WELL-POSEDNESS IN GEVREY FUNCTION SPACE FOR THE PRANDTL EQUATIONS WITH NON-DEGENERATE CRITICAL POINTS

arXiv:1609.08430v3 [math.AP] 29 Aug 2017

EQUATIONS WITH NON-DEGENERATE CRITICAL POINTS

WEI-XI LI AND TONG YANG

Abstract. In the paper, we study the well-posedness of the Prandtl system without monotonicity and analyticity assumption. Precisely, for any indexσ∈[3/2,2],we obtain the local in time well-posedness in the space of Gevrey class G^σ in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. G´erard-Varet and N.

Masmoudi [Ann. Sci. ´Ec. Norm. Sup´er. (4) 48 (2015), no. 6, 1273-1325], in which the case σ= 7/4 is solved.

1. Introduction and main results

The Prandtl equations introduced by Prandtl in 1904 describe the behavior of the incompressible flow near a rigid wall at high Reynolds number:











∂_tu^P +u^P∂_xu^P +v^P∂_yu^P −∂_y²u^P +∂_xp= 0, t >0, x∈R, y >0,

∂_xu^P +∂_yv^P = 0,

u^P|y=0 =v^P|y=0 = 0, lim_y→+∞u=U(t, x), u^P|t=0 =u^P₀(x, y),

(1)

whereu^P(t, x, y) andv^P(t, x, y) represent the tangential and normal velocities of the boundary layer, with y being the scaled normal variable to the boundary, whileU(t, x) and p(t, x) are the values on the boundary of the tangential velocity and pressure of the outflow satisfying the Bernoulli law

∂_tU +U ∂_xU +∂_xp= 0.

We refer to [14, 17] for the mathematical derivation and background of this fundamental system in the field of boundary layer.

By using the divergence free condition, one can representv in terms of uso that the above system is reduced a scalar equation. Moreover, note that the above U and p are known functions coming from the outflow so that the Prandtl system is a degenerate parabolic mix-type equation with loss of derivative in the tangential direction x because of the termv∂_yu. In fact, this is the main difficulty in the study of this boundary layer system.

Up to now, the well-posedness on the Prandtl system is achieved in various function spaces. Pre- cisely, when the initial data satisfy the monotonic condition, that is, when the tangetial velocity is monotonic with respect to y, in the classical work by Oleinik and her collaborators, they obtained the local-in-time well-posedness by using Crocco transformation. And this result together with some of her other works were well presented in the monograph [17]. Recently, Alexandre-Wang-Xu-Yang [1] and Masmoudi-Wong [15] independently obtained the well-posedness in the Sobolev space by the virtue of energy method instead of the Crocco transformation, where the key observation in their proofs is the cancellation of the loss derivative terms. On the other hand, for the initial data without the monotonicity assumption, it is natural to perform estimate in the space of analytic functions, and

2010 Mathematics Subject Classification. 35Q30, 35Q31.

Key words and phrases. Prandtl boundary layer, non-degenerate critical points, Gevrey class.

1

in this context, the well-posedness results were achieved by Sammartino and Caflisch, after the earlier work of Asano [2]; cf also [16, 10] for the improvement. The first result that does not require mono- tonicity and analyticity was established by G´erard-Varet and Masmoudi [6] in which they obtained the well-posedness in the Gevrey space G^7/4. In fact, our paper is motivated by [6] and we give an affirmative answer to an open question raised in it. Also in the very recent work of Chen-Wang-Zhang [3], the well-posedness for the linearized Prandtl equation is studied in Gevrey spaceG^σ for any index 1≤σ <2.

Recall that the Gevrey class, denoted by G^σ, σ ≥ 1, is an intermediate function space between analytic functions and C^∞ functions. Note that the Gevrey space G^σ, σ > 1 contains compactly supported functions that are more physical, and this is the main difference from analytic functions.

We also refer to [12] for the smoothing effects in Gevrey space under the monotonicity assumption, and the global weak solutions by Xin-Zhang [19]. On the other hand, without the monotonicity assumption on the tangential velocity field, the degeneracy may cause strong instability so that the system is ill-posed in Sobolev spaces, cf. [4, 5, 11] and references therein.

Without the assumption on monotonicity and analyticity, in the recent interesting paper [6], the authors establishedG^7/4 well-posedness for Prandtl equation with non-degenerate critical points with respect to the normal variable, and they also conjectured the result should be valid for G². In this paper, we will give an affirmative answer to this conjecture. In fact, we show the well-posedness in all Gevrey space G^σ with σ ∈ [3/2,2] and this includes the case studied in [6]. In addition, we believe the well-posedness result can be extended, with some new technique such as subelliptic estimates, to σ ∈ [1,3/2]. Finally, as the aforementioned works, the present paper also aims at giving insight on the justification of inviscid limit for the Navier-Stokes equation with physical boundary, for this, we refer to [7, 8, 13] and the references therein for the recent progress.

To have a clear presentation, we will construct a solutions u^P that is a small perturbation around a shear flow, that is, u^P(t, x, y) =u^s(t, y) +u(t, x, y). For this, we suppose that the initial datau^P₀ in (1) can be written as

u^P₀(x, y) =u^s₀(y) +u0(x, y),

with u^s₀ being independent of x variable. Then we reduce the original Prandtl equation (1) to the following two time evolutional equations, one of which is the equation for the shear flow (u^s,0) with u^s solving











∂_tu^s−∂_y²u^s= 0, u^s

y=0= 0, lim

y→+∞u^s = 1, u^s

t=0 =u^s₀.

(2)

and the another reads











∂_tu+ (u^s+u)∂_xu+v∂_y(u^s+u)−∂_y²u= 0, u

y=0 = 0, lim

y→+∞u= 0, u

t=0=u₀,

(3)

where

v=− Z y

0

∂_xu(x,y)˜ d˜y.

Note the equation (2) is the heat equation and the well-posedness problem is well studied. In this paper, we assume that the initial datumu^s₀ in (2) admits non-degenerate critical points. Precisely, we impose

Assumption 1.1 (Assumption on the initial data u^s₀). There exists a y₀ >0 such that u^s₀ ∈C⁶(R+) satisfies the following properties (see Figure 1):

(i) ^du_dy^s0 (y₀) = 0 and ^d_dy^2u2^s0 (y₀)6= 0.Moreover, there exist 0< δ < y₀/2 and a constant c₀ such that

∀y ∈

y₀−2δ, y₀+ 2δ ,

d²u^s₀ dy² (y)

≥c₀. (ii) There exists a constant 0< c₁ <1 such that

∀ y∈

0, y₀−δ

∪

y₀+δ, +∞

, c₁hyi^−α ≤

du^s₀(y) dy

≤c⁻¹₁ hyi^−α for some α >1,and that

∀ y ≥0,

d^ju^s₀(y) dy^j

≤c⁻¹₁ hyi^−α−1 for 2≤j ≤6.

(iii) The compatibility condition holds, that is,u^s₀

y=0=∂_y²u^s₀

y=0= 0 andu^s₀(y)→1 as y→+∞.

Figure 1. An example ofu^s0

Remark 1.2. (i) For brevity of presentation, we only consider the case when the initial datum admits one non-degenerate critical point. The result can be generalized to the case when there are several non-degenerate critical points with slight modification.

(ii) The initial datumu^s₀here is not monotonic anymore. Note that in the work [1] the monotonicity condition is required to overcome the loss of derivative in thexvariable.

Proposition 1.3 (Well-posedness for the shear flow). Let the initial data u^s₀ satisfy the conditions in Assumption 1.1. Then there exists a constant Ts > 0 such that the heat equation (2) admits a unique solution u^s inC [0, T_s]; C⁶(R+)

. In addition, for anyt∈[0, T_s],we have, using the notation ω^s=∂_yu^s,

∀ y∈ y₀−7

4δ, y₀+7 4δ

, |∂_yω^s(t, y)| ≥c₀/2,

∀ y∈

0, y₀−5 4δ

∪ y₀+5

4δ, +∞

, 2⁻¹c₁hyi^−α≤ |ω^s(t, y)| ≤2c⁻¹₁ hyi^−α, and

∀y ≥0,

∂_y^jω^s(t, y)

≤2c⁻¹₁ hyi^−α−1 for 1≤j≤5.

Recall c₀, c₁, δ are the constants given in Assumption 1.1.

Observe that the solution to (2) has explicit representation by virtue of heat kernels. Then the above proposition follows from direct estimation. For brevity, we omit its proof and refer to Lemma 2.1 in the second version of [20] for detailed discussion. So it remains to solve (3), which is the main part of the paper. And we will solve the equation in the framework of Gevery space inx and Sobolev space in y.To state the main result, we first introduce the function spaces to be used.

Definition 1.4 (Gevrey space in tangential variable). Let α be the number given in Assumption 1.1, and let ℓbe a fixed number satisfying that

ℓ >3/2, α≤ℓ < α+ 1

2. (4)

With each pair (ρ, σ), ρ >0, σ≥1, we associate a Banach space X_ρ,σ, equipped with the normk·kρ,σ

that consists of all the smooth functions f such that kfkρ,σ <+∞,where kfkρ,σ

def= sup

m≥6

ρ^m−5 (m−6)!σ

hyi^ℓ−1∂_x^mf

L²+ sup

m≥6

ρ^m−5 (m−6)!σ

hyi^ℓ∂_x^m(∂yf) L²

+ sup

0≤m≤5

hyi^ℓ−1∂_x^mf L² +

hyi^ℓ∂_x^m(∂yf) L²

+ sup

1≤j≤4

i+j≥6

ρ^i+j−5 (i+j−6)!σ

hyi^ℓ+1∂_xⁱ∂^j_y(∂_yf)

L² + sup

1≤j≤4

i+j≤5

hyi^ℓ+1∂_xⁱ∂_y^j(∂_yf) L².

(5)

Remark 1.5. For the classical Gevrey spaceG^σ =∪L>0G^σ(L) inxvariable,f ∈G^σ(L) if the following estimates hold:

∀m≥0,

hyi^ℓ−1∂_x^mf L²+

hyi^ℓ∂_x^m(∂_yf)

L² ≤L^m+1(m!)^σ, and

∀i≥0, ∀1≤j≤4,

hyi^ℓ+1∂_xⁱ∂_y^j(∂_yf)

L² ≤L^i+j+1[(i+j)!]^σ.

The space X_ρ,σ given in Definition 1.4 is equivalent to the classical Gevrey spaceG^σ in the following sense. If f ∈X_ρ,σ for some ρ >0 then we can find a constant C such that kfkρ,σ ≤C. Thus direct calculation shows f ∈G^σ(L) if we choose

L= 1

ρ + sup

0≤m≤5

hyi^ℓ−1∂^m_x f L² +

hyi^ℓ∂_x^m(∂_yf) L²

+ sup

1≤j≤4

i+j≤5

hyi^ℓ+1∂_xⁱ∂^j_y(∂_yf)

L² +C.

Conversely, iff ∈G^σ(L), thenf ∈X_ρ,σ, providedρ is chosen in such a way that

∀m≥6, L^m+1(m!)^σ ≤ρ^−(m−5)[(m−6)!]^σ.

In view of the definitionk·kρ,σ,we see the the order ofyderivatives is at most 5.Then, if the equation (3) is well-posed inX_ρ,σ, the initial datau₀should satisfy the following compatibility conditions, using the notation ω₀ =∂_yu₀,

(u₀|y=0 =∂_yω₀|y=0= 0,

∂_y³ω₀

y=0= (ω₀^s+ω₀)∂_xω₀

y=0. (6) Now we state the main result in this paper as follows.

Theorem 1.6. For σ∈[3/2,2], let the initial datum u₀ in (3)belong to X_2ρ0,σ for some ρ₀>0 and moreover

ku₀k2ρ0,σ≤η₀

for some η₀ >0.Suppose that the compatibility condition (6)holds for u₀. Then (3) admits a unique solutionu∈L^∞ [0, T]; X_ρ,σ

for some T >0and some 0< ρ <2ρ₀,provided η₀ is sufficiently small.

Remark 1.7. (i) For clear presentation, we consider the solution as a perturbation around the shear flow. In fact, the method can be applied to the general periodic case studied in [6], and we will clarify further in Section 9 why our result holds for the general case without requiring the small perturbation around a shear flow.

(ii) As pointed out in [6], it is natural, inspired by [5], to ask whether theσ = 2 is the critical Gevrey index for the well-posedness for Prandtl equation.

The methodologies. At the end of the introduction, we will present the main methodologies used in the proof.

(i) After applying∂_x^m to the equation (3) for the velocity, the main difficulty arises from the term (∂^m_x v) (ω^s+ω),

which results in the lost of derivative in x variable. Under Oleinik’s monotonicity assumption on the tangential velocity field, this can be overcome by using the cancellation introduced by AWXY [1] and Masmoudi-Wong [15]. In fact, this cancellation method works at least in the domain where u^s+u admits monotonicity. Precisely, we apply∂_x^m to the equation for the vorticity ω=∂_yu

∂tω+ (u^s+u)∂xω+v∂y(ω^s+ω)−∂_y²ω= 0,

in which the most difficult term is (∂_x^mv) (∂yω^s+∂yω). To capture the cancellation, one can work on the function, introduced in [6],

f_m =χ₁∂_x^mω−χ₁∂_yω^s+∂_yω

ω^s+ω ∂_x^mu, m≥1, whereχ₁ is a smooth function supported in the monotonic region.

(ii) As for the domain near the critical points, we do not have the monotonicity anymore. One of the new observations in this paper is that we can also apply the cancellation to the equation for the vorticity and the equation for∂_yω

∂_t(∂_yω) +u∂_x(∂_yω) +v∂_y(∂_yω)−∂_y²(∂_yω) =−(ω^s+ω)∂_xω+ (∂_yω^s+∂_yω)∂_xu, by using another auxilliary function

h_m=χ₂∂_x^m∂_yω−χ₂∂_y²ω^s+∂_y²ω

∂_yω^s+∂_yω∂_x^mω,

whereχ2 is a cut-off function compactly supported in the region admitting the non-degenerate critical points. However, even with this, we also have the loss of xderivative because

gm+1 def

= ∂_x^m

(ω^s+ω)∂xω−(∂yω^s+∂yω)∂xu

appears in the equation forh_m.Nonetheless, we can use again the cancellation method to the equations for the velocity and the vorticity, to obtain a equation forg_m+1.Precisely, we apply∂_xto the equations for velocity uand for vorticityω =∂_yu, and then multiply respectively the obtained equations by the factors ∂_yω^s+∂_yω and ω^s+ω respectively, and finally subtract one from another. We then obtain the equation forg₁^def= (ω^s+ω)∂_xω−(∂_yω^s+∂_yω)∂_xu as follows.

∂_t+ (u^s+u)∂_x+v∂_y−∂_y²

g₁= 2 ∂_y²ω^s+∂_y²ω

∂_xω−2 (∂_yω^s+∂_yω)∂_x∂_yω.

Note that the order ofx derivative for terms on right hand side is equal to 1 that is the same as in the representation of g1.The above equation allows us to perform estimation on gm+1 =∂_x^mg1 in Gevrey norm by standard energy method.

(iii) From the above procedure, we have the upper bound, by energy method, for the auxilliary functionsf_mandh_m.It remains to control the original∂_x^muand∂_x^mωas well as the mixed derivatives, in terms of the auxilliary functions. This is clear when there is no cut-off functions χ_i involved, by

virtue of the Hardy-type inequality (see [15] for instance under the monotonicity assumption). In case considered in this paper, we first follow the cancellation idea used in [6], by taking L² inner product with _∂^χ2∂mxω

yω^s+∂yω on both sides of the equation for ∂_x^mω, to obtain the estimate on χ2∂^m_x ω.

Then by using the representation of h_m, we can derive similar estimate on χ₂∂_x^m∂_yω from those on h_m.Roughly speaking, this impliesχ₂∂_x^m∂_yω behaves similarly as the terms with morder derivatives involved, rather than them+ 1 order of mixed derivatives in Definition 1.4. And this is the advantage of the new auxilliary function h_m introduced in this paper and this enables us to extend the well- posedness of the Prandtl system from the Gevrey indexσ = 7/4 obtained in [6] to σ∈[3/2,2].

The rest of the paper is organized as follows. Section 2-6 are devoted to the proof of the uniform estimate in Gevrey norm for the approximate solutions to a regularized Prandtl equation. In Section 7, we will give the proof of existence of the regularized Prandtl equation and in Section 8 we will prove the main result of this paper. We explain in Section 9 why the main result in this paper holds for the general initial data rather than the small perturbations around a shear flow. The proofs of some technical lemmas will be given in the Appendix.

2. Regularized Prandtl equation and uniform estimates in Gevrey norm

In this section as well as Sections 3-7, we will study the initial-boundary problem for the following regularized Prandtl type equation of (3) by recallingu^s given in Proposition 1.3,











∂tuε+ (u^s+uε)∂xuε+vε∂y(u^s+uε)−∂²_yuε−ε∂_x²uε= 0, uε

y=0= 0, lim

y→+∞uε= 0, u_ε

t=0=u₀,

(7)

where ε > 0 is an arbitrarily small number and v_ε = −R_y

0 ∂_xu_ε(x,y)˜ d˜y. We remark the regularized equation above shares the same compatibility condition (6) as the original one (3).

The existence of solutions to (7) will be given in Section 7, where the life span T_ε^∗ may depend on the ε. Thus in order to obtain the solution to the original equation by letting ε→ 0,we need an uniform estimate, for example, in the Gevrey norm for u_ε, that will be stated in this section with the proof given in Sections 3-6. To simplify the notations, we will use the notations ωε =∂yuε and ω^s=∂_yu^s from now on.

Throughout the paper, we will work on those solutions u_ε that the properties listed in Proposition 1.3 foru^s can be preserved byu^s+u_ε.Precisely, we suppose that the solution u_ε∈L^∞([0, T]; X_ρ0,σ) to (7) has the following properties. For any t∈[0, T] and any x∈R,we have























|∂_yω^s(t, y) +∂_yω_ε(t, x, y)| ≥ c₀

4, if y∈ y₀−7

4δ, y₀+ 7 4δ

, 4⁻¹c₁hyi^−α ≤ |ω^s(t, y) +ω_ε(t, x, y)| ≤4c⁻¹₁ hyi^−α, if y∈

0, y₀−5 4δ

∪ y₀+ 5

4δ, +∞ ,

|∂_yω^s(t, y) +∂_yω_ε(t, x, y)| ≤4c⁻¹₁ hyi^−α−1 for y≥0, X

1≤j≤2

hyi^ℓ−1∂_x^ju_ε

L^∞+

∂_x^j−1v_ε L^∞+

hyi^ℓ∂_x^jω_ε L^∞

+ X

1≤i,j≤2

hyi^ℓ+1∂_xⁱ∂_y^jω_ε

L^∞ ≤1, (8)

wherec₀, c₁ andδ are the constants given in Assumption 1.1.

According to the properties (8) above, we can divide the normal directiony≥0 into two parts, one is near the critical points of u^s+u_ε, and another one is away from the critical points where u^s+u_ε admits the monotonicity condition. That is, we can find two non-negative C^∞ smooth functions χ₁ and χ₂ depending only ony such that

0≤χ₁≤1, χ₁≡1 on

− ∞, y₀− 3 2δ

∪ y₀+3

2δ, +∞

, χ₁ ≡0 on y₀−5

4δ, y₀+5 4δ

, (9)

and

0≤χ₂≤1, χ₂≡1 on y₀−3

2δ, y₀+ 3 2δ

, suppχ₂⊂ y₀−7

4δ, y₀+7 4δ

. (10)

From the properties listed in (8), it follows that |ω^s+ω| > 0 on suppχ1, and |∂yω^s+∂yω| > 0 on suppχ₂.Moreover,

χ^′₁ =χ^′₁χ2, χ^′₂ =χ^′₂χ1, and (1−χ2) = (1−χ2)χ1, (11) becauseχ₂≡1 on suppχ^′₁, χ₁≡1 on suppχ^′₂,and χ₁ ≡1 on supp (1−χ₂).Here and throughout the paper, f^′ and f^′′ stand for the first and the second order derivatives off.

Definition 2.1. Let χ₁ andχ₂ given above and letu_εsatisfy the properties (8). Form≥1,we define three auxilliary functions fm,ε, hm,ε and gm,ε according to the cancellation property:

f_m,ε=χ₁∂_x^mω_ε−χ₁∂_yω^s+∂_yω_ε

ω^s+ω_ε ∂_x^mu_ε=χ₁(ω^s+ω_ε)∂_y

∂_x^mu_ε ω^s+ω_ε

, (12)

h_m,ε=χ₂∂_x^m∂_yω_ε−χ₂∂_y²ω^s+∂_y²ω_ε

∂_yω^s+∂_yω_ε∂_x^mω_ε, (13) and

g_m,ε=∂_x^m−1

(ω^s+ω_ε)∂_xω_ε−(∂_yω^s+∂_yω_ε)∂_xu_ε

. (14)

Definition 2.2. Let X_ρ,σ be given in Definition 1.4, equipped with the normk·kρ,σ defined by (5). Let χ₁, χ₂ be given by (9)-(10), and let u_ε satisfy the properties listed in (8). We will use the notation

|·|ρ,σ which is given by

|u_ε|ρ,σ = ku_εkρ,σ+ sup

1≤m≤5

mkg_m,εk_L² +

hyi^ℓf_m,ε

L² +kh_m,εk_L²+kχ₂∂_y∂_x^mω_εk_L² + sup

m≥6

ρ^m−5 (m−6)!σ

mkg_m,εk_L²+

hyi^ℓf_m,ε

L²+kh_m,εk_L² +kχ₂∂_y∂_x^mω_εk_L² . Similarly we can define |u0|ρ,σ.

Remark 2.3. (i) Observe there is an extra factormin front of the termkgm,εkL² in the definition of the norm|·|ρ,σ.

(ii) Direct calculation shows that

ku_εk_ρ,σ ≤ |u_ε|_ρ,σ ≤C_ρ,ρ^∗ ku_εk_ρ∗,σ+ku_εk²_ρ∗,σ

(15) for anyρ < ρ^∗,with C_ρ,ρ^∗ being a constant depending only on the differenceρ^∗−ρ.

Theorem 2.4 (uniform estimates in Gevrey space). Let 3/2 ≤ σ ≤ 2. Let uε ∈ L^∞([0, T]; Xρ0,σ) be a solution to (7) such that the properties listed in (8) hold. Then there exists a constant C∗ >1, independent of εand the solution u_ε, such that the estimate

|u_ε(t)|²ρ,σ ≤C∗|u₀|²ρ,σ +C∗

Z t 0

|u_ε(s)|²ρ,σ+|u_ε(s)|⁴ρ,σ

ds+C∗

Z t 0

|u_ε(s)|²ρ,σ˜

˜

ρ−ρ ds (16) holds for any pair (ρ,ρ)˜ with 0 < ρ < ρ < ρ˜ ₀, and for any t ∈ [0,T˜], where [0,T]˜ is the maximal interval of existence for |u_ε|ρ,σ˜ <+∞.

The above theorem is the key part of the paper, and its proof follows from the discussion in Sections 3 to 6.

3. Proof of Theorem 2.4: uniform estimate on g_m

This section along with Sections 4-6 are devoted to proving Theorem 2.4, the uniform estimates for the approximate solutions u^ε. To simplify the notations, we will remove in the following discussion the subscriptε in u_ε, ω_ε if no confusion occurs. Similarly, we write f_m, h_m and g_m for the auxilliary functionsfm,ε, hm,εand gm,ε defined in (12)-(14). Moreover, we will use the capital letterC to denote some generic constants, which may vary from line to line that depend only on the constants c_j, δ, ρ₀ and α in Assumption 1.1 as well as on the Sobolev embedding constants, but are independent of ε and the order m of derivatives.

We begin with a uniform estimate on g_m=g_m,ε withg_m,ε defined by (14), that is, gm=∂_x^m−1

(ω^s+ω)∂xω−(∂yω^s+∂yω)∂xu

, m≥1. (17)

The main result in this section can be stated as follows.

Proposition 3.1. Letm≥6and letu∈L^∞([0, T]; X_ρ0,σ) be a solution to (7)under the assumptions in Theorem 2.4. Then for any pair (ρ,ρ)˜ with 0< ρ <ρ˜≤ρ₀ and for any smallt∈[0, T], we have

m²kg_m(t)k²L² ≤ C

(m−6)!2σ

ρ^2(m−5) |u₀|²ρ,σ+C Z t

0

ε

∂_x^m+1u

2 L²+ε

∂_x^m+1ω

2 L²

ds +Cm²

Z t

0 k∂yfm−1k²L²ds+C

(m−6)!2σ

ρ^2(m−5)

Z t 0

|u(s)|²ρ,σ+|u(s)|⁴ρ,σ

ds+

Z t 0

|u(s)|²ρ,σ˜

˜

ρ−ρ ds

! . 3.1. Preliminaries. Before proving Proposition 3.1, we first list some inequalities used throughout the paper.

Lemma 3.2 (Some inequalities). (i) Given any non-negative integers p and q, we have p!q!≤(p+q)!.

(ii) We have |·|ρ,σ ≤ |·|ρ,σ˜ for ρ≤ρ.˜

(iii) For any integer k≥1 and for any pair (ρ,ρ)˜ with 0< ρ <ρ˜≤1, we have k

ρ

˜ ρ

k

≤ 1

˜ ρk

ρ

˜ ρ

k

≤ 1

˜

ρ−ρ. (18)

(iv) Let χ₂ be given in (10) and let σ≥1.Then for any 0< r≤1, we have

hyi^ℓ−1∂_x^mu L² +

hyi^ℓ∂_x^mω L² ≤







(m−6)!σ

r^(m−5) |u|r,σ, if m≥6,

|u|r,σ, if 0≤m≤5,

(19) and

mkg_mk_L² +

hyi^ℓf_m

L² +kh_mk_L²+kχ₂∂_y∂_x^mωk_L² ≤







(m−6)!σ

r^(m−5) |u|r,σ, if m≥6,

|u|r,σ, if 1≤m≤5,

(20) and

hyi^ℓ+1∂_xⁱ∂_y^jω L² ≤







(i+j−6)!σ

r^(i+j−5) |u|r,σ, if i+j≥6 and 1≤j≤4,

|u|r,σ, if 0≤i+j ≤5 and 1≤j≤4.

(21) (v) Let σ≥1 and letm≥7.Then for any 0< r≤1 we have

∂_x^m−1∂_yω

L² ≤ k∂_yf_m−1k_L² +C

(m−7)!σ

r^m−6 |u|r,σ ≤ k∂_yf_m−1k_L²+Cm^−σ

(m−6)!σ

r^m−5 |u|r,σ. (22)

Proof. The first statement (i) is clear. The second and the fourth statements (ii) and (iv) follow directly from the definition of |·|ρ,σ (See Definition 2.2). As for (iii), we have for any k≥1 and any pair (ρ,ρ) with 0˜ < ρ <ρ˜≤1,

1 1− ^ρ_ρ˜ =

∞

X

j=0

ρ

˜ ρ

j

≥k ρ

˜ ρ

k

, from which the desired inequalities follow.

Now we prove (v). In view of (11) we see χ₁ ≡1 on supp 1−χ₂ so that

∂_x^m−1∂_yω

L² ≤

(1−χ₂)∂_y χ₁∂_x^m−1ω L² +

χ₂∂_y∂^m−1_x ω L²

≤ k(1−χ₂)∂_yf_m−1k_L² +

(1−χ₂)∂_y χ₁∂_yω^s+∂_yω

ω^s+ω ∂_x^m−1u L²

+

χ₂∂_y∂_x^m−1ω L²

≤ k∂yfm−1kL² +C

∂^m−1_x u

L²+C

∂_x^m−1ω L²+

χ2∂y∂_x^m−1ω L².

In the above, the second inequality uses (12), the definition off_m. This along with (19) and (20) yield

∂_x^m−1∂_yω

L² ≤ k∂_yf_m−1k_L² +C

(m−7)!σ

r^m−6 |u|r,σ≤ k∂_yf_m−1k_L² +Cm^−σ

(m−6)!σ

r^m−5 |u|r,σ.

The proof is then completed.

Letgm be given in (17), and we define its key component ˜gm by setting

˜

g_m= (ω^s+ω)∂_x^mω−(∂_yω^s+∂_yω)∂_x^mu, m≥1. (23) The next lemma is concerned with the difference g_m−˜g_m.

Lemma 3.3. Let m≥6 and let 1≤σ≤2.We have kg_m−g˜_mkL² ≤Ck∂_yf_m−1k_L²+Cm^1−σ

(m−6)!σ

˜

ρ^m−5 |u|ρ,σ˜ +Cm^2−2σ

(m−6)!σ

ρ^m−5 |u|²ρ,σ. Proof. First, direct calculation shows

g_m−g˜_m =

m−1

X

j=1

m−1 j

∂_x^jω

∂_x^m−jω−

m−1

X

j=1

m−1 j

∂_x^j∂_yω

∂_x^m−ju. (24) Thus

kg_m−˜g_mkL² ≤

m−1

X

j=1

(m−1)!

j!(m−1−j)!

∂_x^jω

∂_x^m−jω L² +

m−1

X

j=1

(m−1)!

j!(m−1−j)!

∂_x^j∂_yω

∂_x^m−ju L². We first handle the second term on the right side of the above estimate, and write

m−1

X

j=1

(m−1)!

j!(m−1−j)!

∂_x^j∂_yω

∂_x^m−ju

L² ≤R₁+R₂, where

R₁ =

[m/2]

X

j=1

(m−1)!

j!(m−1−j)!

∂_x^j∂_yω L^∞

∂_x^m−ju L²

with [m/2] standing for the largest integer less than or equal to m/2,and R₂=

m−1

X

j=[m/2]+1

(m−1)!

j!(m−1−j)!

∂_x^j∂_yω L²

∂_x^m−ju L^∞.

To estimateR₂, we use (19) and (21) along with the Sobolev inequality (see Lemma A.1 in Appendix), to compute

R₂ ≤

∂_x^m−1∂_yω

L²k∂_xukL^∞+m

∂_x^m−2∂_yω L²

∂_x²u

L^∞+

m−3

X

j=m−4

(m−1)!

j!(m−1−j)!

(j−5)!σ

ρ^j−4 |u|²ρ,σ

+

m−5

X

j=[m/2]+1

(m−1)!

j!(m−1−j)!

(j−5)!σ

ρ^j−4

(m−j−5)!σ

ρ^m−j−4 |u|²ρ,σ. Moreover, by (21) and the last inequality in (8), we have

m

∂_x^m−2∂_yω L²

∂_x²u

L^∞ ≤Cm^1−σ

(m−6)!σ

˜

ρ^m−5 |u|ρ,σ˜ . Direct computation gives

m−3

X

j=m−4

(m−1)!

j!(m−1−j)!

(j−5)!σ

ρ^j−4 |u|²ρ,σ ≤Cm^2−2σ

(m−6)!σ

ρ^m−5 |u|²ρ,σ, and, using the statement (i) in Lemma 3.2,

m−5

X

j=[m/2]+1

(m−1)!

j!(m−1−j)!

(j−5)!σ

ρ^j−4

(m−j−5)!σ

ρ^m−j−4 |u|²ρ,σ

≤ C 1

ρ^m−5|u|²ρ,σ m−5

X

j=[m/2]+1

(m−1)!

(j−5)!σ−1

(m−j−5)!σ−1

j⁵(m−j)⁴

≤ C 1

ρ^m−5|u|²_ρ,σ

m−3

X

j=[m/2]+1

(m−6)!

(m−10)!σ−1

m⁵ j⁵(m−j)⁴

≤ Cm^4−4σ

(m−6)!σ

ρ^m−5 |u|²ρ,σ.

Finally, for any small κ >0, we use (22) and the last inequality in (8) to obtain ∂_x^m−1∂_yω

L²k∂_xuk_L^∞ ≤Ck∂_yf_m−1k_L² +Cm^−σ

(m−6)!σ

˜

ρ^m−5 |u|_ρ,σ˜ . Combining the inequalities above we conclude

R₂≤Ck∂_yf_m−1k_L²+Cm^1−σ

(m−6)!σ

˜

ρ^m−5 |u|ρ,σ˜ +Cm^2−2σ

(m−6)!σ

ρ^m−5 |u|²ρ,σ.

The estimation on R1 is similar as above with simpler so that we omit it for brevity. Then we have R₁≤Cm^1−σ

(m−6)!σ

˜

ρ^m−5 |u|ρ,σ˜ +Cm^2−2σ

(m−6)!σ

ρ^m−5 |u|²ρ,σ.

Thus the desired estimate follows and the proof of Lemma 3.3 is completed.

3.2. Proof of Proposition 3.1. The rest of this section is devoted to proving Proposition 3.1 by energy method, and the proof is inspired by the arguments used in [6]. To do so, we first write the

equation forg_mas follows with its derivation given in the Appendix (see Lemma B.3 in the Appendix).

∂_t+ (u^s+u)∂_x+v∂_y−∂_y²−ε∂_x² g_m

= −

m−1

X

j=1

m−1 j

∂_x^ju

gm−j+1−

m−1

X

j=1

m−1 j

∂^j_xv

∂ygm−j

+2

m−1

X

j=0

m−1 j

∂_x^j∂_y²ω^s+∂_x^j∂_y²ω

∂_x^m−jω+ 2ε

m−1

X

j=0

m−1 j

∂_x^j+1∂_yω

∂_x^m−j+1u

−2

m−1

X

j=0

m−1 j

∂_x^j∂_yω^s+∂_x^j∂_yω

∂^m−j_x ∂_yω−2ε

m−1

X

j=0

m−1 j

∂_x^j+1ω

∂_x^m−j+1ω, Moreover, observe that∂_yω^s|y=0 =∂_yω|y=0 = 0 and then

∂_yg_m|y=0 = 0.

Thus multiplying both sides bym²g_m and then taking integration over R²+, we have 1

2m²kg_m(t)k²_L²+m² Z _t

0 k∂_yg_m(s)k²_L² ds+εm² Z _t

0 k∂_xg_mk²_L² ds≤ 1

2m²kg_m(0)k²_L²+ X

1≤i≤6

Pi (25) with

P1 = − Z _t

0

m−1

X

j=1

m−1 j

∂^j_xu

g_m−j+1, m²g_m

L²

ds,

P2 = − Z _t

0

m−1

X

j=1

m−1 j

∂^j_xv

∂_yg_m−j, m²g_m

L²

ds,

P³ = Z t

0

2

m−1

X

j=0

m−1 j

∂_x^j∂_y²ω^s+∂_x^j∂_y²ω

∂_x^m−jω, m²gm

L²

ds,

P4 = − Z _t

0

2

m−1

X

j=0

m−1 j

∂_x^j∂_yω^s+∂_x^j∂_yω

∂_x^m−j∂_yω, m²g_m

L²

ds,

P5 = Z _t

0

2ε

m−1

X

j=0

m−1 j

∂_x^j+1∂_yω

∂_x^m−j+1u, m²g_m

L²

ds,

P6 = − Z _t

0

2ε

m−1

X

j=0

m−1 j

∂_x^j+1ω

∂_x^m−j+1ω, m²g_m

L²

ds.

In the following lemmas, we will estimate Pi,1≤i≤6 respectively.

Lemma 3.4 (Estimate onP3). Let 3/2 ≤ σ ≤ 2. Then for any small κ >0 and for any pair (ρ,ρ)˜ with 0< ρ <ρ,˜ we have

P3 ≤ κm² Z _t

0 k∂_yg_mk²_L²ds+Cκ⁻¹m² Z _t

0 k∂_yf_m−1k²_L²ds +Cκ⁻¹

(m−6)!2σ

ρ^2(m−5)

Z t 0

|u(s)|²ρ,σ+|u(s)|⁴ρ,σ

ds+

Z t 0

|u(s)|²ρ,σ˜

˜

ρ(s)−ρds

! .

Proof. We write

P³ = 2 Z t

0

m−1

X

j=0

m−1 j

∂_x^j∂_y²ω^s+∂_x^j∂_y²ω

∂_x^m−jω, m²gm

L²

dt≤J1+J2+J3+J4

with

J₁ = 2m Z _t

0

∂_y²ω^s+∂_y²ω

L^∞k∂_x^mωkL²(mkg_mkL²) dt, J₂ = 2m(m−1)

Z _t

0

∂_x∂_y²ω L^∞

∂_x^m−1ω

L²(mkg_mkL²) dt, J₃ = 2m

[m/2]

X

j=2

(m−1)!

j!(m−1−j)!

Z _t

0

∂^j_x∂_y²ω L^∞

∂^m−j_x ω

L²(mkg_mkL²)dt, J₄ = 2

m−1

X

j=[m/2]+1

m−1 j

Z t 0

∂_x^j∂_y²ω

∂_x^m−jω, m²g_m

L²

.

Estimate on J₁ and J₂: ForJ₁, we use the third estimate in (8) as well as (20) to obtain

J₁ ≤ C

(m−6)!2σ

ρ^2(m−5)

Z _t

0 |u(s)|²ρ,σ˜

mρ^2(m−5)

˜

ρ^2(m−5) ds≤ C

(m−6)!2σ

ρ^2(m−5)

Z _t

0

|u(s)|²ρ,σ˜

˜

ρ(s)−ρds, where the last inequality follows from (18) in Lemma 3.2. Similarly,

J2≤ C

(m−6)!2σ

ρ^2(m−5)

Z t

0 |u(s)|²ρ,σ˜

m^2−σρ^2(m−5)

˜

ρ^2(m−5) ds≤ C

(m−6)!2σ

ρ^2(m−5)

Z t 0

|u(s)|²ρ,σ˜

˜

ρ(s)−ρds, where the last inequality follows from the fact that σ≥3/2≥1.

Estimate on J₃: By using the statement (iv) in Lemma 3.2 as well as the Sobolev inequality (see Lemma A.1 in the Appendix), we have

J₃ ≤ Cm

[m/2]

X

j=2

(m−1)!

j!(m−1−j)!

(j−2)!σ

ρ^j−1

(m−j−6)!σ

ρ^m−j−5

(m−6)!σ

ρ^m−5

Z t

0 |u(s)|³ρ,σ ds

≤ Cm

(m−6)!σ+1

ρ^2(m−5)

[m/2]

X

j=2

m⁵

(j−2)!σ−1

(m−j−6)!σ−1

j²(m−j)⁵

Z t

0 |u(s)|³_ρ,σ ds

≤ Cm

(m−8)!σ−1

(m−6)!σ+1

ρ^2(m−5)

Z t

0 |u(s)|³_ρ,σ ds

[m/2]

X

j=2

1

j² (using (i) in Lemma 3.2)

≤ C

(m−6)!2σ

ρ^2(m−5)

Z t

0 |u(s)|³ρ,σ ds,

where in the last inequality, we have usedm^−2(σ−1) ≤m⁻¹ becauseσ ≥3/2.