FENG CHENG, WEI-XI LI, AND CHAO-JIANG XU

Transcription

1 GEVERY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE arxiv: v2 [mathap] 23 Nov 2016 FENG CHENG WEI-XI LI AND CHAO-JIANG XU Abstract In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid The method presented here is based on the basic weighted - estimate the main difficulty arises from the estimate on the pressure term due to the appearance of weight function 1 Introduction In this paper we study the Gevrey propagation of solutions to incompressible Euler equation Gevrey class is a stronger concept than the C -smoothness In fact it is an intermediate space between analytic space C space There have been extensive mathematical investigations (cf[2] [7] [8] [9] [10] [11] [13] [14] [15] for instance the references therein on Euler equation in different kind of frames such as Sobolev space analytic space Gevrey space In this work we will consider the problems of Gevrey regularity with weight this is motivated by the study of the vertical boundary layer problem introduced in [1] which remains still open up to now The related preliminary work for the well-posedness of vertical boundary problem is to establish the Gevrey regularty with weight this is the main result of the present paper In the future work we hope to investigate the vertical boundary layer problem basing on the weighted Gevrey regularity of Euler equations Without loss of generality we consider the incompressible Euler equation in half plane R 2 + where R2 + = {(xy;x Ry R+ } our results can be generalized to 3-D Euler equation The velocity (u(txyv(txy the pressure p(txy satisfy the following equation: t u+u x u+v y u+ x p = 0 in R 2 + (0 ( Mathematics Subject Classification 35M33 35Q31 76N10 Key words phrases Gevery class incompressible Euler equation weight Sobolev space 1

2 2 F CHENG W-X LI AND C-J XU with initial data t v +u x v +v y v + y p = 0 in R 2 + (0 (12 x u+ y v = 0 in R 2 + (0 (13 v y=0 = 0 in R (0 (14 u t=0 = u 0 v t=0 = v 0 on R 2 + {t = 0} (15 Here the initial data (u 0 v 0 satisfy the compatibility condition: x u 0 + y v 0 = 0; v 0 y=0 = 0 Before stating our main result we first introduce the (global weighted Gevrey space Definition 11 Let l x l y 0 be real constants that independent of xy we say that f G s τl xl y (R 2 + if sup α 0 τ α x l x α! s y ly α f < L2 (R 2 + where throughout the paper we use the notation = ( In this work we present the persistence of weighted Gevrey class regularity of the solution ie we prove that if the initial datum (u 0 v 0 is in some weighted Gevrey space satisfy the compatible condition then the global solution belongs to the same space With only minor changes these results can also extend to 3-D Euler equation the global solution here will be replaced by a local solution Precisely Theorem 12 Suppose the initial data u 0 G s τ 00l y v 0 G s τ 0l x0 for some s 1τ 0 > 0 0 l x l y 1 Then the Euler equation (11-(15 admits a solution uvp such that u ([0+ ; G s τ0l y v ([0+ ; G s τl x0 p satisfies x p ([0+ ; G s τ0l y y p ([0+ ; G s τl x0 where τ > 0 depends on the initial radius τ 0 We remark that the existence of smooth solutions to (11-(15 is well developed (cf[2] [7] [10] [13] [15] for instance in two-dimensional case smooth initial data can yield global solutions while in the three-dimensional case the solution may be local in general condition The appearance of the weight function increases the difficulty of estimating the pressure term for this part it is different from

3 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 3 [8] We also point out that in the whole space R 2 or two dimensional torus T 2 the classical approach to analyticity or Gevrey regularity is that it makes crucial use of Fourier transformation which can t apply to our case Instead we will use the basic estimate (cf [ ] for instance The paper is organized as follows In section 2 we introduce the notation used to define the weighted Sobolev norms we prove the persistence of the weighted Sobolev regularity In section 3 we state the priori estimate to prove the main theorem Section 4 5 are consist of the proofs of these lemmas 2 Notations Preliminaries In the following context we use the conventional symbols for the stard Sobolev spaces H m (R 2 + with m N let H m be its norm For the case m = 0 it was usually written as (R 2 + Denote be the norm inner product in (R 2 + We usually write a vector function in bold type as u a scalar function in it s conventional way as u For a vector function u = (uv we denote u H m = u 2 H m + v 2 H m And when we say that u H m we mean that uv H m With the notations above we introduce the weighted Sobolev spaces H m l x (R 2 + Hl m y (R 2 + where l x l y are real constants Let } Hl m x (R 2 + {v = H m (R 2 + ; x lx α v 1 α m it s norm is defined by v H m = v 2 lx L + 2 Similarly let H m l y (R 2 + = 1 α m equipped with the norm u H m = u 2 ly L + 2 x lx α v 2 { } u H m (R 2 +; y ly α u 1 α m 1 α m y ly α u 2 We then define space Hl m xl y of vector functions by } Hl m xl y = {u = (uv H m : u H m ly v H m lx

4 4 F CHENG W-X LI AND C-J XU which is equipped with the norm u H m = u 2 H + v 2 ly m H lx m It s well known that the corresponding Cauchy problem to (11-(15 is globally well posed in H k if k > 2 with dimension d = 2 see eg[[12] Chapter 17 Section 2] t u(t H m u 0 H m exp (C 0 u(s ds 0 Where C 0 is a constant depending on u 0 Now we will show that if the initial data u 0 H m l xl y for 0 l x l y 1 the solution is also in H m l xl y This is the first step for the Gevery regularity Proposition 21 For fixed m 3 0 l x l y 1 let the initial data u 0 Hl m xl y (R 2 + suppose the compatibility condition is fulfilled Then the H m -solution u to the Euler equation (11-(15 is also in weighted Sobolev space: ( u(t C [0 ; Hl m xl y (R 2 + Moreover t ( u(t H m u 0 H m exp [C 0 u(s + y ly u(s 0 ] + x lx v(s ds (21 where C 0 is a constant depending on m Proof It suffices to show (21 holds We begin with proving a priori estimate First we have 1 d ( u(t 2L 2dt 2 + v(t 2L = 0 (22 2 Now let α N 2 0 be the multi-index such that 1 α m We apply α on both sides of (11 take inner product with y 2ly α u 1 d y ly α u 2 + 2dt y ly α (u u y ly α u + y ly α x p y ly α u = 0 And similarly for (12 taking inner product with x 2lx α v (23 1 d x lx α v 2 + x lx α (u v x lx α v + x lx α y p x lx α v = 0 2dt (24

5 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 5 Taking sum over 1 α m in (23 (24 combining (22 we have 1 d 2dt u(t 2 H + m 1 α m x lx α (u v x lx α v [ y ly α (u u y ly α u + ]+ ] + x lx α y p x lx α v = 0 1 α m [ y ly α x p y ly α u (25 It remains to estimate I 1 I 2 with I j defined by I 1 = I 2 = 1 α m 1 α m [ ] y ly α (u u y ly α u + x lx α (u v x lx α v [ ] y ly α x p y ly α u + x lx α y p x lx α v The estimate on I 1 : Using Hölder inequality divergence-free condition we have I 1 u H m 1 α m [ y l y α (u u u ( y ly α u + x lx α (u v u ( x lx α v ] Note the fact that β y ly β x lx C m for 1 β α C m depending on m then the weight function can be put in the bracket And with the application of [12 (322Chapter 13 Section 3] we have ( I 1 C u + y ly u + x lx v u 2 H (26 m The estimate on I 2 : In order to estimate I 2 we need to use Lemma 51 Lemma 53 in Section 5 Observe p satisfies the following Neumann problem p = 2( y u x v 2( x u y v in R 2 + {0 } y p y=0 = 0 on R {0 } We proceed to estimate I 2 through two cases

6 6 F CHENG W-X LI AND C-J XU (a If α = 1 then we use Lemma 51 classical argument of H 2 -regularity result of the above Neumann problem to get ( y l y x p α x + l x y α p α =1 y C y ly y u x v +C l y x x u y v +C l x y u x v +C x lx x u y v +C xp +C y p C u H m u where we used the Hodge decomposition of (R 2 + to estimate p L2 C is a constant (b If 2 α = k m then we use Lemma 53 similar arguments as [12 Proposition 36 Chapter 13 Section 3]; this gives ( y l y x p α + x lx y α p 2 α m C m k=2 β =k 1 ( y l y β ( y u x v + + x lx β x ( y u x v + l x β ( x u y v + k 2 x ( x u y v ( y l C u y x H m u + l x v y ly β ( x u y v +C m k=2 Thus we combine the above two cases to conclude that ( I 2 C u 2 x H u m + y ly u + l x v where C is a constant depending only on m And then by (25 (26 (27 we have d ( dt u(t x H C u m + y ly u + l x v Then with Grownwall inequality we obtain (21 ( k 2 x ( y u x v (27 u(t H m Now consider u H m Repeating the above arguments with y ly x lx replaced respectively by y ly εy ly l x x εx lx where 0 < ε < 1 then we can also deduce (21 by letting ε 0 We complete the proof of the proposition

7 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 7 3 Weighted Gevrey regularity We inherit the notations that used in [8] for X τ Y τ That is to say for a multi-index α = (α 1 α 2 in N 2 a vector function u = (uv define the Sobolev semi-norms as follows: u mlxl y = α =m u mlxl y = α =m ( y l y u α x + l x α v ( y l y u α + where u m = u m00 u m = u m00 x lx α v For s 1 τ > 0 define a new weighted Gevrey spaces which is equivalent to that in Definition 11 by where And let where X τlxl y = u Xτ = Y τlxl y = u Yτ = { } u C : u Xτ < u mlxl y τ m 3 (m 3! s { } u C : u Yτ < We will denote X τ = X τ00 Y τ = Y τ00 u mlxl y (m 3τ m 4 (m 3! s In order to show the main result Theorem 12 it suffices to show the following Theorem 31 Let the initial data u 0 = (u 0 v 0 satisfy u 0 X τ0l xl y for some s 1τ 0 > 0 0 l x l y 1 Then the Euler system (11-(13 admits a solution u(t C([0 ; X τ(tlxl y where τ(t depends on the initial radius τ 0 We will prove Theorem 31 using the method of [8] with main difference from the estimate on pressure By Proposition 21 we see u H m < + for each m With notations above we have d dt u(t X τ(t = τ u(t Yτ(t + d dt u(t τ(t m 3 ml xl y (m 3! s (31

8 8 F CHENG W-X LI AND C-J XU Recalling from (24 (25 using Hölder inequality we obtain d dt u(t ml xl y ( α ( y l y β u α β u β α =m β αβ 0 + x lx β u α β v + ( y l y x p α Set P = C = + x lx y α p α =mβ αβ 0 + x lx β u α β v α =m Combined with (31 we have α =m + u u m ( α ( y l y β u α β u β ( y l y x p α + τ m 3 (m 3! s x lx y α p τ m 3 (m 3! s d dt u(t X τ(t τ u(t Yτ(t +C +P +Cτ u u Yτ (32 4 We give the following Lemma to estimate C the proof is postponed to Section Lemma 32 There exists a sufficiently large constant C > 0 such that C C (C 1 +C 2 u Yτ where C 1 = u u 1lxl y 3l xl y + u 2 u 2lxl y +τ u u 2lxl y 3l xl y +τ 2 u 3 u 3lxl y C 2 = τ u 1lxl y +τ2 u 2lxl y +τ2 u Xτ +τ 3 u 3lxl y The following lemmas shall be used to estimate P The proof is postponed to Section 5 below Lemma 33 There exists a sufficiently large constant C > 0 such that P l1l 2 C (P 1 +P 2 u Yτ

9 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 9 where P 1 = u u 1lxl y 3l xl y + u u 2lxl y 2l xl y + u u 1lxl y 2l xl y +τ ( u u 2 + u u u u τ 2 u u 2lxl y 3l xl y +τ 3 u u 3lxl y 3l xl y P 2 = τ u +τ2( 1lxl y u + u 2lxl y 1l xl y +τ 3( u 3lxl y ( + τ 2 +τ 5/2 +τ 3 u Xτ +τ 4 u 3lxl y + u 2lxl y Let m 6 be fixed With Sobolev embedding theorem the lemmas above (32 we have d dt u(t X τ(tlxτy τ(t u(t Yτ(t +C(1+τ(t 3 u 2 H m ( +Cτ(t u + u 1lxl y u Yτ +C(τ(t 2 +τ(t 4 u H m u Yτ +C(τ(t 2 +τ(t 3 u Xτ(t u Yτ (33 where the constant C is independent of uv If τ(t decreases fast enough such that ( τ(t+cτ(t u + u 1lxl y +C(τ(t 2 +τ(t 4 u H m (34 +C(τ(t 2 +τ(t 3 u Xτ 0 Then (33 implies Therefore d dt u(t X τ(t C(1+τ(0 3 u 2 H m t u(t Xτ(t u 0 Xτ0 +C τ(0 u(s 2 H ds m for all 0 < t < where C τ(0 = C ( 1+τ(0 3 As τ is chosen to be a decrease function a sufficient condition for (34 to hold is that ( τ(t+c u + u(t 1lxl y τ(t +Cτ(t 2[ C τ(0 u(t H m where C τ(0 = ( 1+τ(0 2 C τ(0 = 1+τ(0 Denote 0 +C τ(0 M(t ] 0 t M(t = u 0 Xτxly +C τ(0 u(s 2 H ds m [ G(t = exp C t 0 0 ( ] u(s + u(s 1lxl y ds (35

10 10 F CHENG W-X LI AND C-J XU By Proposition 21 we can choose the constant C > 0 is taken largely enough such that u(t 2 H u m 0 2 H G(t m It then follows that (35 is satisfied if we let τ(t = G(t 1 1 [ C τ(0 u(s H +C m τ(0 ]G(s M(s 1 ds 1 τ(0 +C t 0 The proof of Theorem 31 is complete 4 the commutator estimate In this section we will prove Lemma 32 the method here is similar with [8] except for the parts involving the weight function Proof of Lemma 32 We first write the sum as m C = C mj where we denote C mj = τm 3 (m 3! s j=1 α =m β =jβ α + x lx β u α β v ( α ( y l y β u u α β β Then we split the right side of the above inequality into seven terms according to the values of m j prove the following estimates For small j we have C m1 C u 1 u 3lxl y +Cτ u u 1lxl y Y τ C m2 C u 2 u 2lxl y +Cτ u 2 u 3lxl y +Cτ 2 u 2 u Yτ For intermediate j we have m 3 m=7j=[m/2]+1 [ m 2 ] C mj Cτ 2 u Xτ u Yτ m=6 j=3 C mj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ (41

11 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 11 For higher j we have C m Cτ 2 u 3 u 3lxl y +Cτ3 u u 3lxl y Y τ m=5 C mm 1 Cτ u 3 u 2lxl y +Cτ2 u u 2lxl y Y τ m=4 C mm C u u 1lxl y 3 +Cτ u 1l u xl y Y τ The proof of the above estimates is similar as in [8] we just point out the difference due to the weight function The main difference may be caused by the weight function is the estimation of (41 Note that for [m/2] + 1 j m 3 with Hölder inequality [Proposition 38 Chapter 13Section 3[12]] one have y ly β u α β u C β u y ly α β u D 2( y ly α β 1/2 u where we used the notation D k u = { α u : α = k} D k u = α =k 1/2 α u (42 Note that by Leibniz formula D 2( y ly α β L2 y l u C( y α β u + y ly D 1 α β u (43 + y ly D 2 α β u where C is a constant And here we used the fact that observing 0 l y 1 y 2 y ly C y y ly 1 1 y ly for some constant C And similar arguments also applied to x lx β u α β v With (42 (43 we have m 3 m=7j=[m/2]+1 C mj C m 3 m=7j=[m/2]+1 u j u 1/2 m j+1l xl y ( u 1/2 m j+1l xl y + u 1/2 m j+2l xl y + u 1/2 m j+3l xl y ( m j τ m 3 (m 3! s And the estimation of the right side of the above inequality is similar as in [8] So we omit the details here The proof of Lemma 43 is complete

12 12 F CHENG W-X LI AND C-J XU 5 the pressure estimate It can be deduced from the Euler system (11-(13 that the pressure term p satisfies p = h(uv in R 2 + (51 where h(uv = 2( y u x v 2( x u y v Take the values of (12 on R 2 + use (14 to obtain y p y=0 = 0 on R 2 + (52 In order to estimate y ly x α p x lx y α p we first consider the following Neumann problem here we hope to obtain a weighted H 2 -regularity result Lemma 51 Suppose φ is the smooth solution of the following equation with Neumann boundary condition ψ C φ = ψ in R 2 + y φ y=0 = 0 on R 2 + (53 Then there exist a constant C such that for α N 2 0 with α = 2 y ly α y φ C l y ψ +C xφ (54 x lx α x φ C l x ψ +C yφ (55 y ly y α φ C y ly y ψ +C ψ (56 x lx y α φ C x lx y ψ +C ψ (57 Proof The proof is similar with the classical H 2 - regularity arguments Due to the symmetry it suffices to prove (54 (55 since (56 (57 can be proved similarly The method is to use integration by parts We first multiply the first equation of (53 by y 2ly xx φ integrate over R 2 + to obtain y ly xx φ 2 + y 2ly xx φ yy φdxdy = y ly xx φ y ly ψ R 2 + Integrating by parts with the second term we have y ly xx p 2 y + l y xy p 2 y = ly xx φ y ly ψ 2 ( y y ly x φ y ly xy φ

13 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 13 Using Cauchy-Schwarz inequality noticing that y y ly 1 for 0 l y 1 we can obtain for any 0 < εε < 1 y ly xx φ 2 + y ly xy φ 2 y l C y ǫ ψ 2 +ǫ y ly xx φ L 2 +ǫ y l y xy φ 2 +C ǫ xφ 2 L 2 thus y ly xx φ + y ly xy φ C y ly ψ +C x φ for some constant C > 0 Now if we multiply y 2ly yy φ on both sides of ((53 do the procedure as above we can obtain y y ly yy p + l y y xy φ C l y ψ +C xφ Then we have proven (54 (55 To prove (56 (57 we first apply y on equation (53 to get y φ = y ψ in R 2 + y φ y=0 = 0 on R (58 And with this Dirichlet equation for y φ we can also proceed like before In this time we multiply y 2ly xxy φ y 2ly yyy φ on both sides of (58 calculate as above then with the use of the classical H 2 regularity result (56 ((57 can be obtained Remark 52 The terms of order one on right side of (54-(55 are created by differentiating on the weight functions x lx y ly when integrating by parts And this is the main reason why we need the constants l x l y to be in the interval [01] For higher order regularity estimates we need the following lemma Lemma 53 Suppose g is a smooth solution of g = f in R 2 + y g y=0 = 0 on R 2 + (59 with f C Then there exist a universal constant C > 0 such that the following estiamtes y ly x α g C l N 0 β =m 1 β α =2l+1 y ly β f +C x f (510

14 14 F CHENG W-X LI AND C-J XU x lx y α g C l N 0 β =m 1 β α =2l x lx β f +C x f (511 hold for any m 3 any multi-index α N 2 0 such that α = m In (510 (511 we have summation over the set {β N 2 0 : β = m 1 l N 0 such that β α = 2l+1} {β N 2 0 : β = m 1 l N 0 such that β α = 2l} similar conventions are used throughout this section Proof First by (59 we use the following induction equality from [8]: 2k+2 y g = ( 1 k+1 2k+2 g applying y on the above equation gives 2k+3 y g = ( 1 k+1 x 2k+2 y g x k ( 1 k j x 2k 2j y 2j f (512 j=0 k ( 1 k j x 2k 2j y 2j+1 f (513 Then for given α = m we discuss the situations as the value of α 2 varies j=0 Case 1 If α 2 = 0 then y ly x α g = y ly x m+1 g x lx y α g = x lx y m x g Letting φ = x m 1 g applying Lemma 51 we obtain y ly x α g C y ly m 1 x f +C x m g C y ly x m 1 f +C x f x l x y g α C x lx m 1 x f +C y x m 1 g C x lx x m 1 f +C x f In such case Lemma 53 is proved Case 2 If α 2 = 1 then y ly x α g = y ly m x y g x lx y α g = x lx y 2 m 1 x g Letting φ = x m 1 g we can obtain the same result by Lemma 51 as above

15 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 15 Case 3 If α 2 = 2k+2 2 then by the induction (512 we have y ly x α g = y ly 2k+2 Letting φ = 2k y x g = y ly ( 1 k+1 2k+2 x x y ly 2k+2 x x x x g y ly k ( 1 k j x 2k 2j y 2j x f j=0 g we apply Lemma 51 to obtain y g C l y 2k C y ly 2k Substituting (515 into (514 we have y ly x α g C C x x x x k y ly 2k 2j j=0 l N 0 β =m 1 β α =2l+1 x f +C m x g f +C x f f y 2j x y ly β f And similarly from the induction (513 equality x lx y α g = x lx y 2k+3 x α1 g = x lx ( 1 k+1 x 2k+2 y x α1 g x lx Letting φ = 2k x α1 x g we apply Lemma 51 to get x lx x 2k+2 x y x α1 g C l x x 2k y α1 C x lx x 2k y α1 Substituting (517 into (516 yields x lx y α g C C k x lx 2k 2j j=0 l N 0 β =m 1 β α =2l+1 Thus in such case the lemma is also proved x f x f x y 2j+1 +C x f +C x f k ( 1 k j 2k 2j j=0 x lx β f x y 2j+1 +C 2 y x 2k x α1 g +C x f x α1 f L +C x f 2 +C x f (514 (515 α1 x f (516 (517

16 16 F CHENG W-X LI AND C-J XU Case 4 If α 2 = 2k+3 3 then by the induction we have y ly x α g = y ly 2k+3 y x = y ly ( 1 k+1 x 2k+2 y g y ly g k ( 1 k j 2k 2j j=0 x x y 2j+1 x f (518 Letting φ = x 2k x then applying Lemma 51 we have y ly x 2k+2 y x y g C l y y x 2k x f +C x y x 2k+ g C y ly y x 2k x f +C x f Thus substituting the above estimate into (518 yields y ly x α k g C y ly 2k 2j C On the other h observe j=0 l N 0 β =m 1 β α =2l+1 x lx y α g = x lx y 2(k+1+2 α1 g x y 2j+1 x y ly β f k+1 = x lx ( 1 k+2 x 2k+4 x α1 g x lx j=0 f +C x f +C x f ( 1 k+1 j 2k+2 2j x 2j y α1 x f Then letting φ = x 2k+2 x α1 g applying Lemma 51 we obtain x lx x 2k+4 x α1 g L C x lx x 2k+2 x α1 f L +C y x 2k+2+α1 g 2 2 C x lx x 2k+2 x α1 f L +C x f 2 which along with (519 yields k+1 x lx y α g C x lx x 2k+2 2j y 2j α1 x f L +C x f 2 C j=0 l N 0 β =m 1 β α =2l+1 x lx β f +C x f (519 So in this case Lemma 53 is also proved Thus for all α such that α = m we have proved Lemma 53 Now we come to the proof of Lemma 33

17 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 17 Proof of Lemma 33 Apply Lemma 53 with equation (51-(52 we have P = C + C τ m 3 (m 3! s l N 0 β =m 1 β α =2l + β =m 1 α =m τ m 3 (m 3! s α =m ( y l y x p α + l N 0 β =m 1 β α =2l+1 x lx y α p y ly β h x lx β h +C x h mτ m 3 (m 3! s β =m 1 y ly β h x lx β h + x h where we denote h = 2( x v y u 2( x u y v If we exchange the order of the summation we can obtain by direct verification α =m l N 0 β =m 1 β α =2l+1 y ly β h Cm β =m 1 y ly β h α =m l N 0 β =m 1 β α =2l x lx β h Cm β =m 1 x lx β h And direct computation also gives α =m x h m x h Recall h = 2( x u y v +2( y u x v then we have for arbitrary β N 2 0 y ly β h C γ β ( β y l y γ u β γ v γ x lx β h C γ β ( β x l x γ u β γ v γ

18 18 F CHENG W-X LI AND C-J XU So with these inequalities we have P w = C C P x = C mτ m 3 (m 3! s mτ m 3 (m 3! s β =m 1 β =m 1 0 γ β + x lx γ u β γ v C Then we have ( y l y h β x + l x β h ( β γ mτ m 3 (m 3! s x h mτ m 3 (m 3! s 0 j The rest part is to estimate P w P x ( y l y γ u β γ v ( j j x u x j v P P w +P x We first estimate P w To do so we split the summation into where P wmj = mτm 3 (m 3! s P w C m 1 j=0 P wmj (520 ( m 1 ( y l y γ u v β γ j β =m 1 γ =j + x lx γ u β γ v Moreover we split the right side of (520 into seven terms according to the values of m j For lower j we have P wm0 C u u 1lxl y 3l xl y +Cτ u u 1lxl y Y τ P wm1 C u u 2lxl y 2l xl y +Cτ u u 2lxl y 3l xl y +Cτ 2 u 2lxl y u Y τ P wm2 Cτ 2 u u 3lxl y 3l xl y +Cτ 3 u u 3lxl y Y τ m=5

19 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 19 For intermediate j we have m=8 [m/2] 1 j=3 m 3 m=6j=[m/2] For higher j we have P wmj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ P wmj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ m=4 P wm Cτ u 2lxl y u 3l xl y +Cτ 2 u 2lxl y u Y τ P wmm 1 C u u 1lxl y 3l xl y +Cτ u u 1lxl y Y τ In these estimations we used the fact that for vector function u = (uv the norm of u or v can be bounded by the norm of u for example y ly γ u u γ +1lxl y With this consideration the estimations can be proved similarly by the method of [8] the arguments of the commutator estimates we omit the details To estimate P x we proceed as above write where P x C P xmj = mτm 3 (m 3! s j=0 ( j P xmj j x u m j 2 x v For lower j we have P xm0 C u 1 u 2 +Cτ u 1 u 3 +Cτ 2 u 1 u Yτ P xm1 Cτ u 2 u 2 +Cτ 2 u 2 u 3 +Cτ 3 u 2 u Yτ m=4 P xm2 Cτ 3 u 3 u 3 +Cτ 4 u 3 u Yτ m=6 For mediate j we have m=8 [m/2] 1 j=3 P xmj Cτ 3 u Xτ u Yτ

20 20 F CHENG W-X LI AND C-J XU m 3 m=6j=[m/2] P xmj Cτ 3 u Xτ u Yτ Finally for higher j we have P xm Cτ 2 u 1 u Yτ m=5 These estimations can be proved similarly as [8] with the fact that u Xτ u Xτ u Yτ u Yτ u H m u H m With all these esti- mations we can complete the proof Lemma 33 Acknowledgments W Li would like to appreciate the support from NSF of China (No C-J Xu was partially supported by the Fundamental Research Funds for the Central Universities the NSF of China(No References [1] J-Y Chemin B Desjardins I Gallagher E Grenier Mathematical geophysics An introduction to rotating fluids the Navier-Stokes equations Oxford Lecture Series in Mathematics its Applications 32 The Clarendon Press Oxford University Press Oxford2006 [2] JPBourguignon HBrezis Remarks on the Euler equation J Functional Analysis 15( [3] H Chen W-X Li C-J Xu Gevrey regularity of subelliptic Monge-Ampére equations in the plane Advances in Mathematics 228( [4] H Chen W-X Li C-J Xu Gevrey hypoellipticity for a class of kinetic equations Communications in Partial Differential Equations 36 ( [5] H Chen W-X Li C-J Xu Analytic smoothness effect of solutions for spatially homogeneous Lau equation Journal of Differential Equations 248 ( [6] H Chen W-X Li C-J Xu Gevrey hypoellipticity for linear non-linear Fokker- Planck equations Journal of Differential Equations 246 ( [7] DGEbin JEMarsden Groups of diffeomorphisms the solution of the classical Euler equations for a perfect fluid BullAmerMathSoc75( [8] IKukavica VVicol The domain of analyticity of solutions to three-dimensional euler equations in a half space Discrete Continuous Dynamical Systems Volume 29 Number 1( [9] IKukavica VVicol On the analyticity Gevrey class regularity up to the boundary for the Euler equation Nonlinearity Volume 24 Number 3 ( [10] TKato Nonstationary flows of viscous idear fluids in R 3 J Functional Analysis 9( [11] TKato on classical solutions of two dimensional nonstationary Euler equations Arch Rat Mech Anal Vol 25(

21 GEVERY REGULARITY WITH WEIGHT FOR EULER EQUATION 21 [12] MTaylor Partial Differential Equations III Nonlinear Equations Springer-Verlag New York 1996 [13] RTemam On the Euler equations of incompressible perfect fluids J Functional Analysis 20(1975no [14] CFoias UFrisch RTemam Existence de solutions C des équations d Euler CRAcadSciParisSérA-B 280 (1975A505-A508 [15] VIYudovich Non stationary flow of an ideal incompressible liquid Zh Vych Mat 3( China Feng Cheng School of Mathematics Statistics Wuhan University Wuhan address: chengfengwhu@whueducn Wei-Xi Li School of Mathematics Statistics Computational Science Hubei Key Laboratory Wuhan University Wuhan China address: wei-xili@whueducn Chao-Jiang Xu School of Mathematics Statistics Wuhan University Wuhan China address: chjxumath@whueducn

22 GEVREY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE arxiv: v2 [mathap] 23 Nov 2016 FENG CHENG WEI-XI LI AND CHAO-JIANG XU Abstract In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid The method presented here is based on the basic weighted - estimate the main difficulty arises from the estimate on the pressure term due to the appearance of weight function 1 Introduction In this paper we study the Gevrey propagation of solutions to incompressible Euler equation Gevrey class is a stronger concept than the C -smoothness In fact it is an intermediate space between analytic space C space There have been extensive mathematical investigations (cf[2] [7] [8] [9] [10] [11] [13] [14] [15] for instance the references therein on Euler equation in different kind of frames such as Sobolev space analytic space Gevrey space In this work we will consider the problems of Gevrey regularity with weight this is motivated by the study of the vertical boundary layer problem From a physical point of view as well as from a mathematical point of view when the direction of rotation is perpendicular to the boundaries the boundary equation is well developed called Ekman layers Up to now the Ekman layers (horizontal layer are well understood (cf [1] for instance the references therein When the direction of rotation is parallel to the boundaries the situation is however different from the perpendicular case above: the vertical layers are very different much more complicated from a physical analytical mathematical point of view many open questions in all these directions remain open We refer to [1] for detailed discussion on the vertical layers Recently we try to investigate the well-posedness problem for vertical layer the related preliminary step is to establish 2010 Mathematics Subject Classification 35M33 35Q31 76N10 Key words phrases Gevrey class incompressible Euler equation weight Sobolev space 1

23 2 F CHENG W-X LI AND C-J XU the Gevrey regularty with weight for the outer flow which is described by Euler equation This is the main result of the present paper Without loss of generality we consider the incompressible Euler equation in half plane R 2 + where R 2 + = {(xy;x Ry R + } our results can be generalized to 3-D Euler equation The velocity (u(txyv(txy the pressure p(txy satisfy the following equation: t u+u x u+v y u+ x p = 0 in (0 R 2 + (11 t v +u x v +v y v + y p = 0 in (0 R 2 + (12 x u+ y v = 0 in (0 R 2 + (13 The boundary condition v y=0 = 0 in (0 R; uv 0 as x 2 +y 2 (14 With initial data u t=0 = u 0 v t=0 = v 0 in R 2 + (15 Here the initial data (u 0 v 0 satisfy the compatibility condition: x u 0 + y v 0 = 0 v 0 y=0 = 0 u 0 v 0 0 as x2 +y 2 Before stating our main result we first introduce the (global weighted Gevrey space Definition 11 Let l x l y 0 be real constants that independent of xy we say that f G s τl xl y (R 2 + if sup α 0 τ α x l x α! s y ly α f < (R 2 + where throughout the paper we use the notation = ( In this work we present the persistence of weighted Gevrey class regularity of the solution ie we prove that if the initial datum (u 0 v 0 is in some weighted Gevrey space satisfy the compatible condition then the global solution belongs to the same space With only minor changes these results can also be extended to 3-D Euler equation the global solution here will be replaced by a local solution Precisely Theorem 12 Suppose the initial data u 0 G s τ 00l y v 0 G s τ 0l x0 for some s 1τ 0 > 0 0 l x l y 1 Then the Euler equation (11-(15 admits a solution uvp such that u(t ( [0+ ; G s τ(t0l y v(t ( [0+ ; G s τ(tl x0

24 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 3 p satisfies x p(t ( [0+ ; G s τ(t0l y y p(t ( [0+ ; G s τ(tl x0 where τ(t is a decreasing function of t with initial value τ 0 Remark 13 If the initial data (u 0 v 0 were posed both horizontal vertical weights namely u 0 G s τ 0l xl y v 0 G s τ 0l xl y the result of Theorem 12 is also valid We remark that the existence of smooth solutions to (11-(15 is well developed (cf[2] [7] [10] [13] [15] for instance in two-dimensional case smooth initial data can yield global solutions while in the three-dimensional case the solution may be local in general condition The appearance of the weight function increases the difficulty of estimating the pressure term for this part it is different from [8] We also point out that in the whole space R 2 or two dimensional torus T 2 the classical approach to analyticity or Gevrey regularity is that it makes crucial use of Fourier transformation which can t apply to our case Instead we will use the basic estimate (cf [ ] for instance The paper is organized as follows In section 2 we introduce the notation used to define the weighted Sobolev norms we prove the persistence of the weighted Sobolev regularity In section 3 we state the priori estimate to prove the main theorem Section 4 5 are consist of the proofs of these lemmas 2 Notations Preliminaries In the following context we use the conventional symbols for the stard Sobolev spaces H m (R 2 + with m N let Hm be its norm For the case m = 0 it was usually written as (R 2 + Denote be the norm inner product in (R 2 + We usually write a vector function in bold type as u a scalar function in it s conventional way as u For a vector function u = (uv we denote u H m = u 2 H m + v 2 H m And when we say that u H m we mean that uv H m With the notations above we introduce the weighted Sobolev spaces H m l x (R 2 + H m l y (R 2 + where l x l y are real constants Let H m l x (R 2 + = { } v H m (R 2 +; x lx α v 1 α m

25 4 F CHENG W-X LI AND C-J XU it s norm is defined by v H m = v 2 lx L α m x lx α v 2 Similarly let } Hl m y (R 2 + {u = H m (R 2 + ; y ly α u 1 α m equipped with the norm u H m = u 2 ly L α m y ly α u 2 We then define space Hl m xl y (R 2 + of vector functions by { } Hl m xl y (R 2 + = u = (uv Hl m xl y (R 2 + : u Hm l y (R 2 + v Hm l x (R 2 + which is equipped with the norm u H m = u 2 H + v 2 ly m H lx m It s well known that the corresponding Cauchy problem to (11-(15 is globally well posed in H m if m > 2 with dimension d = 2 see eg[[12] Chapter 17 Section 2] t u(t H m u 0 H m exp (C 0 u(s ds 0 Where C 0 is a constant depending on u 0 Now we will show that if the initial data u 0 = (u 0 v 0 H m l xl y for 0 l x l y 1 thesolutionisalsoinh m l xl y ThisisthefirststepfortheweightedGeveryregularity Proposition 21 For fixed m 3 0 l x l y 1 let the initial data u 0 H m l xl y (R 2 + suppose the compatibility condition is fulfilled Then the H m -solution u to the Euler equation (11-(15 is also in weighted Sobolev space: u(t ( [0 ; Hl m xl y (R 2 + Moreover t ( u(t H m u 0 H m exp [C 0 u(s + y ly u(s 0 + x lx v(s ] ds (21 where C 0 is a constant depending on m

26 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 5 Proof It suffices to show (21 holds When no ambiguity arises we suppress the time dependence of u v on t We begin with proving a priori estimate First we have 1 d ( u(t 2L 2dt 2 + v(t 2L = 0 (22 2 Now let α N 2 0 be the multi-index such that 1 α m We apply α on both sides of (11 take inner product with y 2ly α u 1 d y ly α u 2 + y ly α (u u y ly α u + y ly α x p y ly α u = 0 2dt (23 And similarly taking inner product with x 2lx α v on both sides of (12 1 d x lx α v 2 + x lx α (u v x lx α v + x lx α y p x lx α v = 0 2dt (24 Taking sum over 1 α m in (23 (24 combining (22 we have 1 d 2dt u(t 2 H m + [ y ly α (u u y ly α u 1 α m + 1 α m ] + x lx α (u v x lx α v [ ] y ly α x p y ly α u + x lx α y p x lx α v = 0 It remains to estimate I 1 I 2 with I j defined by [ ] I 1 = y ly α (u u y ly α u + x lx α (u v x lx α v I 2 = 1 α m 1 α m The estimate on I 1 : [ ] y ly α x p y ly α u + x lx α y p x lx α v Using Hölder inequality divergence-free condition we have [ y l I 1 u y H m α (u u u ( y ly α u 1 α m (25 ] + x lx α (u v u ( x lx α v Note the fact that β y ly β x lx Cm for 1 β α C m depending on m then the weight function can be put into the bracket And with the application of [12 (322Chapter 13 Section 3] we have ( I 1 C u + y ly u + x lx v u 2 H (26 m Where C is a constant depending on m In the following C denotes a generic positive constant depending on m

27 6 F CHENG W-X LI AND C-J XU The estimate on I 2 : In order to estimate I 2 we need to use Lemma 51 Lemma 53 in Section 5 Observe p satisfies the following Neumann problem p = 2( x u y v 2( y u x v in R 2 + {0 } y p y=0 = 0 on R {0 } We proceed to estimate I 2 through two cases (a If α = 1 then we use Lemma 51 classical argument of H 2 -regularity result of the above Neumann problem to get ( y l y x p α + x lx y α p α =1 C y ly y u x v +C y ly x u y v +C x lx y u x v +C x lx x u y v +C xp +C y p C u H m u where we used the Hodge decomposition of (R 2 + to estimate p C is a constant (b If 2 α = k m then we use Lemma 53 similar arguments as [12 Proposition 36 Chapter 13 Section 3]; this gives ( y l y x p α + x lx y α p C 2 α m m k=2 β =k 1 ( y l y β y ( y u x v + l y β ( x u y v + x lx β x ( y u x v + l x β ( x u y v +C m k=2 ( k 2 x ( y u x v + k 2 x ( x u y v ( y l C u y x H m u + l x v Thus we combine the above two cases to conclude that ( I 2 C u 2 H u m + y ly u + x lx v where C is a constant depending only on m And then by (25 (26 (27 we have d ( dt u(t H C u m + y ly u + x lx v (27 u(t H m

28 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 7 Then with Grownwall inequality we obtain (21 Now we consider u H m Repeating the above arguments with y ly x lx replaced respectively by y ly εy ly l x x εx lx where 0 < ε < 1 then we can also deduce (21 by letting ε 0 We then complete the proof of the proposition 3 Weighted Gevrey regularity We inherit the notations that used in [8] for X τ Y τ That is to say for a multi-index α = (α 1 α 2 in N 2 a vector function u = (uv define the Sobolev semi-norms as follows: u mlxl y = ( y l y u α x + l x α v α =m u mlxl y = α =m ( y l y u α + where u m = u m00 u m = u m00 x lx α v For s 1 τ > 0 define a new weighted Gevrey spaces which is equivalent to that in Definition 11 by X τlxl y = { } u C : u Xτ < where And let u Xτ = Y τlxl y = u mlxl y τ m 3 (m 3! s { } u C : u Yτ < where u Yτ = We will denote X τ = X τ00 Y τ = Y τ00 u mlxl y (m 3τ m 4 (m 3! s In order to show the main result Theorem 12 it suffices to show the following Theorem 31 Let the initial data u 0 = (u 0 v 0 satisfy u 0 X τ0l xl y

29 8 F CHENG W-X LI AND C-J XU for some s 1τ 0 > 0 0 l x l y 1 Then the Euler system (11-(15 admits a solution u(t ( [0 ; X τ(tlxl y where τ(t is a decreasing function depending on the initial radius τ 0 the Hl m xl y solution u for m 6 We will prove Theorem 31 using the method of [8] with main difference from the estimate on pressure By Proposition 21 we see u H m < + for each m > 2 When no ambiguity arises we suppress the time dependence of τ uv on t With notations above we have d dt u(t X τ(t = τ (t u(t Yτ(t + d dt u(t τ(t m 3 ml xl y (m 3! s Recalling from (24 (25 using Hölder inequality we obtain d dt u(t ml xl y ( ( α y l y β u α β u β Set P = C = α =mβ αβ 0 + x lx β u α β v + x lx y α p α =m β αβ 0 + x lx β u α β v α =m Combined with (31 we have + α =m + u u m ( ( α y l y β u α β u β ( y l y x p α + τ(t m 3 (m 3! s x lx y α p ( y l y x α p τ(t m 3 (m 3! s (31 d dt u(t X τ(t τ (t u(t Yτ(t +C+P+Cτ(t u(t u(t Yτ (32 4 We give the following Lemma to estimate C the proof is postponed to Section Lemma 32 There exists a sufficiently large constant C > 0 such that C C (C 1 +C 2 u Yτ

30 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 9 where C 1 = u 1lxl y u 3l xl y + u 2 u 2lxl y +τ u 2lxl y u 3l xl y +τ 2 u 3 u 3lxl y C 2 = τ u 1lxl y +τ2 u 2lxl y +τ2 u Xτ +τ 3 u 3lxl y The following lemmas shall be used to estimate P The proof is postponed to Section 5 below Lemma 33 There exists a sufficiently large constant C > 0 such that P C (P 1 +P 2 u Yτ where P 1 = u u 1lxl y 3l xl y + u u 2lxl y 2l xl y + u u 1lxl y 2l xl y +τ ( u u 2 + u u u u τ 2 u 2lxl y u 3l xl y +τ 3 u 3lxl y u 3l xl y P 2 = τ u +τ2( 1lxl y u + u 2lxl y 1l xl y +τ 3( u 3lxl y ( + τ 2 +τ 5/2 +τ 3 u Xτ +τ 4 u 3lxl y + u 2lxl y Let m 6 be fixed With Sobolev embedding theorem the lemmas above (32 we have d dt u(t X τ(tlxτy τ (t u(t Yτ(t +C(1+τ(t 3 u(t 2 H m ( +Cτ(t u + u 1lxl y u Yτ +C(τ(t 2 +τ(t 4 u H m u Yτ (33 +C(τ(t 2 +τ(t 3 u Xτ(t u Yτ where the constant C is independent of uv If τ(t decreases fast enough such that ( τ (t+cτ(t u + u 1lxl y +C(τ(t 2 +τ(t 4 u H m (34 +C(τ(t 2 +τ(t 3 u Xτ 0 Then (33 implies d dt u(t X τ(t C(1+τ(0 3 u 2 H m

31 10 F CHENG W-X LI AND C-J XU Therefore t u(t Xτ(t u 0 Xτ0 +C τ(0 u(s 2 H ds (35 m for all 0 < t < where C τ(0 = C ( 1+τ(0 3 As τ(t is chosen to be a decrease function a sufficient condition for (34 to hold is that ( τ (t+c u + u(t 1lxl y τ(t +Cτ(t 2[ C τ(0 u(t H m where C τ(0 = ( 1+τ(0 2 C τ(0 = 1+τ(0 Set denote 0 +C τ(0 M(t ] 0 t M(t = u 0 Xτxly +C τ(0 u(s 2 H ds m [ G(t = exp C t 0 0 ( ] u(s + u(s 1lxl y ds (36 By Proposition 21 we can choose the constant C > 0 is taken largely enough such that u(t 2 H u m 0 2 H G(t m It then follows that (36 is satisfied if we let τ(t = G(t 1 1 [ C τ(0 u(s H +C m τ(0 ]G(s M(s 1 ds 1 τ(0 +C t 0 With this decreasing function τ(t we can conclude the a priori estimates that are used to prove of Theorem 31 4 the commutator estimate In this section we will prove Lemma 32 the method here is similar with [8] except for the parts involving the weight function Proof of Lemma 32 We first write the sum as m C = C mj where we denote C mj = τm 3 (m 3! s j=1 α =m β =jβ α + x lx β u α β v ( ( α y l y β u u α β β

32 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 11 Then we split the right side of the above inequality into seven terms according to the values of m j prove the following estimates For small j we have C m1 C u 1 u 3lxl y +Cτ u u 1lxl y Y τ C m2 C u 2 u 2lxl y +Cτ u 2 u 3lxl y +Cτ 2 u 2 u Yτ For intermediate j we have m 3 m=7j=[m/2]+1 For higher j we have [ m 2 ] C mj Cτ 2 u Xτ u Yτ m=6 j=3 C mj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ (41 C m Cτ 2 u 3 u 3lxl y +Cτ3 u u 3lxl y Y τ m=5 C mm 1 Cτ u 3 u 2lxl y +Cτ2 u u 2lxl y Y τ m=4 C mm C u u 1lxl y 3 +Cτ u 1l u xl y Y τ The proof of the above estimates is similar as in [8] we just point out the difference due to the weight function The main difference may be caused by the weight function is the estimation of (41 Note that for [m/2] + 1 j m 3 with Hölder inequality [Proposition 38 Chapter 13Section 3[12]] one have y ly β u α β u C β u y ly α β u D 2( y ly α β 1/2 u where we used the notation D k u = { α u : α = k} D k u = α =k 1/2 α u (42

33 12 F CHENG W-X LI AND C-J XU Note that by Leibniz formula D 2( ( y y ly u α β L C l y α β y u + l y D 1 α β u 2 + y ly D 2 α β u (43 where C is a constant And here we used the fact that observing 0 l y 1 y 2 y ly y C y ly l 1 1 y y for some constant C And similar arguments also applied to x lx β u α β v With (42 (43 we have m 3 m=7j=[m/2]+1 C mj C m 3 m=7j=[m/2]+1 u j u 1/2 m j+1l xl y ( u 1/2 m j+1l xl y + u 1/2 m j+2l xl y + u 1/2 m j+3l xl y ( m j τ m 3 (m 3! s And the estimation of the right side of the above inequality is similar as in [8] So we omit the details here The proof of Lemma 43 is complete 5 the pressure estimate It can be deduced from the Euler system (11-(14 that the pressure term p satisfies p = h in R 2 + (51 where h = 2( x u y v 2( y u x v Taking the values of (12 on R 2 + using (14 we have the following Neumann boundary condition y p y=0 = 0 on R 2 + (52 In order to estimate y ly x α p x lx y α p we first consider the following Neumann problem here we hope to obtain a weighted H 2 -regularity result Lemma 51 Suppose φ is the smooth solution of the following equation with Neumann boundary condition ψ C φ = ψ in R 2 + y φ y=0 = 0 on R 2 + (53 Then there exist a constant C such that for α N 2 0 with α = 2 y ly α y φ C l y ψ +C xφ (54

34 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 13 x lx α x φ C l x ψ +C yφ (55 y ly y α y φ C l y y ψ +C ψ L (56 2 x lx y α φ C x lx y ψ +C ψ (57 Proof The proof is similar with the classical H 2 - regularity arguments Due to the symmetry it suffices to prove (54 (56 since (55 (57 can be proved similarly The method is to use integration by parts We first multiply the first equation of (53 by y 2ly xx φ integrate over R 2 + y ly xx φ 2 + y 2ly xx φ yy φdxdy = R 2 + y ly xx φ y ly ψ Integrating by parts with the second term we have y ly xx φ 2 y + l y xy φ 2 y = ly xx φ y ly ψ 2 ( y y ly x φ y ly xy φ Using Cauchy-Schwarz inequality noticing that y y ly 1 for 0 ly 1 we can obtain for some 0 < εε < 1 y ly xx φ 2 + y ly xy φ 2 y l C y ǫ ψ 2 +ǫ y ly xx φ 2 L 2 +ǫ y l y xy φ 2 +C ǫ x φ 2 L 2 thus y ly xx φ + y ly xy φ C y ly ψ +C x φ for some constant C > 0 Now if we multiply y 2ly yy φ on both sides of (53 do the procedure as above we can obtain y ly yy φ + y ly xy φ C y ly ψ +C x φ Then we have proven (54 To prove (56 we first apply y on equation (53 to get y φ = y ψ in R 2 + y φ y=0 = 0 on R 2 + (58 Denote Φ y φ then we have Φ = y ψ in R 2 + Φ y=0 = 0 on R 2 + (59

35 14 F CHENG W-X LI AND C-J XU And with this Dirichlet boundary problem for Φ we multiply (59 by y 2ly xx Φ integrate over R 2 + y ly xx Φ 2 + R 2 + yy Φ y 2ly xx Φdxdy = R 2 + y ψ y 2ly xx Φdxdy (510 Since Φ vanish at infinity Φ y=0 then y Φ y 2ly xx Φ = 0 We can inte- y=0 grate by parts to obtain yy Φ y 2ly xx Φdxdy = R 2 + R 2 + xy Φ y ( y 2ly x Φ dxdy = y ly xy Φ 2 +2 R 2 + ( y ly xy Φ y y ly x Φ dxdy Substituting the above equality into (510 using Hölder inequality on the right h side we then obtain y ly xx Φ 2 + y ly xy Φ 2 C y ly y ψ 2 +C x Φ 2 If we multiply (59 by y 2ly yy Φ then we can proceed as above to obtain y ly yy Φ 2 y + l y xy Φ 2 y C l y y ψ 2 L +C x Φ 2 2 With the use of the classical H 2 regularity result we have x Φ = xy φ C ψ Combining the above three equalities we can prove (56 (55 (57 can be proved similarly Remark 52 The terms of order one on right side of (54-(55 are created by differentiating on the weight functions x lx y ly when integrating by parts And this is the main reason why we need the constants l x l y to be in the interval [01] For higher order regularity estimates we need the following lemma Lemma 53 Suppose g is a smooth solution of g = f in R 2 + y g y=0 = 0 on R 2 + (511 with f C Then there exist a universal constant C > 0 such that the following estiamtes y ly x α g C l N 0 β =m 1 β α =2l+1 y ly β f +C x f (512

36 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 15 x lx y α g C l N 0 β =m 1 β α =2l x lx β f hold for any m 3 any multi-index α N 2 0 such that α = m +C x f (513 In (512 (513 we have summation over the set { β N 2 0 : β = m 1 l N 0 such that β α = 2l+1 } { β N 2 0 : β = m 1 l N 0 such that β α = 2l } similar conventions are used throughout this section Proof First by (511 we use the following induction equality from [8]: 2k+2 y g = ( 1 k+1 2k+2 x g + applying y on the above equation gives 2k+3 y g = ( 1 k+1 2k+2 x y g + k ( 1 k j x 2k 2j y 2j f (514 j=0 k ( 1 k j x 2k 2j y 2j+1 f (515 Then for given α N 2 0 with α = m we discuss the situations as the value of α 2 varies j=0 Case 1 If α 2 = 0 then y ly x α g = y ly x m+1 g x lx y α g = x lx y m x g Letting φ = x m 1 g applying Lemma 51 we obtain y ly x α y g C l y m 1 x f +C m x g C y ly x m 1 f +C x f x l x y g α x C l x m 1 x f +C y x m 1 g C x lx x m 1 f +C x f In such case Lemma 53 is proved Case 2 If α 2 = 1 then y ly x α g = y ly m x y g x lx y α g = x lx y 2 x m 1 g Letting φ = x m 1 g we can obtain the same result by Lemma 51 as above

37 16 F CHENG W-X LI AND C-J XU Case 3 If α 2 = 2k+2 2 then by the induction (514 we have y ly x α g = y ly 2k+2 Letting φ = 2k y x g = y ly ( 1 k+1 2k+2 x x y ly 2k+2 x x x x g + y ly k ( 1 k j x 2k 2j y 2j x f j=0 g we apply Lemma 51 to obtain y g C l y 2k C y ly 2k Substituting (517 into (516 we have y ly x α g C C x x x x k y ly 2k 2j j=0 l N 0 β =m 1 β α =2l+1 x f +C m x g f +C x f f y 2j x y ly β f And similarly from the induction (515 equality x lx y α g = x lx y 2k+3 x α1 g = x lx ( 1 k+1 x 2k+2 y x α1 g + x lx Letting φ = 2k x α1 x g we apply Lemma 51 to get x lx x 2k+2 x y x α1 g C l x x 2k y α1 C x lx x 2k y α1 Substituting (519 into (518 yields x lx y α g C C k x lx 2k 2j j=0 l N 0 β =m 1 β α =2l+1 Thus in such case the lemma is also proved x f x f x y 2j+1 +C x f +C x f k ( 1 k j 2k 2j j=0 x lx β f x y 2j+1 +C 2 y x 2k x α1 g +C x f x α1 f L +C x f 2 +C x f (516 (517 α1 x f (518 (519

38 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 17 Case 4 If α 2 = 2k+3 3 then by the induction we have y ly x α g = y ly 2k+3 y x = y ly ( 1 k+1 x 2k+2 y g + y ly g x k ( 1 k j 2k 2j j=0 x y 2j+1 x f (520 Letting φ = x 2k x then applying Lemma 51 we have y ly x 2k+2 y x y g C l y y x 2k x f +C x y x 2k+ g C y ly y x 2k x f +C x f Thus substituting the above estimate into (520 yields y ly x α k g C y ly 2k 2j C On the other h observe j=0 l N 0 β =m 1 β α =2l+1 x lx y α g = x lx y 2(k+1+2 α1 g x y 2j+1 x y ly β f k+1 = x lx ( 1 k+2 x 2k+4 x α1 g + x lx j=0 f +C x f +C x f ( 1 k+1 j 2k+2 2j x 2j y α1 x f Then letting φ = x 2k+2 x α1 g applying Lemma 51 we obtain x lx x 2k+4 x α1 g L C x lx x 2k+2 x α1 f L +C y x 2k+2+α1 g 2 2 C x lx x 2k+2 x α1 f L +C x f 2 which along with (521 yields k+1 x lx y α g C x lx x 2k+2 2j y 2j α1 x f L +C x f 2 C j=0 l N 0 β =m 1 β α =2l+1 x lx β f +C x f (521 So in this case Lemma 53 is also proved Thus for all α such that α = m we have proved Lemma 53 Now we come to the proof of Lemma 33

39 18 F CHENG W-X LI AND C-J XU Proof of Lemma 33 Apply Lemma 53 with equation (51-(52 we have τ m 3 ( y l P = y (m 3! s x p α x + l x y α p C + C α =m τ m 3 (m 3! s l N 0 β =m 1 β α =2l + β =m 1 mτ m 3 (m 3! s α =m ( x lx β h ( l N 0 β =m 1 β α =2l+1 β =m 1 y ly β h + x h y ly β h x lx β h + x h If we exchange the order of the summation we can obtain by direct verification y ly β h Cm y ly β h α =m α =m l N 0 β =m 1 β α =2l+1 l N 0 β =m 1 β α =2l β =m 1 x lx β h Cm β =m 1 And direct computation also gives x h m x h α =m x lx β h Since h = 2( x u y v 2( y u x v then we have for arbitrary β N 2 0 y ly β h C ( β y l y γ u β γ v γ L γ β 2 x lx β h C γ β So with these inequalities we have mτ m 3 P w = C (m 3! s C mτ m 3 (m 3! s β =m 1 β =m 1 0 γ β ( β x l x γ u β γ v γ ( y l y h β + ( β γ + x lx γ u β γ v L 2 x lx β h ( y l y γ u β γ v

40 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 19 P x = C C mτ m 3 (m 3! s x h mτ m 3 (m 3! s 0 j ( j j x u j x v Then we have P P w +P x The rest part is to estimate P w P x We first estimate P w To do so we split the summation into where P wmj = mτm 3 (m 3! s P w C m 1 j=0 ( m 1 j β =m 1 γ =j + x lx γ u β γ v L 2 P wmj (522 ( y l y γ u β γ v Moreover we split the right side of (522 into seven terms according to the values of m j For lower j we have P wm0 C u u 1lxl y 3l xl y +Cτ u u 1lxl y Y τ P wm1 C u u 2lxl y 2l xl y +Cτ u u 2lxl y 3l xl y +Cτ 2 u 2lxl y u Y τ P wm2 Cτ 2 u u 3lxl y 3l xl y +Cτ 3 u u 3lxl y Y τ m=5 For intermediate j we have m=8 [m/2] 1 j=3 P wmj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ m 3 m=6j=[m/2] P wmj C(τ 2 +τ 5/2 +τ 3 u Xτ u Yτ

41 20 F CHENG W-X LI AND C-J XU For higher j we have m=4 P wm Cτ u 2lxl y u 3l xl y +Cτ 2 u 2lxl y u Y τ P wmm 1 C u u 1lxl y 3l xl y +Cτ u u 1lxl y Y τ In these estimations we used the fact that for vector function u = (uv the norm of u or v can be bounded by the norm of u for example y ly γ u u γ +1l xl y With this consideration the estimations can be proved similarly by the method of [8] the arguments of the commutator estimates we omit the details To estimate P x we proceed as above write where For lower j we have P x C P xmj = mτm 3 (m 3! s j=0 ( j P xmj j x u m j 2 x v P xm0 C u 1 u 2 +Cτ u 1 u 3 +Cτ 2 u 1 u Yτ P xm1 Cτ u 2 u 2 +Cτ 2 u 2 u 3 +Cτ 3 u 2 u Yτ m=4 P xm2 Cτ 3 u 3 u 3 +Cτ 4 u 3 u Yτ m=6 For mediate j we have m=8 [m/2] 1 j=3 m 3 m=6j=[m/2] Finally for higher j we have P xmj Cτ 3 u Xτ u Yτ P xmj Cτ 3 u Xτ u Yτ P xm Cτ 2 u 1 u Yτ m=5

42 GEVREY REGULARITY WITH WEIGHT FOR EULER EQUATION 21 These estimations can be proved similarly as [8] with the fact that u Xτ u Xτ u Yτ u Yτ u H m u H m With all these estimations we can complete the proof Lemma 33 Acknowledgments W Li would like to appreciate the support from NSF of China (No C-J Xu was partially supported by the Fundamental Research Funds for the Central Universities the NSF of China(No References [1] J-Y Chemin B Desjardins I Gallagher E Grenier Mathematical geophysics An introduction to rotating fluids the Navier-Stokes equations Oxford Lecture Series in Mathematics its Applications 32 The Clarendon Press Oxford University Press Oxford2006 [2] JPBourguignon HBrezis Remarks on the Euler equation J Functional Analysis 15( [3] H Chen W-X Li C-J Xu Gevrey regularity of subelliptic Monge-Ampére equations in the plane Advances in Mathematics 228( [4] H Chen W-X Li C-J Xu Gevrey hypoellipticity for a class of kinetic equations Communications in Partial Differential Equations 36 ( [5] H Chen W-X Li C-J Xu Analytic smoothness effect of solutions for spatially homogeneous Lau equation Journal of Differential Equations 248 ( [6] H Chen W-X Li C-J Xu Gevrey hypoellipticity for linear non-linear Fokker- Planck equations Journal of Differential Equations 246 ( [7] DGEbin JEMarsden Groups of diffeomorphisms the solution of the classical Euler equations for a perfect fluid BullAmerMathSoc75( [8] IKukavica VVicol The domain of analyticity of solutions to three-dimensional euler equations in a half space Discrete Continuous Dynamical Systems Volume 29 Number 1( [9] IKukavica VVicol On the analyticity Gevrey class regularity up to the boundary for the Euler equation Nonlinearity Volume 24 Number 3 ( [10] TKato Nonstationary flows of viscous idear fluids in R 3 J Functional Analysis 9( [11] TKato on classical solutions of two dimensional nonstationary Euler equations Arch Rat Mech Anal Vol 25( [12] MTaylor Partial Differential Equations III Nonlinear Equations Springer-Verlag New York 1996 [13] RTemam On the Euler equations of incompressible perfect fluids J Functional Analysis 20(1975no [14] CFoias UFrisch RTemam Existence de solutions C des équations d Euler CRAcadSciParisSérA-B 280 (1975A505-A508