Study of Anisotropic MHD system in Anisotropic Sobolev spaces

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

JAMELBENAMEUR, RIDHASELMI

Study of Anisotropic MHD system in Anisotropic Sobolev spaces

Tome XVII, n^o1 (2008), p. 1-22.

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Annales de la Facult´e des Sciences de Toulouse Vol. XVII, n^◦1, 2008 pp. 1–22

Study of Anisotropic MHD system in Anisotropic Sobolev spaces

Jamel Ben Ameur⁽¹⁾, Ridha Selmi⁽²⁾

ABSTRACT. — Three-dimensional anisotropic magneto-hydrodynamical system is investigated in the whole spaceR³. Existence and uniqueness results are proved in the anisotropic Sobolev space H^0,s for s > 1/2.

Asymptotic behavior of the solution when the Rossby number goes to zero is studied. The proofs, where the incompressibility condition is crucial, use the energy method, an appropriate dyadic decomposition of the frequency space, product laws in anisotropic Sobolev spaces and Strichartz-type estimates.

R^´ESUM ´E.— On ´etudie un syst`eme magneto-hydro-dynamique tridimen- sionnel dans le cas de l’espace entier R³. On d´emontre l’existence et l’unicit´e de la solution pour des donn´ees initiales dans les espaces de Sobolev anisotropes ;H^0,s,s >1/2. On ´etudie le comportement asymp- totique de la solution lorsque le nombre de Rossby tends vers z´ero. Les preuves se basent essentiellement sur des m´ethodes d’energie, une d´ecompo- sition ad´equate de l’espace de fr´equences, les lois produit dans les espaces de Sobolev anisotropes et une estimation de type Stichartz. La condition d’incompressibilit´e joue un rˆole crucial dans les d´emonstrations.

(∗) Re¸cu le 17 janvier 2006, accept´e le 27 septembre 2006

(1) Facult´e des Sciences de Bizerte, Tunisie.

jamel.BenAmeur@fsb.rnu.tn

(2) Institut Sup´erieur d’Informatique, l’Ariana 2080, Tunisie.

Ridha.Selmi@isi.rnu.tn

1. Introduction

The purpose of this paper is to study the incompressibleM HD model with anisotropic diffusion in the limit of small Rossby number, namely the following system

(M HD_ν^ε_h)













































∂tu−νh∆hu+u· ∇u−b· ∇b+1

εcurlb×e3

+1

εu×e3=−∇p inR⁺×R³

∂tb−ηh∆hb+u· ∇b−b· ∇u +1

εcurl (u×e3) = 0 in R⁺×R³ divu= 0 in R⁺×R³

divb= 0 in R⁺×R³ (u, b)|t=0= (u0, b0),

where the velocity fieldu,the induced magnetic perturbationband the pres- surepare unknown functions of timet and space variablex= (x1, x2, x3), e3is the third vector of the cartesian coordinate system,νhis the anisotropic dynamic viscosity,ηhis the anisotropic magnetic diffusivity, ∆hdenotes the horizontal Laplace operator defined by ∆h =∂₁²+∂²₂ andε is the Rossby number which is the ratio between the fluid’s typical velocity to the earth rotation velocity around the axise3.

The above system is a simplified version of a general one given in [4] to modelise the magneto-hydrodynamical process in the earth’s core which is believed to support a self-excited dynamo process generating the earth’s magnetic field.

Our first motivation follows from Taylor Proudman Theorem [10], that the three-dimensional fluid is inclined to behave as a two-dimensional one. The second motivation follows from the fact that large turbulent eddies move preferably in horizontal layers and do rarely move vertically, which makes vertical turbulent diffusion negligible compared to the horizontal one (see [3] and references therein). So, it makes sense to consider an anisotropic diffusive process. Without loss of generality,−e3is supposed to be the time independent component of the earth magnetic fieldB defined in [4] by

B=−e3+θb,

where θ denotes the Reynolds’ number. According to [5], the magnetic Prandtl number Pm defined as the ratio of the dynamic viscosity by the

magnetic viscosity is equal to 10⁻⁶in the case of the earth’s core. However, to simplify the mathematical computation we will takeνh=ηh. Moreover, we remark that

curl (u×e3) =∂3u and thanks to the divergence free condition,

curlb×e3=∂3b− ∇b3.

Since the proofs are based in taking the L²-scaler product either in (u, b) or in their associated dyadic blocs (∆^q_vu,∆^q_vb) (see section 2 for notation), terms like ∇f, for any function f, have no contribution in the L²-energy estimate. Thus, we can reformulate the (M HD_ν^ε_h) system to obtain the following noted also (M HD^εν_h)

(M HD^ε_νh)





















∂tu−νh∆hu+u· ∇u−b· ∇b+1 ε∂3b+1

εu×e3

=−∇P inR⁺×R³

∂tb−νh∆hb+u· ∇b−b· ∇u+1

ε∂3u= 0 in R⁺×R³ divu= 0 in R⁺×R³

divb= 0 in R⁺×R³ (u, b)|t=0= (u0, b0), where

P =p+b3.

This system can be written in the following abstract form

(S^ε)









∂tU+Q(U, U) +a2,h(D)U+L^ε(U) = ^t(−∇p,0) in R⁺×R³ divu= 0 in R⁺×R³

divb= 0 in R⁺×R³ U|t=0=U0

where

U = u

b

, the quadratic term Qis defined by

Q(U, U) =

u· ∇u−b· ∇b u· ∇b−b· ∇u , the viscous term is

a2,h(D)U =−νh∆hU and the linear perturbationL^ε is given by

L^ε(U) =1

εL(U) := 1 ε

∂3b+u×e3

∂3u .

We note thatL^ε is a skew-symmetric linear operator. This skew-symmetry is an important property for the existence results since the perturbation disappears in the energy estimate.

In the case of anisotropic diffusion process, existence, uniqueness and asymp- totic behavior of some magneto hydrodynamical systems have been studied by several authors ([1], [2], [11]). According to our knowledge, no paper is concerned by the anisotropic case. Nevertheless, anisotropic Navier-Stokes equation was studied in [3], where it was proved the local existence for ar- bitrary initial data and global existence for small initial data inH^0,s(R³) for s >1/2 and uniqueness for s > 3/2. In [8], the author closed the gap between existence and uniqueness and proved that uniqueness holds when existence does; that is fors >1/2.

In this paper, dealing with all coupled nonlinearities and singular per- turbation, we establish local well-posedness (existence and uniqueness) for arbitrary initial data, global existence for small initial data inH^0,s(R³) for s >1/2 and we investigate the asymptotic behavior of the solution as the Rossby number εtends to zero.

Precisely, we establish global existence for small initial data and local existence of solution on uniform time for arbitrary initial data. Namely, solution given by the following theorems.

Theorem 1.1. — Let s > 1/2 be a real number and U0 = (u0, b0) ∈ H^0,s(R³) such that divu0 = 0 and divb0 = 0. There exists a positif time T such that for all ε > 0, there exists a unique solution U^ε of (M HD^ε_νh) satisfying U^ε ∈ L^∞_T (H^0,s(R³)) and ∇hU^ε ∈ L²_T(H^0,s(R³)). Moreover, U^ε satisfies the following energy estimate

U^ε(t, .)²H^0,s(R³⁾+νh

t 0

∇hU^ε(τ, .)²H^0,s(R³⁾dτ U0²H^0,s. (1.1) Furthermore, there exists a constantcsuch that ifU0H^0,s(R³⁾cνh, then the solution is global.

To prove Theorem 1.1, we will rearrange the nonlinear terms in a suit- able way to apply Lemma 2.2. The proof, where the divergence free con- dition plies a crucial role, uses the energy method, an appropriate dyadic decomposition of the frequency space, compactness argument and classical analysis.

The following theorem gives an uniqueness result.

Theorem 1.2. — LetU^εandV^εbe two solutions of(M HD^ε_νh)on[0, T] belonging to

L^∞_T (H^0,s(R³))∩L²_T(H^1,s(R³)),

wheres >1/2. IfU^ε andV^ε have the same initial data, thenU^ε=V^ε. To prove Theorem 1.2, we will estimate theH^0,−1/2 norm of the differ- enceU^ε − V^εinstead of theH^0,1/2one. This will be done via Lemma 3.3 and standard arguments.

Once the existence and the uniqueness results are proved we turn to the asymptotic behavior of the solution when the initial data belongs toH^0,s(R³), s >2. This will be the aim of the following theorem.

Theorem 1.3. — Lets >2,U0= (u0, b0)∈H^0,s(R³)such that divu0= 0 and divb0= 0. Let (U^ε)be the family of solution given by Theorem 1.1, then

U^ε −−−→

ε−→0 0 in L²_T(C^−3/2(R³)). (1.2) That is, for χ inD(R),U_R^ε =χ(|∇h|/R))U^ε andU˜_R^ε =U^ε−U_R^ε, it holds

lim sup

ε→0 UR^εL⁴_T(L^∞) −−−−−→

R−→+∞0 (1.3)

lim sup

ε→0 U˜_R^εL²_T(H1−η,0∩H^0,s) −−−−−→

R−→+∞ 0, ∀s < sandη >0. (1.4) To prove Theorem 1.3, we will use a Strichartz type inequality and stan- dard Fourier analysis results. In fact, dispersive effects are of great impor- tance in the study of nonlinear partial differential equations, since they yield decay estimates on waves when the domain is the whole space R³, chosen here essentially for mathematical convenience.

2. Proof of existence results

2.1. Anisotropic Littlewood-Paley theory

To introduce the anisotropic Sobolev spaces, we use an anisotropic dyadic decomposition of the frequency space and we introduce, for any functiona, the following operators of localization in Fourier space :

∆^hja = F⁻¹(ϕ(2^−j|ξh|)F(a)) for j∈Z,

∆^v_qa = F⁻¹(ϕ(2^−q|ξ3|)F(a)) for q∈N,

∆^v₋₁a = F⁻¹(ϑ(|ξ3|)F(a))

and

∆^v_qa=0 for q−2.

The functionsϕandϑrepresent a dyadic partition of unity inR. This means that these functions are regular non-negative and satisfy

supp (ϑ)⊂B(0,4/3), supp (ϕ)⊂ C(0,3/4,8/3) and for allt∈R,

ϑ(t) +

q0

ϕ(2^−qt) = 1.

Here,

C(0,3/4,8/3) ={ξ, ξ∈R³, 3/4|ξ|8/3}. We define, also, for any functionu, the operatorsS_q^vuandS_j^huby

S_q^vu=

qq−1

∆^v_qu

and

S_j^hu=

jj−1

∆^h_ju.

By this way, we note that we consider an homogeneous decomposition in the horizontal variable and an inhomogeneous one in the vertical.

We define the corresponding Sobolev spaces by the following.

Definition 2.1. — Lets ands be two real numbers and ua tempered distribution. Let

uH^s,s =

j,q

2^2(js+qs)∆^hj∆^v_qu²L² 1/2

.

The space H^s,s(R³)is the closure of D(R³)for the above semi-norm.

The interest of the dyadic decomposition lies in the fact that any vertical derivative of a function localized in vertical frequencies of size 2^q acts as a multiplication by 2^q. We refer to [8] for a precise construction of anisotropic Sobolev spaces.

2.2. Global existence result for small initial data

In this section, the following lemma, a detailed proof of which is given in [3], will be useful to estimate the nonlinear part.

Lemma 2.2. — For any real numbers s0>1/2 andss0a constantC exists such that for any divergence free vectors fieldsaandb, we have

|(∆^v_q(a· ∇b)|∆^v_qb)L²| Cdq2^−2qsbH^1/2,s

aH^1/2,s0∇hbH^0,s

+aH^1/2,s∇hbH^0,s0 + ∇haH^0,s0bH^1/2,s+∇haH^0,sbH^1/2,s

where the positive numbersdq verify

q∈Z

dq = 1.

To prove Theorem 1.1, apply the operator ∆^v_q to (S^ε), denote Uq = (∆^v_qu^ε,∆^v_qb^ε) and use an L² energy estimate to obtain

1 2

d

dtU_q^ε(t)L²+νh∇hU_q^ε(t)L² ∆^v_q(u^ε· ∇u^ε)|u^ε_q

L² + ∆^v_q(u^ε· ∇b^ε)|b^ε_q

L² + ∆^v_q(b^ε· ∇b^ε)|u^εq

L²

+

∆^vq(b^ε· ∇u^ε)|b^εq

L².

In order to estimate the left hand side of the above inequality, rearrange the nonlinearity as

(b^ε· ∇b^ε|u^ε)L²+ (b^ε· ∇u^ε|b^ε)L² =

b^ε· ∇(u^ε+b^ε)|(u^ε+b^ε)

L²

− (b^ε· ∇b^ε|b^ε)L²−(b^ε· ∇u^ε|u^ε)L². Note that the H^s,s norm of any vector field is the Eucldiean norm of the H^s,s norms of its components. That is why

u^ε,b^εU^ε,

∇hu^ε,∇hb^ε∇hU^ε and so on.

Use Lemma 2.2, in the cases=s0, to deduce that 1

2 d

dtU_q^ε2L²+νh∇hU_q^ε2L²Cdq2^−2qs0U^ε2_H^1/2,s0∇hU^εH^1/2,s0. (2.1) Multiply by 2^2qs0 and take the sum overqto obtain

1 2

d

dtU^ε2_H^0,s0 +νh∇hU^ε2_H^0,s0 U^ε2_H1/2,s0∇hU^εH^1/2,s0.

An interpolation inequality yields

U^ε2_H1/2,s0 U^εH^0,s0∇hU^εH^0,s0.

So, one concludes that ifU0H^0,s0 is sufficiently small thenU^ε(t)²_H^0,s0 is a decreasing function. More precisely, if

U0H^0,s0 νh

2C then for any positive timet, one gets

U^ε(t)²H^0,s0+νh

t 0

∇hU^ε(τ)²H^0,s0dτ U0²H^0,s0. (2.2) Finally, a standard compactness argument gives the global existence result.

2.3. Local existence result for arbitrary initial data

Now, we turn to the local existence result of (M HD^ε_νh). Decompose the frequency space in order to investigate the high vertical frequencies then the low vertical-high horizontal ones.

For the high vertical frequencies term, one multiplies inequality (2.1) by 2^2qs0 then takes the sum for q greater than n−1 in order to obtain after integration in time

(Id−S_n^v)U^εL^∞_T(H^0,s0) + 2νh∇h(Id−S_n^v)U^ε2L²_T(H^0,s0)

C(Id−S_n^v)U^ε2H^0,s0 (2.3)

+ C

T 0

U^ε(t)²_H1/2,s0∇hU^ε(t)H^0,s0dt.

To study the low vertical-high horizontal frequency, letmandntow cutoff integers such that mnand defineU_m,n^ε by

U_m,n^ε =(Id−S_m^h)S_n^vU^ε.

Denote, respectively, byQ^h andQ^v the horizontal and the vertical part of the nonlinear term defined by

Q^h(U, U) =

uh· ∇hu−bh· ∇hb uh· ∇hb−bh· ∇hu

and

Q^v(U, U) =

u3·∂3u−b3·∂3b u3·∂3b−b3∂3u·

.

By standardL² energy estimate, one has 1

2 d

dtU_m,n^{ε 2}L²+νh∇hU_m,n^{ε 2}L² =

(Id−S_m^h)S_n^v(Q^h(U^ε, U^ε))|U_m,n^ε

L²

+

(Id−S_m^h)S_n^v(Q^v(U^ε, U^ε))|U_m,n^ε

L², where

(Id−S_m^h)S_n^v(Q^h(U^ε, U^ε))|U_m,n^ε

L² =

(Id−S_m^h)S_n^v(u^ε_h· ∇hu^ε)|u^ε_m,n

L²

−

(Id−S_m^h)S_n^v(b^ε_h· ∇hb^ε)|u^εm,n

L²

+

(Id−S_m^h)S_n^v(u^ε_h· ∇hb^ε)|b^ε_m,n

L²

−

(Id−Sm^h)Sn^v(b^εh· ∇hu^ε)|b^εm,n

L²

and

(Id−S_m^h)S_n^v(Q^v(U^ε, U^ε))|Um,n^ε

L² =

(Id−S_m^h)S_n^v(u^ε₃·∂3u^ε)|u^εm,n

L²

−

(Id−S_m^h)S_n^v(b^ε₃·∂3b^ε)|u^ε_m,n

L²

+

(Id−S_m^h)S_n^v(u^ε₃·∂3b^ε)|b^εm,n

L²

−

(Id−S_m^h)S_n^v(b^ε₃·∂3u^ε)|b^ε_m,n

L². By product law in Sobolev spaces, one infers that

(Id−S^hm)Sn^v(Q^h(U^ε, U^ε)(., x3))H−1/2(R²_xh⁾

CU_h^ε(., x3)H^1/2(R²_xh⁾∇hU^ε(., x3)L²(R²_xh⁾. Sinces0>1/2, one obtains

(Id−S^h_m)S_n^v(Q^h(U^ε, U^ε))_L2

Rx3;H^−1/2(R²⁾CU_h^εH^1/2,s0∇hU^εH^0,s0. By Cauchy-Schwarz inequality, one has

(Id−S_m^h)S_n^v(Q^h(U^ε, U^ε))|U_m,n^ε

L²(R³⁾CU^ε2_H^1/2,s0∇hU^εH^0,s0. To estimateQ^v(U^ε, U^ε), use the divergence free condition; that is

∂3u^ε₃=−divhu^ε_h and

∂3b^ε₃=−divhb^ε_h.

These equalities allow one to reformulate the vertical nonlinearity as Q^v(U^ε, U^ε) =Q^v₁(U^ε, U^ε) +Q^v₂(U^ε, U^ε),

where

Q^v1(U^ε, U^ε) =

∂3(u^ε₃u^ε)−∂3(b^ε₃b^ε)

∂3(u^ε₃b^ε)−∂3(b^ε₃u^ε)

and

Q^v₂(U^ε, U^ε) =

u^εdivhu^ε_h−b^εdivhb^ε_h b^εdivhu^ε_h−u^εdivhb^ε_h

. The nonlinear term,

(Id−S_m^v)S_n^v(Q^v₂(U^ε, U^ε))|U_m,n^ε

L² can be estimated exactly as the term

(Id−S_m^v)S^v_n(Q^h(U^ε, U^ε))|Um,n^ε

L². To study the other term, one observes that

Id−S_m^h = 2^−m(Id−S^h_m)(χ1(D)∂1+χ2(D)∂2),

where χ1 and χ2 are homogeneous functions of degree 0. Then, it is clear that, whenmn,

(Id−S_m^v)S_n^v(Q^v₁(U^ε, U^ε))|Um,n^ε

L² 2^n−mU3^εU^εL²(R³⁾∇hU_m,n^ε L²(R³⁾

CU^ε2_H1/2,s0∇hUm,n^ε H^0,s0

CU^ε2_H^1/2,s0∇hU^εH^0,s0. Finally, by integration in time, one has

Um,n^{ε 2}L^∞_T(H^0,s0) + 2νh∇hUm,n^{ε 2}L²_T(H^0,s0)

CUm,n^ε (0)²_H^0,s0 (2.4)

+ C

T 0

U^ε(t)²_H1/2,s0∇hU^ε(t)H^0,s0dt.

To conclude the proof of the local part, one begins by observing that since U0 belongs toL²(R³) the basicL² energy estimate holds

U^ε(t)²L²+ 2νh

t 0

∇hU^ε(τ)²L²dτ U0^ε2L². (2.5) Thus, it is obvious that

S_n^vS_m^hU^ε(t)²H^0,s0+2νh

t 0

∇hS_n^vS_m^hU^ε(τ)²H^0,s0dτ C2^2m+2ns0TU₀^ε2L². Using an interpolation inequality, estimations (2.3) and (2.4) one infers that

(2νh)^1/2U^ε2_L4

T(H^1/2,s0) +2νh∇hU^ε2_L2

T(H^0,s0)C

(Id−S_n^v)U₀^ε2_H0,s0

+S_n^v(Id−S_m^h)U₀^ε2_H^0,s0+ 2^2m+2ns0TU0²L²

+ T

0

U^ε(t)²_H1/2,s0∇hU^ε(t)H^0,s0dt .

It follows that ν_h^1/2U^ε2_L4

T(H^1/2,s0)+νh∇hU^ε2_L2

T(H^0,s0)C

(Id−S_n^v)U₀^ε2_H0,s0+ Sn^v(Id−S_m^h)U₀^ε2_H0,s0 + 2^2m+2ns0TU0²_L2+ C

νhU^ε(t)⁴_L4

T(H^1/2,s0) . Choose, in this order, an integer n, then an integer m greater than nand finally a strictly positive real numberT0such that

(Id−S_n^v)U₀^ε2H^0,s0 η, Sn^v(Id−S_m^h)U₀^ε2H^0,s0 η and

2^2m+2ns0T0U₀^ε2L² η to obtain, for anyT T0,

ν_h^1/2U^ε2_L4

T(H^1/2,s0)+νh∇hU^ε2_L²

T(H^0,s0)Cη+ C νh

U^ε(t)⁴_L4

T(H^1/2,s0). So, for any positive real numberη, such that

0< η < ν_h² 4C² a positive real numberT1 exists, such that

ν_h^1/2U^ε2_L⁴

T1(H^1/2,s0)+νh∇hU^ε2L²_T

1(H^0,s0)η.

That proves Theorem 1.1.

3. Proof of uniqueness result

In this section we denote byxthe quantity (1 +|x|²)^1/2and by Λ3

the operator defined by

Λ3=(1−∂₃²)^1/2,

that is, the operator of multiplication byξ3in the frequency space. Clearly, Λ3is an isometry fromH^s,s toH^s,s−1 for all real numberssands.

The following lemmas will be useful in the sequel. The first one and the second one are proved in [7], the third one is proved in [8].

Lemma 3.1. — Lets, t, s, t∈Randα∈[0,1]. Iff ∈H^s,s∩H^t,t then f belongs to H^{αs+(1−αt),αs+(1−αt)} and

f gHαs+(1−αt),αs+(1−αt) f^α_H^s,sg¹_H⁻t,t^α.

Lemma 3.2. — Let s, t <1,s+t >0 and s, t <1/2, s+t >0. Iff belongs to H^s,s and g belongs toH^t,t thenf g belongs to H^{s+t−1,s+t−1/2} and there exists a constant C such that

f gH^{s+t−1,s+t−1/2} CfH^s,sgH^t,t.

Lemma 3.3. — Lets, t <1,s+t >0ands >1/2. Iff belongs toH^s,s andg belongs toH^s,−1/2thenf g belongs toH^{s+t−1,−1/2} and there exists a constant C such that

f gH^s+t−1,−1/2CfH^s,sgH^t,−1/2.

The originality of Lemma 3.3, as said in [8], is to give a product law for the anisotropic Sobolev spaces where the regularities in the vertical direction are supercritical for one of the terms and subcritical for the other.

3.1. Proof of Theorem 1.2

Let U^ε = ^t(u^ε, b^ε) and V^ε = ^t(v^ε, c^ε) be two solutions of (M HD^ε_νh) satisfying Theorem 1.1 and having the same initial data. It is explained, in Remark 2, that this makes sense.

Denote byW^ε= ^t(w^ε, β^ε) the difference W^ε=U^ε−V^ε. One has



















∂tW^ε+νh∇hW^ε+Q(W^ε, W^ε+ 2U^ε) +1 εL(W^ε)

= ^t(−∇p^ε,0) in R⁺×R³ divw^ε = 0 in R⁺×R³ divβ^ε= 0 in R⁺×R³.

(3.1)

Without loss of generality, assume that s < 1. Suppose that U^ε and V^ε belong toC_T⁰(H^0,r) for allr < s, a fact on which we will return in Remark 1. By this assumption, one is able to multiply equation (3.1) by Λ⁻₃¹W^ε then integrate on (ε, t)×R³, letεtends to zero and use the continuity in time ofW^εH^0,−1/2 to obtain

W^ε2_H0,−1/2 + 2νh

t 0

∇hW^ε2_H0,−1/2dτ

= −2 t

0

Q(W^ε, W^ε+ 2U^ε)Λ⁻₃¹W^εdxdτ, (3.2) where

Q(W^ε, W^ε+ 2U^ε) =

u^ε· ∇w^ε+w^ε· ∇v^ε−b^ε· ∇β^ε−β^ε· ∇c^ε u^ε· ∇β^ε+w^ε· ∇c^ε−b^ε· ∇w^ε−β^ε· ∇v^ε

.

Here, note that for every multi-indexsand sufficiently smooth functionU,

one has

∂^αU L^ε(∂^αU^ε) = 0.

To estimate the left hand side of equation (3.2), one restricts itself to the two first terms, the same holds for the others since that

u^ε, b^εU^ε and so on.

Following [8], decompose the nonlinearity as follows

(u^ε· ∇w^ε+w^ε· ∇v^ε)·Λ⁻₃¹w^εdx= 4 i=1

Li, where

L1 =

u^ε_h· ∇hw^ε·Λ⁻₃¹w^εdx, L2 =

u^ε₃·∂3w^ε·Λ⁻₃¹w^εdx, L3 =

w^ε_h· ∇hv^ε·Λ⁻₃¹w^εdx and

L4=

w₃^ε·∂3v^ε·Λ⁻₃¹w^εdx.

To estimateL1, note that

|L1|u^ε_h· ∇hw^εH^{−1/2,−1/2}Λ⁻₃¹w^εH^1/2,1/2,

apply Lemma 3.3 to the first factor, use properties of Λ3 and Lemma 3.1 for the second to obtain

|L1|CU^εH^1/2,sW^ε1/2_H0,−1/2W^ε3/2_H1,−1/2. The same holds forL3, just begin by

|L3|w^εh· ∇hv^εH^{−3/4,−1/2}Λ⁻3¹w^εH^3/4,1/2

to obtain

|L3|CV^εH^1/2,sW^ε1/2_H0,−1/2W^ε3/2_H1,−1/2. ForL4, remark that

|L4|w^ε3∂3v^εH−1/2,(2s−3)/4Λ⁻3¹w^εH1/2,(3−2s)/4,