A dispersive property of the Euler-Korteweg system Corentin Audiard - - PowerPoint PPT Presentation

A dispersive property of the Euler-Korteweg system

Corentin Audiard

Laboratoire Jacques-Louis Lions (UMR 7598) Universit´ e Pierre et Marie Curie

28 june 2012

Corentin Audiard A dispersive property of the Euler-Korteweg system

The Euler-Korteweg system consists of the following equations ∂tρ + div(ρu) = 0, ∂tu + (u · ∇)u + ∇g0(ρ) = ∇ (K(ρ)∆ρ + 1

2K ′(ρ)|∇ρ|2) ,

(EK) It is a perturbation of the classical Euler equations, that takes into account the capillarity effects. The quantities involved are the density ρ and the velocity u, K is the so called Korteweg stress tensor. On the opposite of the Navier-Stokes equations, the perturbation is dispersive.

Corentin Audiard A dispersive property of the Euler-Korteweg system

The linearized system near a constant state (u, ρ) admits the following dispersion relation (τ + iu · ξ)2 + ρ(g′|ξ|2 + K|ξ|4) = 0. At high frequencies it amounts to τ ∼ ±iKρ|ξ|2 and thus bears some similarity with the usual Schr¨

• dinger equation

i∂tu + a∆u = 0, whose dispersion relation is τ + ia|ξ|2 = 0.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Local well-posedness of the Euler-Korteweg system in any dimension was obtained in 2007. Theorem (Benzoni-Danchin-Descombes ′07) Given s > d/2 + 1, u0 ∈ Hs(Rd), ρ0 ∈ C 0

b (Rd) such that ∇ρ ∈ Hs,

there exists T > 0 and a unique solution (ρ, u) ∈ CTCb × CTHs, ∇ρ ∈ CTHs of the Cauchy problem    ∂tρ + div(ρu) = 0, ∂tu + (u · ∇)u + ∇g0(ρ) = ∇ (K(ρ)∆ρ + 1

2K ′(ρ)|∇ρ|2) ,

(ρ, u)|t=0 = (ρ0, u0) (1)

Corentin Audiard A dispersive property of the Euler-Korteweg system

The proof relied on rather involved a priori estimates, for an extended (formally equivalent) system displaying a better structure : ∂tζ + u · ∇ζ + a(ζ)divu = 0, ∂tz + (u · ∇)z + i(∇z) · w + i∇(adivz) = −g′(ζ)Re(z). (2) where ζ = R(ρ), R is a primitive of the application ρ →

• K(ρ)/ρ,

w = ∇ζ, z = u + iw, a(ζ) =

• R−1(ζ)K(R−1(ζ)).

The second equation actually looks like a quasi-linear degenerate Schr¨

• dinger equation.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Opposedly to the Schr¨

• dinger equation, no dispersive estimate was

proved yet for the Euler-Korteweg system. Our main result is a local smoothing property. Theorem Under the assumptions of the local well-posedness theorem, if moreover u0 is irrotational, the curves associated to the hamiltonian a(x, 0)|ξ|2 are unbounded and ∇x,ta(0, x) ≤ C/(1 + |x|2), then any solution (u, ∇ρ) ∈ (CTHs)2 additionally satisfies for some T (u, ∇ρ)/(1 + |x|) ∈ L2([0, T]; Hs+1/2(Rd)).

Corentin Audiard A dispersive property of the Euler-Korteweg system

A few comments on the assumptions : the irrotationality seems natural since ∂tz + ia∇divz = 0 admits trivial stationary solution, which precisely correspond to “purely rotational” initial data, div(z0) = 0. The second assumption means that the solutions of the differential equation X(t)′ = ∇ξ(a(X(t), 0)|Ξ(t)|2) = 2a(X(t), 0)Ξ(t), Ξ(t)′ = −∇x(a(X(t), 0)|Ξ|2), satisfy limt→∞ X(t) = +∞. It is standard in the frame of linear Schr¨

• dinger equations with variable coefficients.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Some elements of proof : Since the local gain of regularity is only of 1/2 derivative, it can hardly be obtained by basic multiplier techniques. It is necessary to use slightly more sophisticated tools. Nonlinearities appear even in the highest order derivatives, thus the pseudo-differential calculus is not well-suited as it usually requires a lot of smoothness from the coefficients. A more fitted tool would be Bony’s paradifferential calculus. Para-differential calculus allows to replace a product uv by Tuv + R(u, v), where R is (hopefully) smooth, and Tu acts Hs → Hs. It is even possible for a class of symbol s(x, ξ) that satisfy minimal regularity assumptions to define the paradifferential

• perators Ts.

Corentin Audiard A dispersive property of the Euler-Korteweg system

A very basic sketch of proof : fix some symbol p(x, ξ) and consider the derivative d dt Tpz, z = Tp(−i∇adivz), z + Tpz, −i∇adivz + l.o.t. = [Tp, −i∇adiv]z, z + l.o.t. use then the rules of para-differential calculus and the irrotationality · · · = [Tp, T−i|ξ|2a]z, z + l.o.t. = T{ip, |ξ|2a}z, z + l.o.t. where {ip, |ξ|2a} is the Poisson bracket

d

• j=1

∂ξjp∂xj(a|ξ|2) − ∂xjp∂ξj(a|ξ|2).

Corentin Audiard A dispersive property of the Euler-Korteweg system

We have obtained (roughly) d dt Tpz, z ≃ T{ip, |ξ|2a}z, z. If p is a zeroth order operator, and {ip, |ξ|2a} ≥ c|ξ|/(1 + |x|2), it is then possible to use a G¨ arding-like inequality to deduce d dt Tpz, z cz/

• 1 + |x|22

H1/2(Rd)

⇒

T

z(t)/

• 1 + |x|22

H1/2dt zL∞([0, T],L2(Rd)),

which is the expected smoothing effect.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Why it is not that simple : The construction of p is complicated (but similar to the one

• f Doi for Schr¨
• dinger like equations),

The G¨ arding inequality is not standard, The lower order terms are actually not neglectible, Instead of a gain L2 → H1/2 we want a gain Hs → Hs+1/2, thus instead of working on z one has to study the quantity T|ξ|sz, it raises new commutators and more “bad” lower order terms.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Some more details on the gauge method : set Zs = Tϕ|ξ|sz, the equation satisfied by Zs is ∂tZs+Tu·∇Zs+i(∇Zs) · w + i[diva∇, Tϕ|ξ|s]z−idivTa∇Zs = l.o.t. We have i(∇Zs) · w ≃ T−w·ξ|ξ|sϕz, and i[diva∇, Tϕ|ξ|s]z ≃ T{a|ξ2, |ξ|sϕ}z. To suppress the bad terms, it is sufficient to have {a|ξ|2, ϕ|ξ|s} = ϕ|ξ|sξ · w, and the “miraculous” function ϕ = √ρas/2 works (as in Benzoni-Danchin-Descombes) !

Corentin Audiard A dispersive property of the Euler-Korteweg system

The Euler-Korteweg system admits traveling waves (ρ, u) that only depend on x1 − ct, the assumptions for the smoothing effect are usually not satisfied even for the linearized equations near such waves : ∂tz + u∇z + i∇z · w + i∇adivz = l.o.t. Nevertheless, it is still possible to prove local smoothing in special cases. Proposition Assume that 2

• a0(x1) − a0′(x1)
• a0(x1) ≥ α > 0,

then the same local smoothing property is still true.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Principle of proof : Doi’s construction of p does not work, but a simpler one is actually available. Essentially, local smoothing is reduced again to the positivity of {a|ξ|2, p}, now the choice p = f (x1 − ct)x · ξ |ξ| gives {a|ξ|2, p} = |ξ|(2af − x1a′f ) + ξ1 x · ξ |ξ|

• a′f + 2f ′a
• .

For f = c/√a the bad term cancels, and the positivity condition becomes {a|ξ|2, p} = |ξ|

• 2√a − x1

a′ √a

• > 0,

which is precisely the assumption.

Corentin Audiard A dispersive property of the Euler-Korteweg system

Thank you for you attention !

Corentin Audiard A dispersive property of the Euler-Korteweg system