Local Well Posedness of the Euler–Korteweg Equations on T

https://doi.org/10.1007/s10884-020-09927-3

Local Well Posedness of the Euler–Korteweg Equations on T

M. Berti¹·A. Maspero¹ ·F. Murgante¹ Dedicated to the memory of Walter Craig

Received: 1 July 2020 / Revised: 15 October 2020 / Accepted: 20 December 2020 / Published online: 22 March 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021

Abstract

We consider the Euler–Korteweg system with space periodic boundary conditionsx ∈T^d. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

Keywords Euler–Korteveg equations·Local well-posedness·Energy estimates· Para-differential calculus

1 Introduction

In this paper we consider the compressible Euler–Korteweg (EK) system ∂tρ+div(ρu) =0

∂tu+ u· ∇ u+ ∇g(ρ)= ∇

K(ρ)ρ+¹₂K(ρ)|∇ρ|²

, (1.1)

which is a modification of the Euler equations for compressible fluids to include capillary effects, under space periodic boundary conditionsx ∈T^d:=(R/2πZ)^d. The scalar variable ρ(t,x) >0 is the density of the fluid andu(t, x)∈R^d is the time dependent velocity field.

The functionsK(ρ),g(ρ)are defined onR⁺, smooth, andK(ρ)is positive.

The quasi-linear equations (1.1) appear in a variety of physical contexts modeling phase transitions [17], water waves [14], quantum hydrodynamics whereK(ρ)=κ/ρ[4], see also [15].

B

alberto.maspero@sissa.it M. Berti

berti@sissa.it F. Murgante fmurgant@sissa.it

1 International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Local well posedness results for the (EK)-system have been obtained in Benzoni-Gavage, Danchin and Descombes [8] for initial data sufficiently localized in the space variablex∈R^d. Then, ford ≥3, thanks to dispersive estimates, global in time existence results have been obtained for small irrotational data by Audiard–Haspot [7], assuming the sign condition g(ρ) >0. The case of quantum hydrodynamics corresponds toK(ρ)=κ/ρand, in this case, the (EK)-system is formally equivalent, via Madelung transform, to a semilinear Schrödinder equation onR^d. Exploiting this fact, global in time weak solutions have been obtained by Antonelli–Marcati [4,5] also allowingρ(t,x)to become zero (see also the recent paper [6]).

In this paper we prove a local in time existence result for the solutions of (1.1), with space periodic boundary conditions, under natural minimal regularity assumptions on the initial datum in Sobolev spaces, see Theorem1.1. Relying on this result, in a forthcoming paper [10], we shall prove a set of long time existence results for the (EK)-system in 1-space dimension, in the same spirit of [11,12].

We consider irrotational velocity fieldsu, i.e. with vorticity := curl(u) := ∇ u− (∇ u) equal to zero. Note that, since ∂t = −curl(u), if is initially zero, then remains zero under the evolution of (1.1). An irrotational vector field onT^d reads (Helmotz decomposition)

u= c(t)+ ∇φ, c(t)∈R^d, c(t) = 1 (2π)^d

T^dudx, (1.2)

whereφ:T^d →Ris a scalar potential. By the second equation in (1.1) and curlu=0, we get

∂tc(t)= − 1 (2π)^d

T^du· ∇ udx = 1 (2π)^d

T^d−1

2∇(|u|²)dx =0 ⇒ c(t)= c(0) is independent of time. Note that if the dimensiond=1, the average _2π¹

Tu(t,x)dxis an integral of motion for (1.1), and thus any solutionu(t,x),x∈T, of the (EK)-system (1.1) has the form (1.2) withc(t)=c(0)independent of time, that isu(t,x)=c(0)+φx(t,x).

The (EK) system (1.1) is Galilean invariant: if(ρ(t,x),u(t,x))solves (1.1) then ρc(t,x):=ρc(t,x+ ct), u_c(t,x):= u(t,x+ ct)− c

solve (1.1) as well. Thus, regarding the Euler–Korteweg system in a frame moving with a constant speedc(0), we may always consider in (1.2) that

u= ∇φ, φ:T^d →R.

The Euler–Korteweg equations (1.1) read, for irrotational fluids, ∂tρ+div(ρ∇φ)=0

∂tφ+¹₂|∇φ|²+g(ρ)=K(ρ)ρ+¹₂K(ρ)|∇ρ|². (1.3) The main result of the paper proves local well posedness for the solutions of (1.3) with initial data(ρ0, φ0)in Sobolev spaces

H^s(T^d):=

u(x)=

j∈Z^d

u_je^ij·x : u²_s :=

j∈Z^d

|u_j|²j^2s<+∞

wherej :=max{1,|j|}, under the natural mild regularity assumptions>2+(d/2). Along the paper,H^s(T^d)may denote either the Sobolev space of real valued functionsH^s(T^d,R) or the complex valued onesH^s(T^d,C).

Theorem 1.1 (Local existence onT^d) Let s>2+^d₂ and fix^d₂ <s₀≤s−2. For any initial data

(ρ0, φ0)∈H^s(T^d,R)×H^s(T^d,R) with ρ0(x) >0, ∀x ∈T^d,

there exists T :=T((ρ0, φ0)s0+2,min_xρ0(x)) >0and a unique solution(ρ, φ)of (1.3) such that

(ρ, φ)∈C⁰

[−T,T],H^s(T^d,R)×H^s(T^d,R)

∩C¹

[−T,T],H^s−2(T^d,R)×H^s−2(T^d,R)

andρ(t,x) >0for any t ∈ [−T,T]. Moreover, for|t| ≤T , the solution map(ρ0, φ0)→ (ρ(t,·), φ(t,·))is locally defined and continuous in H^s(T^d,R)×H^s(T^d,R).

We remark that it is sufficient to prove the existence of a solution of (1.3) on[0,T] because system (1.3) is reversible: the Euler–Korteweg vector fieldXdefined by (1.3) satisfies X◦S = −S◦X, whereSis the involution

S ρ

φ

:=

ρ^∨

−φ^∨

, ρ^∨(x):=ρ(−x). (1.4)

Thus, denoting by(ρ, φ)(t,x)=^t(ρ0, φ0)the solution of (1.3) with initial datum(ρ0, φ0) in the time interval[0,T], we have thatS^−t(S(ρ0, φ0))solves (1.3) with the same initial datum but in the time interval[−T,0].

Let us make some comments about the phase space of system (1.3). Note that the average

(2π)1 ^d

T^dρ(x)dxis a prime integral of (1.3) (conservation of the mass), namely 1

(2π)^d

T^dρ(x)dx=m, m∈R, (1.5)

remains constant along the solutions of (1.3). Note also that the vector field of (1.3) depends only onφ− _(2π)¹d

T^dφdx. As a consequence, the variables(ρ−m, φ)belong naturally to some Sobolev space H₀^s(T^d)× ˙H^s(T^d), whereH₀^s(T^d)denotes the Sobolev space of functions with zero average

H₀^s(T^d):=

u∈H^s(T^d):

T^du(x)dx=0

andH˙^s(T^d),s∈R, the corresponding homogeneous Sobolev space, namely the quotient space obtained by identifying all theH^s(T^d)functions which differ only by a constant. For simplicity of notation we denote the equivalent class[u] := {u+c,c∈R}, just byu. The homogeneous norm ofu ∈ ˙H^s(T^d)isu²_s :=

j∈Z^d\{0}|uj|²|j|^2s. We shall denote by

s either the Sobolev norm inH^s or that one in the homogenous space H˙^s, according to the context.

Let us make some comments about the proof. First, in view of (1.5), we rewrite system (1.3) in terms ofρm+ρwithρ∈H₀^s(T^d), obtaining

∂tρ= −mφ−div(ρ∇φ)

∂tφ= −¹₂|∇φ|²−g(m+ρ)+K(m+ρ)ρ+¹₂K(m+ρ)|∇ρ|². (1.6) Then Theorem1.1follows by the following result, that we are going to prove

Theorem 1.2 Let s>2+^d₂,^d₂ <s₀ <s−2and0<m₁ <m₂. For any initial data of the form(m+ρ0, φ0)with(ρ0, φ0)∈H₀^s(T^d)× ˙H^s(T^d)andm₁<m+ρ0(x) <m₂,∀x∈T^d, there exist T =T

(ρ0, φ0)s0+2,min_x(m+ρ0(x))

>0 and a unique solution(m+ρ, φ) of (1.6)such that

(ρ, φ)∈C⁰

[0,T],H₀^s(T^d,R)× ˙H^s(T^d,R)

∩C¹

[0,T],H₀^s−2(T^d,R)× ˙H^s−2(T^d,R)

andm₁ < m+ρ(t,x) <m₂ holds for any t ∈ [0,T]. Moreover, for|t| ≤T , the solution map(ρ0, φ0)→(ρ(t,·), φ(t,·))is locally defined and continuous in H₀^s(T^d)× ˙H^s(T^d).

We consider system (1.6) on the homogeneous spaceH˙^s× ˙H^s, that is we study ∂tρ= −mφ−div(( ^⊥₀ρ)∇φ)

∂tφ= −¹₂|∇φ|²−g(m+ ^⊥₀ρ)+K(m+ ^⊥₀ρ)ρ+¹₂K(m+ ^⊥₀ρ)|∇ρ|² (1.7) where ^⊥₀ is the projector onto the Fourier modes of index=0. For simplicity of notation we shall not distinguish between systems (1.7) and (1.6), which are equivalent via the iso- morphism ^⊥₀ : ˙H^s(T^d) → H₀^s(T^d). In Sect.3, we paralinearize (1.6), i.e. (1.7), up to bounded semilinear terms (for which we do not need Bony paralinearization formula). Then, introducing a suitable complex variable, we transform it into a quasi-linear type Schrödinger equation, see system (3.4), defined in the phase space

H˙^s:=

U = u

u

: u∈ ˙H^s(T^d,C) , U²_s := U²_H˙s = u²_s+ u²_s. (1.8) We use paradifferential calculus in the Weyl quantization, because it is quite convenient to prove energy estimates for this system. Since (3.4) is a quasi-linear system, in order to prove local well posedness (Proposition4.1) we follow the strategy, initiated by Kato [20], of constructing inductively a sequence of linear problems whose solutions converge to the solution of the quasilinear equation. Such a scheme has been widely used, see e.g. [1,8,18,22]

and reference therein.

The equation (1.3) is a Hamiltonian PDE. We do not exploit explicitly this fact, but it is indeed responsible for the energy estimate of Proposition4.4. The method of proof of Theorem1.1is similar to the one in Feola–Iandoli [19] for Hamiltonian quasi-linear Schrödinger equations onT^d (and Alazard–Burq–Zuily [1] in the case of gravity-capillary water waves inR^d). The main difference is that we aim to obtain the minimal smoothness assumptions>2+(d/2). This requires to optimize several arguments, and, in particular, to develop a sharp para-differential calculus for periodic functions that we report in the Appendix in a self-contained way. Some other technical differences are in the use of the modified energy (Sect.4.2), the mollifiers (4.17) which enables to prove energy estimates independent ofεfor the regularized system, the argument for the continuity of the flow in H^s. We expect that our approach would enable to extend the local existence result of [19] to initial data fulfilling the minimal smoothness assumptionss>2+(d/2).

We now set some notation that will be used throughout the paper. SinceK :R+→Ris positive, given 0<m₁ <m₂, there exist constantscK,CK >0 such that

c_K ≤K(ρ)≤C_K, ∀ρ∈(m₁,m₂). (1.9)

Since the velocity potentialφis defined up to a constant, we may assume in (1.6) that

g(m)=0. (1.10)

From now onwe fix s₀so that d

2 <s0<s−2. (1.11)

The initial datumρ0(x)belongs to the open subset ofH₀^s0(T^d)defined by Q:=

ρ∈H₀^s0(T^d) : m₁<m+ρ(x) <m₂

(1.12) and we shall prove that, locally in time, the solution of (1.6) stays in this set.

We writea bwith the meaninga ≤Cbfor some constantC > 0 which does not depend on relevant quantities.

2 Functional Setting and Paradifferential Calculus

The Sobolev norms ssatisfy interpolation inequalities (see e.g. Sect. 3.5 in [9]):

(i) for alls≥s0> ^d₂,u, v∈H^s,

uvsus₀vs+ usvs₀. (2.1)

(ii) Lets0> ^d₂. For all 0≤s≤s0,v∈H^s,u∈H^s0,

uvsus0vs. (2.2)

(iii) For alls₁<s₂,θ ∈ [0,1]andu∈H^s2,

uθs1+(1−θ)s2 ≤ u^θs₁u^1−θs₂ . (2.3) (iv) For alla≤α≤β≤b, withα+β=a+b,u, v∈H^b,

uαvβ ≤ uavb+ ubva. (2.4) Paradifferential calculus.We now introduce the notions of paradifferential calculus that will be used in the proof of Theorem1.1. We develop it in the Weyl quantization since it is more convenient to get the energy estimates of Sect.4. The main results are the continuity Theorem2.4and the composition Theorem2.5, which require mild regularity assumptions of the symbols in the space variable (they are deduced by the sharper results proved in TheoremsA.7andA.8in the Appendix). This is needed in order to prove the local existence Theorem1.1with the natural minimal regularity on the initial datum(ρ0, φ0) ∈H^s×H^s withs>2+^d₂.

Along the paperW may denote either the Banach spaceL^∞(T^d), or the Sobolev spaces H^s(T^d), or the Hölder spacesW^,∞(T^d), introduced in DefinitionA.3. Given a multi-index β∈N^d₀we define|β| :=β1+ · · · +βd.

Definition 2.1 (Symbols with finite regularity) Given m ∈ Rand a Banach space W ∈ {L^∞(T^d),H^s(T^d),W^,∞(T^d)}, we denote by^m_W the space of functionsa:T^d×R^d →

C,a(x, ξ), which areC^∞with respect toξ and such that, for anyβ ∈N^d₀, there exists a constantC_β >0 such that

∂_ξ^βa(·, ξ)

W ≤C_β ξ^m−|β|, ∀ξ∈R^d. (2.5)

We denote by_W^m the subclass of symbolsa∈^m_W which arespectrally localized, that is

∃δ∈(0,1): a(j, ξ)=0, ∀|j| ≥δξ, (2.6) wherea(j, ξ):= (2π)^−d

T^da(x, ξ)e^−ij·xdx, j ∈ Z^d, are the Fourier coefficients of the functionx →a(x, ξ).

We endow_W^m with the family of norms defined, for anyn∈N0, by

|a|_m,W,n:= max

|β|≤n sup

ξ∈R^d

ξ^−m+|β| ∂_ξ^βa(·, ξ)

W. (2.7)

WhenW = H^s, we also denote _s^m ≡ ^m_Hs and|a|m,s,n ≡ |a|m,H^s,n. We denote by ^m_s ⊗M2(C)the 2×2 matrices A =

a1 a2

a₃ a₄

of symbols in_s^m and|A|m,W,n :=

max_i=1,...,4{|a_i|m,W,n}. Similarly we denote by^ms ⊗R^dthed-dimensional vectors of sym- bols in^m_s.

Let us make some simple remarks:

• (i) given a functiona(x)∈W thena(x)∈⁰_W and

|u|_0,W,n = u_W ,∀n∈N0. (2.8)

• (ii) For anys₀> ^d₂ and 0≤≤, we have that

|a|m,L^∞,n|a|_m,W,∞,n |a|m,W^,∞,n|a|_m,H^s0+,n, ∀n∈N0. (2.9)

• (iii) Ifa∈^m_W, then, for anyα∈N^d₀, we have∂_ξ^αa∈_W^m−|α|and

|∂_ξ^αa|m−|α|,W,n|a|m,W,n+|α|, ∀n∈N0. (2.10)

• (iv) Ifa∈^m_Hs, resp.a∈^m_W,∞, then∂_x^αa∈^m_Hs−|α|, resp.∂^α_xa∈^m_W−|α|,∞, and

|∂_x^αa|m,s−|α|,n |a|m,s,n, resp.|∂_x^αa|m,W^−|α|,∞,n |a|m,W^,∞,n, ∀n∈N0. (2.11)

• (v) Ifa,b ∈ ^m_W then ab ∈ _W^m with |ab|m+m,W,n |a|m,W,N|b|m,W,n for any n∈N0. In particular, ifa,b∈^m_s withs>d/2 thenab∈^m+m_s and

|ab|_m+m,s,n|a|_m,s,n|b|m,s0,n+ |a|_m,s0,n|b|m,s,n, ∀n∈N0. (2.12) Let∈(0,1)and consider aC^∞, even cut-off functionχ:R^d → [0,1]such that

χ(ξ)=

1 if|ξ| ≤1.1

0 if|ξ| ≥1.9, χ(ξ):=χ ξ

. (2.13)

Given a symbolain_W^m we define theregularizedsymbol a_χ(x, ξ):=χξ(D)a(x, ξ)=

j∈Z^d

χ

j ξ

a(j, ξ)e^ij·x. (2.14)

Note thata_χ is analytic inx(it is a trigonometric polynomial) and it is spectrally localized.

In order to define the Bony–Weyl quantization of a symbola(x, ξ)we first remind the Weyl quantization formula

Op^W(a)[u] :=

j∈Z^d k∈Z^d

a

j−k,k+j 2

u_k

e^ij·x. (2.15)

Definition 2.2 (Bony–Weyl quantization) Given a symbola∈_W^m, we define theBony–Weyl paradifferential operatorOp^BW(a)=Op^W(a_χ)that acts on a periodic functionuas

Op^BW(a)[u]

(x):=

j∈Z^d k∈Z^d

a_χ

j−k, j+k 2

u_k

e^ij·x

=

j∈Z^d k∈Z^d

a

j−k, j+k 2

χ

j−k j+k

uk

e^ij·x.

(2.16)

If A =

a₁ a₂ a₃ a₄

is a matrix of symbols in_s^m, then Op^BW(A)is defined as the matrix valued operator

Op^BW(a1) Op^BW(a2) Op^BW(a₃) Op^BW(a₄)

.

Given a symbola(ξ)independent ofx, then Op^BW(a)is the Fourier multiplier operator Op^BW(a)u=a(D)u=

j∈Z^d

a(j)uje^ij·x.

Note that ifχ

k−j k+j

=0 then|k−j| ≤j+kand therefore, for∈(0,1), 1−

1+|k| ≤ |j| ≤1+

1−|k|, ∀j,k∈Z^d. (2.17) This relation shows that the action of a para-differential operator does not spread much the Fourier support of functions. In particular Op^BW(a)sends a constant function into a constant function and therefore Op^BW(a)sends homogenous spaces into homogenous spaces.

Remark 2.3 Actually, ifχ_k−j

k+j

=0, ∈ (0,1/4), then|j| ≤ |j+k| ≤ 3|j|, for all j,k∈Z^d.

Along the paper we shall use the following results concerning the action of a paradiffer- ential operator in Sobolev spaces.

Theorem 2.4 (Continuity of Bony–Weyl operators)Let a∈^m_s0, resp. a∈^m_L∞, with m∈R. Then Op^BW(a)extends to a bounded operator H˙^s → ˙H^s−mfor any s ∈Rsatisfying the estimate, for any u∈ ˙H^s,

Op^BW(a)u

s−m |a|_m,s0,2(d+1) us (2.18) Moreover, for any≥0, s∈R, u∈ ˙H^s(T^d),

Op^BW(a)u_s−m− |a|_{m,s0−,2(d+1)} us. (2.19)

Proof Since Op^BW(a) = Op^W a_χ

, the estimate (2.18) follows by (A.35), (A.21) and

|a|_m,L^∞_,N |a|_m,s0,N. Note that the condition on the Fourier support ofa_χin TheoremA.7 is automatically satisfied providedin (2.13) is sufficiently small. To prove (2.19) we use

also (A.22).

The second result of symbolic calculus that we shall use regards composition for Bony–

Weyl paradifferential operators at the second order (as required in the paper) with mild smoothness assumptions for the symbols in the space variablex. Given symbolsa∈s^m₀+, b∈_s^m₀₊withm,m∈Rand∈(0,2]we define

a#b:=

ab, ∈(0,1]

ab+_2i¹{a,b}, ∈(1,2], where {a,b} := ∇_ξa· ∇xb− ∇xa· ∇_ξb,(2.20) is the Poisson bracket betweena(x, ξ)andb(x, ξ). By (2.10) and (2.12) we have thatabis a symbol ins^m+m₀+ and{a,b}is in^m+m_s0+−⁻¹₁ . The next result follows directly by Theorem A.8and (2.9).

Theorem 2.5 (Composition) Let a ∈ ^m_s0+, b ∈ ^m_s0+ with m,m ∈Rand ∈ (0,2].

Then

Op^BW(a)Op^BW(b)=Op^BW a#b

+R⁻(a,b) (2.21) where the linear operator R⁻(a,b): ˙H^s → ˙H^s−(m+m)+, ∀s ∈ R, satisfies, for any u∈ ˙H^s,

R⁻(a,b)u_s−(m+m)+

|a|_m,s0+,N |b|_m,s0,N + |a|_m,s0,N |b|_m,s0+,N u_s

(2.22) where N ≥3d+4.

A useful corollary of Theorems2.5and2.4[using also (2.10)–(2.12)] is the following:

Corollary 2.6 Let a∈^m_s0+2, b∈^m_s0+2, c∈_s^m₀₊₂with m,m,m∈R. Then

Op^BW(a)◦Op^BW(b)◦Op^BW(c)=Op^BW(abc)+R₁(a,b,c)+R₀(a,b,c), (2.23) where

R1(a,b,c):=Op^BW

{a,c}b+ {b,c}a+ {a,b}c

(2.24) satisfies R1(a,b,c) = −R1(c,b,a) and R0(a,b,c) is a bounded operator H˙^s → H˙^{s−(m+m+m)+2}, ∀s∈R, satisfying, for any u∈ ˙H^s,

R₀(a,b,c)s−(m+m+m)+2|a|m,s0+2,N|b|m,s0+2,N |c|m,s0+2,Nus (2.25) where N ≥3d+5.

We now provide the Bony-paraproduct decomposition for the product of Sobolev functions in the Bony–Weyl quantization. Recall that ^⊥₀ denotes the projector on the subspaceH₀^s.

Lemma 2.7 (Bony paraproduct decomposition) Let u∈H^s,v∈H^r with s+r≥0. Then uv=Op^BW(u) v+Op^BW(v)u+R(u, v) (2.26) where the bilinear operator R: H^s×H^r →H^s+r−s0is symmetric and satisfies the estimate

R(u, v)_s+r−s0 us vr. (2.27)

Moreover R(u, v)=R( ^⊥₀u, ^⊥₀v)−u₀v0and then

⊥0R(u, v)s+r−s₀ ^⊥₀us ⊥

0vr. (2.28)

Proof Introduce the functionθ(j,k)by 1=χ

j−k j+k

+χ

k 2j−k

+θ(j,k). (2.29)

Note that|θ(j,k)| ≤1. Let:= {(j,k)∈Z^d×Z^d : θ(j,k)=0}denote the support of θ. We claim that

(j,k)∈ ⇒ |j| ≤Cmin(|j−k|,|k|). (2.30) Indeed, recalling the definition of the cut-off functionχin (2.13), we first note that¹

= {(0,0)} ∪

|j−k| ≥j+k,|k| ≥2j−k . Thus, for any(j,k)∈,

|j| ≤ 1

2|j−k| +1

2|j+k| ≤ 1

2 + 1 2

|j−k|,

|j| ≤ 1

2|2j−k| +1 2|k| ≤

1 2+ 1

2

|k|

proving (2.30). Using (2.29) we decompose

uv=

j,k

u_j−kχ

j−k j+k

vke^ij·x

+

j,k

vkχ

k 2j−k

u_j−ke^ij·x+

j,k

θ(j,k)u_j−kvke^ij·x

=Op^BW(u) v+Op^BW(v)u+R(u, v).

By (2.30),s+r≥0, and the Cauchy–Schwartz inequality, we get R(u, v)²_s+r−s0 ≤

j

j^2(s+r−s0)

k

θ(j,k)uj−kvk²

j

j^−2s0

k

j−k^s |u_j−k| k^r |vk|²u²_s v_r² proving (2.27). Finally, since on the support ofθwe have or(j,k)=(0,0)or j−k=0 andk=0, we deduce that

1Forδsufficiently small, if|j−k| ≤δj+kand|k| ≤δ2j−kthen(j,k)=(0,0).

R(u, v)=θ(0,0)u₀v0+

j−k=0,k=0

θ(j,k)u_j−kvke^ij·x= −u₀v0+R( ^⊥0u, ^⊥0v)

and we deduce (2.28).

Composition estimates.

We will use the following Moser estimates for composition of functions in Sobolev spaces.

Theorem 2.8 Let I ⊆ Rbe an open interval and F ∈ C^∞(I;C)a smooth function. Let J ⊂I be a compact interval. For any function u, v∈H^s(T^d,R), s> ^d₂, with values in J , we have

F(u)s≤C(s,F,J) (1+ us) ,

F(u)−F(v)s ≤C(s,F,J) (u−vs+(us+ vs)u−vL^∞)

F(u)s≤C(s,F,J)us if F(0)=0. (2.31)

Proof Take an extension F˜ ∈C^∞(R;C)such thatF˜_|I = F. ThenF(u)= ˜F(u)for any u∈H^s(T^d;R)with values inJ, and apply the usual Moser estimate, see e.g. [3], replacing the Littlewood–Paley decomposition onR^d with the one onT^din (A.12).

3 Paralinearization of (EK)-System and Complex Form

In this section we paralinearize the Euler–Korteweg system (1.6) and write it in terms of the complex variable

u:= 1

√2 m

K(m) _−1/4

ρ+ i

√2 m

K(m) _1/4

φ, ρ∈ ˙H^s, φ∈ ˙H^s. (3.1) The variableu∈ ˙H^s. We denote this change of coordinates inH˙^s× ˙H^sby

u u

=C⁻¹ ρ

φ

, C:= 1

√2

⎛

⎜⎝ m

K(m)

¹

4

m K(m)

¹

4

−i m

K(m)

₋¹

4 i

m K(m)

₋¹

4

⎞

⎟⎠, C⁻¹ = 1

√2

⎛

⎜⎝ m

K(m)

₋¹

4 i

m K(m)

¹

4

m K(m)

₋¹

4 −i

m K(m)

¹

4

⎞

⎟⎠.

(3.2) We also define the matrices

J :=

0 1

−1 0

, J:=

−i 0 0 i

, 1:=

1 0 0 1

. (3.3)

Proposition 3.1 (Paralinearized Euler–Korteweg equations in complex coordinates) The (EK)-system(1.6)can be written in terms of the complex variable U:=

u u

with u defined in(3.1), in the paralinearized form

∂tU = J

Op^BW(A₂(U;x, ξ)+A₁(U;x, ξ))

U+R(U) (3.4)

where, for any function U ∈ ˙H^s0+2such that ρ(U):= 1

√2 m

K(m) _1/4

⊥0(u+u)∈Q (see(1.12)), (3.5) one has

(i) A₂(U;x, ξ)∈²_s0+2⊗M2(C)is the matrix of symbols A₂(U;x, ξ):=

mK(m)|ξ|²

1+a₊(U;x) a₋(U;x) a₋(U;x) 1+a₊(U;x)

(3.6) wherea_±(U;x)∈⁰_s0+2are theξ-independent functions

a_±(U;x):= 1 2

K(ρ+m)−K(m) K(m) ±ρ

m

. (3.7)

(ii) A₁(U;x, ξ)∈¹_s0+1⊗M2(C)is the diagonal matrix of symbols A₁(U;x, ξ):=

b(U;x)·ξ 0 0 −b(U;x)·ξ

, b(U;x):= ∇φ∈⁰_s0+1⊗R^d. (3.8) Moreover for anyσ ≥ 0there exists a non decreasing functionC( ) : R+ →R+

(depending on K ) such that, for any U,V ∈ ˙H^s0withρ(U), ρ(V)∈Q, W ∈ ˙H^σ+2 and j=1,2, we have

Op^BW A_j(U)

Wσ ≤C Us0

Wσ+2 (3.9)

Op^BW

A_j(U)−A_j(V)

Wσ ≤C

Us0,Vs0

Wσ+2U−Vs0 (3.10) where in(3.10)we denoted byC(·,·):=C(max{·,·}).

(iii) The vector field R(U)satisfies the following “semilinear” estimates: for anyσ≥s₀>

d/2there exists a non decreasing functionC( ):R+→R+(depending also on g,K ) such that, for any U,V ∈ ˙H^σ+2such thatρ(U), ρ(V)∈Q, we have

R(U)σ ≤C

Us0+2

Uσ, R(U)σ ≤C Us0

Uσ+2, (3.11) R(U)−R(V)σ≤C

Us0+2,Vs0+2

U−Vσ

+C(Uσ,Vσ)U−Vs0+2 (3.12) R(U)−R(V)s0 ≤C

Us0+2,Vs0+2

U−Vs0, (3.13)

where in(3.12)and(3.13)we denoted again byC(·,·):=C(max{·,·}).

Proof We first paralinearize the original equations (1.6), then we switch to complex coordi- nates.

Step 1: paralinearization of (1.6). We apply several times the paraproduct Lemma 2.7 and the composition Theorem2.5. In the following we denote by R^p the remainder that comes from Lemma2.7, and by R⁻, = 1,2, the remainder that comes from Theorem 2.5. We shall adopt the following convention: givenR^d-valued symbolsa =(a_j)j=1,...,d, b=(bj)j=1,...,d in some class_s^m⊗R^d, we denoteR^p(a,b):=_d

j=1R^p(aj,bj), R⁻(a,b):=

d j=1

R⁻(aj,bj) and Op^BW(a)·Op^BW(b):=

d j=1

Op^BW aj

Op^BW bj

.

We paralinearize the terms in the first line of (1.6). We haveφ = −Op^BW

|ξ|² φ and div(ρ∇φ)= ∇ρ· ∇φ+ρφcan be written as

ρφ= −Op^BW

ρ|ξ|²+ ∇ρ·iξ φ

+Op^BW(φ) ρ+R^p(ρ, φ)+R⁻²(ρ,|ξ|²)φ, (3.14)

∇ρ· ∇φ=Op^BW(∇ρ·iξ) φ+Op^BW(∇φ·iξ) ρ

+R^p(∇ρ,∇φ)+R⁻¹(∇ρ,iξ)φ+R⁻¹(∇φ,iξ)ρ. (3.15) Then we paralinearize the terms in the second line of (1.6). We have

1

2|∇φ|²=Op^BW(∇φ·iξ) φ +1

2R^p(∇φ,∇φ)+R⁻¹(∇φ,iξ)φ. (3.16) Using (1.10) we regard the semilinear term

g(m+ρ)=g(m+ρ)−g(m)=:R(ρ) (3.17) directly as a remainder. Moreover, writingρ = −Op^BW

|ξ|²

ρ, we get

K(m+ρ)ρ=Op^BW(K(m+ρ)) ρ+Op^BW(ρ)K(m+ρ)+R^p(ρ,K(m+ρ))

= −Op^BW

K(m+ρ)|ξ|²+K(m+ρ)∇ρ·iξ ρ

+Op^BW(ρ)K(m+ρ)+R^p(ρ,K(m+ρ))−R⁻²(K(m+ρ),|ξ|²)ρ.

(3.18) Finally, using for¹₂|∇ρ|²the expansion (3.16) forρinstead ofφ, we obtain

1

2K(m+ρ)|∇ρ|² =1 2Op^BW

K(m+ρ)

|∇ρ|²+1 2Op^BW

|∇ρ|²

K(m+ρ) +1

2R^p(|∇ρ|²,K(m+ρ))=Op^BW

K(m+ρ)∇ρ·iξ

ρ+R(ρ) where

R(ρ):= 1 2Op^BW

|∇ρ|²

K(m+ρ)+1

2R^p(|∇ρ|²,K(m+ρ)) (3.19) +1

2Op^BW

K(m+ρ)

R^p(∇ρ,∇ρ) (3.20)

+Op^BW

K(m+ρ)

R⁻¹(∇ρ,iξ)ρ+R⁻¹(K(m+ρ),i∇ρ·ξ)ρ. (3.21) Collecting all the above expansions and recalling the definition of the symplectic matrix J in (3.3), the system (1.6) can be written in the paralinearized form

∂t

ρ φ

= JOp^BW

K(m+ρ)|ξ|² ∇φ·iξ

−∇φ·iξ (m+ρ)|ξ|² ρ φ

+R(ρ, φ) (3.22) where we collected inR(ρ, φ)all the terms in lines (3.14)–(3.21).

Step 2: complex coordinates.We now write system (3.22) in the complex coordinates U = C⁻¹

ρ φ

. Note thatC⁻¹ conjugates the Poisson tensor J toJdefined in (3.3), i.e.

C⁻¹J=JC^∗and therefore system (3.22) is conjugated to

∂tU =JC^∗Op^BW

K(m+ρ)|ξ|² ∇φ·iξ

−∇φ· iξ ρ|ξ|²

CU +C⁻¹R(CU). (3.23) Using (3.2), system (3.23) reads as system (3.4)–(3.8) withR(U):=C⁻¹R(CU).

We note also that estimates (3.9) and (3.10) forj=2 follow by (2.18) and (2.31), whereas in case j=1 follow by (2.19) applied withm=1,=1.

Step 3: Estimate of the remainderR(U). We now prove (3.11)–(3.13). Sinceρσ,φσ∼ Uσ for anyσ ∈ Rby (3.2), the estimates (3.11)–(3.13) directly follow from those of R(ρ, φ)in (3.22). We now estimate each term in (3.14)–(3.21). In the sequelσ ≥s0>d/2.

Estimate of the term in line(3.14). Applying first (2.18) withm =0, and then (2.19) with=2, we have

Op^BW(φ) ρσ φs0+2ρσ, Op^BW(φ) ρσ φs0ρσ+2. (3.24) By (2.27), the smoothing remainder in line (3.14) satisfies the estimates

R^p(ρ, φ)σ φs0+2ρσ, R^p(ρ, φ)σ φs0ρσ+2, (3.25) and, by (2.22) with=2, and the interpolation estimate (2.4),

R⁻²(ρ,|ξ|²)φσ ρs0+2φσ φs0ρσ+2+ ρs0φσ+2. (3.26) By (3.24)–(3.26) andρ_σ,φ_σ ∼ U_σwe deduce that the terms in line (3.14), written in function ofU, satisfy (3.11). Next we write

Op^BW(φ1) ρ1−Op^BW(φ2) ρ2=Op^BW(φ1)[ρ1−ρ2] +Op^BW(φ1−φ2) ρ2

and, applying (2.18) withm=0, and (2.19) with=2 to Op^BW(φ1−φ2) ρ2, we get Op^BW(φ1) ρ1−Op^BW(φ2) ρ2σ φ1s0+2ρ1−ρ2σ+ φ1−φ2s0+2ρ2σ

Op^BW(φ1) ρ1−Op^BW(φ2) ρ2σ φ1s0+2ρ1−ρ2σ+ φ1−φ2s0ρ2σ+2. (3.27) Concerning the remainderR^p(ρ, φ), we writeR^p(ρ1, φ1)−R^p(ρ2, φ2)= R^p(ρ1− ρ2, φ1)+R^p(ρ2, φ2−φ1)and, applying (2.27), we get

R^p(ρ1, φ1)−R^p(ρ2, φ2)σ φ1s0+2ρ1−ρ2σ+ ρ2σφ1−φ2s0+2

R^p(ρ1, φ1)−R^p(ρ2, φ2)_σ φ1s₀+2ρ1−ρ2_σ+ ρ2_σ+2φ1−φ2s₀. (3.28) Finally we writeR⁻²(ρ1,|ξ|²)φ1−R⁻²(ρ2,|ξ|²)φ2=R⁻²(ρ1−ρ2,|ξ|²)φ1+R⁻²(ρ2,|ξ|²) [φ1−φ2]. Using (2.22) we get

R⁻²(ρ1,|ξ|²)φ1−R⁻²(ρ2,|ξ|²)φ2σ φ1σρ1−ρ2s0+2

+φ1−φ2σρ2s0+2. (3.29)

We also claim that

R⁻²(ρ1,|ξ|²)φ1−R⁻²(ρ2,|ξ|²)φ2σ

ρ1−ρ2s0φ1_σ+2+ φ1−φ2_σρ2s0+2. (3.30)