(s, ξ − η)dηds. Remark This last term is very hard to deal with since it appears a time power s, how to have enough decay in time to apply Strichartz estimate. The idea is to use non space-time resonance, more precisely we are going to use integration by parts in space and in time: eisΩ = ∇ηΩ is|∇ηΩ|2 · ∇ηeisΩ. and: eisΩ = 1 iΩ ∂seisΩ.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Remark Let us mention that this integration by parts allows us to absorb the term in s. Definition We say that the phase Ω is non time resonant if Ω = 0. We say that the phase Ω is non space resonant if ∇ηΩ = 0 Remark Ω is both space and time resonant when ξ = 0 which makes the analysis intricate. A first task will be to prove that the quadratic Duhamel integrals are controlled as follows IL∞

T (L2) ψ2

XT .

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

We have then different phases Ω with: Ω(ξ, η) =

• H(ξ) ± H(η) ± H(ξ − η)
• .

Let us consider the phase Ω1(ξ, η) = H(ξ) + H(η) + H(ξ − η) which has no

• cancellation. By integration by parts we have

I = e−itH(ξ) t

• RN s∇ξΩ1 eisΩ1(ξ)B(η, ξ − η)

f(s, η) g(s, ξ − η)dηds = −e−itH(ξ) t

• RN s ∇ξΩ1

iΩ1 eisΩ(ξ)B(η, ξ − η)∂s f(, η) g(s, ξ − η)dηds + · · · and from the identity ∂tf = −ie−itHN(ψ), we see that this procedure effectively replaces the quadratic nonlinearity by a cubic one, ie acts as a Normal Form. On the other hand, if N(ψ) = ψ2 the phase associated corresponds to Ω2(ξ, η) = H(ξ) − H(η) − H(ξ − η), which cancels on a large set. The remedy is to use an integration by part in the η variable using eisΩ =

∇ηΩ is|∇ηΩ|2 ∇η(eisΩ), it

does not improve the nonlinearity but of course there is a gain of decay 1/s, i.e it acts as a ”Klainerman vector fields.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

We define then the following set. Let us call T the time resonant set: T = {(ξ, η) : Ω(ξ, η) = 0} = {η ⊥ ξ − η}. (13) Let us call S the “space resonant set” as S = {(ξ, η) : ∇ηΩ(ξ, η) = 0} = {η = −ξ − η}. (14) as well as ”the space-time resonant set ” R = S ∩ T = {(ξ, η) : Ω(ξ, η) = 0, ∇ηΩ(ξ, η) = 0}. (15) It is thus essential to study where and at which order we have a cancellation of Ω±,±(ξ, η) = H(ξ) ± H(η) ± H(ξ − η) or ∇ηΩ±± . We note H′(ξ) =

2+2|ξ|2

√

2+|ξ|2 the

radial derivative of H and ∇H(ξ) = H′(ξ)ξ/|ξ|.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Ω++ = H(ξ) + H(η) + H(ξ − η) (|ξ| + |η| + |ξ − η|)(1 + |ξ| + |η| + |ξ − η|), the time resonant set is reduced to T = {ξ = η = 0}, Ω−− = H(ξ) − H(η) − H(ξ − η), we have ∇ηΩ−− = H′(η) η

|η| + H′(ξ − η) η−ξ |η−ξ| . From basic computations

∇ηΩ−− = 0 ⇒

• H′(η) = H′(ξ − η)

ξ−η |η−ξ| = η |η|

⇒ |η| = |ξ − η| ξ = 2η On the other hand Ω−−(2η, η) = H(2η) − 2H(η) = 0 ⇔ η = 0, thus R = {ξ = η = 0}. Ω−+ = H(ξ) − H(η) + H(ξ − η), from similar computations we find that the space-time resonant set is R = S = {ξ = 0}. The case Ω+− is symmetric. As was pointed out by S. Gustafson et al for their study of the Gross-Pitaevskii equation, the small frequency “parallel” resonances are worse than for the nonlinear Schr¨

• dinger equation. Namely near ξ = εη , η << 1 we have

H(εη) − H(η) + H((ε − 1)η) ∼ −3ε|η|3 2 √ 2 = −3|ξ| |η|2 2 √ 2 , while |εη|2 − |η|2 + |(1 − ε)η|2 ∼ −2|η| |ξ|, thus integrating by parts in time causes twice more loss of derivatives, and there is no hope for 1/Ω to belong to any standard class of multipliers.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Due to the limited regularity of our multipliers, we will need a multiplier theorem with loss from Guo and Pausader. In view of the discussion above, the frequency set {(ξ, η) : ξ = 0} is expected to raise some special difficulty. Let su look again at the nonlinearities: N(ψ) = U

• (a(1 + l1) − 1)∆l1 − 1

2

• |∇U−1φ1|2 − |∇l1|2

+ (2l1 − g(1 + l1))

• +i
• − ∇U−1φ1 · ∇l1 −
• 1 − a(1 + l1)
• Hφ
• .

(16) The real part of the nonlinearity is better behaved than the imaginary part since it has the operator U(ξ) in factor whose cancellation near ξ = 0 should compensate the resonances. In the spirit of S. Gustafson et al we introduce a Normal form in order to have a similar cancellation on the imaginary part. We look at new unknowns such that φ1 = Uφ, l1 = l − B(φ, φ) + B(l, l) = l + b(φ, l), with B a bilinear operator. When we make the calculus a natural choice is: B(η, ξ − η) = (α − 1)η · (ξ − η) 2 + |η|2 + |ξ − η|2 , with α = a′(1).

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Finally, if we replace in the quadratic terms l = l1 − b(φ, l) and set z = φ1 + il1 we obtain ∂tz − iHz = U

• α l1∆l1 − 1

2

• |∇U−1φ1|2 − |∇l1|2

+ (−∆ + 2)b(φ, l1)

• −iαdiv(l1∇φ) + U
• α(−b∆l1 − l1∆b + b∆b − 2∇b · ∇l + |∇b|2

+(−∆ + 2)(−2B(l1, b) + B(b, b))

• + iαdiv(b∇φ) + R

(17) = Q(z) + R, (18) where Q(z) contains the quadratic terms, R the cubic and quartic terms. The idea consists in applying Littlewood-Paley decomposition to the integral I with:

• a,b,c

t eisH∆aB1[∆b ¯ Z, ∆cZ]ds, where a ∼ |ξ|, b ∼ |η| and c ∼ |ξ − η|. It remains to split the frequency space in non time and non resonant region. We have the following decomposition in temporally an in space non resonant regions which is due to S. Gustafson, K. Nakanishi and T.-P. Tsai for the non linearity z¯ z . |η| ∼ |ξ| >> |ζ| (temporally non resonant) α = | ξ−η

|ξ−η − ξ |ξ| | >

√ 3 (temporally non resonant) · · ·

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

We define as follows a norm on the bilinear operator: B(η, ξ − η)

L∞

ξ

˙ Bs

2,1,η = 2jsχj(∇)ηB(η, ξ − η)l1(,L∞ ξ L2 η)

The norm B(ξ − ζ, ζ)

L∞

ξ

˙ Bs

2,1,ζ is defined similarly.

Theorem (Guo-Pausader) Let 0 ≤ s ≤ d/2, q1, q2 such that 1 q2 + 1 2 = 1 q1 + 1 2 − s d

• and

2 ≤ q′

1, q2 ≤

2n n − 2s , then B(f, g)Lq1 B

L∞

ξ

˙ Bs

2,1,η+

L∞

ξ

˙ Bs

2,1,ζ fLq2 gL2.

Lemma (S. Gustafson, K. Nakanishi, T.-P. Tsai) Denoting M = max(a, b, c), m = min(a, b, c) and l = min(b, c) we have for non time resonant region: Ba,b,c

1

Hs ( M M )sl

3 2 −sa−1,

for 0 < s < 2.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Remark We prove slight improvement of the previous lemma of S. Gustafson, K. Nakanishi, T.-P. Tsai n order to estimate sups1+ε∈R+ szW k,p How to control the norm sups1+ε∈R+ szW k,p ? We are going to consider the simple case N ≥ 5. Proposition Let N ≥ 5, T > 0, k ≥ 2, N ≥ k + 2 + d/2, we set ψXT = ψL∞([0,T ]HN ) + sup

t∈[0,T ]

(1 + t)d/4ψ(t)W k,4 , then the solution of (16) satisfies ∀ t ∈ [0, T], ψ(t)W k,4 ψ0W k,4/3 + ψ0HN + ψ2

XT

(1 + t)d/4 . The only issue is thus to bound the nonlinear part. Let f, g be a placeholder for l1 or U−1φ1 , we are going to consider a term of the form ∇f · ∇g . The estimates for 0 ≤ t ≤ 1 are easy, so we assume t ≥ 1 and we split the integral from (16) between [0, t − 1] and [t − 1, t].

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

For the first kind we have from the dispersion estimate

• t−1

ei(t−s)H∇f · ∇gds

• W k,4

t−1 ∇f · ∇gW k,4/3 (t − s)N/4 ds

• t−1
• ∇fHk∇gW k−1,4 + ∇gHk∇fW k−1,4
• (t − s)N/4

ds ψ2

XT

t−1 1 (t − s)d/4(1 + s)d/4 ds

• ψ2

XT

td/4 . For the second part on [t − 1, t] we use the Sobolev embedding Hd/4 ֒ → L4 ad Gagliardo Niremberg estimates:

• t

t−1

ei(t−s)H(∇f · ∇g)ds

• W k,4

t

t−1

• ∇f · ∇g
• Hk+d/4ds
• t

t−1

• ∇fL4∇gHk+d/2 + ∇gL4∇fHk+d/2
• ds
• ψ2

XT

(1 + t)d/4 . It concludes the proof.

Corentin Audiard and Boris Haspot

Presentation of the results Idea of the Proof Perspectives

Perspectives:

• 1. Extend this result to the case N = 2?
• 2. Quantify the size of the small initial data when N = 3 for κ(ρ) = 1

ρ .

THANK YOU FOR YOUR ATTENTION!!!

Corentin Audiard and Boris Haspot