On the local wellposedness of 3-D water wave problem with vorticity

Science in China Series A: Mathematics。2007SCIENCE IN CHINA PRESSAug, 2007, Vol. 50, No. 8, 1065- 1077www .scichina.com包Springerwww .springerlink.comBRIEF REPORTSOn the local wellposedness of 3-D water waveproblem with vorticity*Ping ZHANG' & Zhi-fei ZHANG21 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China;2 School of Mathematical Sciences, Peking University, Beijing 100871, China(email: zp@amss.ac.cn, zfzhang@math.pku.edu.cn)AbstractIn this article, we first present an equivalent formulation of the free boundary problemto 3-D incompressible Euler equations, then we announce our local wellposedness result concerning thefree boundary problem in Sobolev space provided that there is no self- intersection point on the initialsurface and under the stability assumption that咒(ξ)|t≈0 ≤-2co < 0 with ξ being restricted to theinitial surface.Keywords: water. -waves, free boundary, incompressible Euler equationsMSC(2000): primary 35Q35, 76B03, secondary 35J67, 35L80IntroductionWe consider in 3-D the motion of a general inviscid, incompressible fluid with a free interfacewhich separates the fuid region from the vacuum. We assume that the fuid region is below thevacum and there is no surface tension on the free surface. We assume further that the densityof mass of the fAuid is one, the gravitational field is (0,0, -1). And for any fixed timet≥0, wedenote the free interface and the fuid region by S(t) and Q(t) respectively. Then the motionof the fuid is described byUt+U.VU =-(0,0,1)-Vp on 8(t)， t≥0,divU=0 on S(t)， t≥0,(1)U(t=0,x)=U(x)，x∈ So,where U = (U,U2,U3) is the fluid velocity, p is the fuid pressure. Since we neglect the .surface tension, the pressure is zero on the interface. So on the interface, we have the followingboundary condition:p=0 on 2(t)， t≥0,(2)(1, U) is tangential to the free surface (t, (),t≥0.Received May 26, 2007; accepted June 13, 2007中国煤化工DOI:10. 1007/s11425-007-0111-7MYHCNMHG* Communicated by Lo YANGThis work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10525101,10421101 and 10601002) and the innovation grant from Chinese Academy of Sciences1066Ping ZHANG & Zhi-fei ZHANGFurthermore, we assume that there is no self-intersection point on the initial surface, and thereholds the stability condition that21()t=o≤-2co<0，ξ∈ 2(0).(3)We want to find a unique local smooth solution to (1)-(2) such that for any fixed t≥0, E(t)approaches the xy-plane at infinity, and |U(t,x, y,z)|→0,|Ut(t,x,y,z)|-→0as|(x,y,z)l→∞.When curlU = 0, the above problem is known as the water wave problem. Concerning the 2-D water wave problem, when the surface tension is neglected and the motion of free surface is asmall perturbation of still water, one could check [1- 3]. In general, the local wellposedness of the2-D full water wave problem was solved by Wu[4 and Ambrose and Masmoudil5. Concerningthe 3-D water wave problem, among them, we should mention the celebrated paper6), where Wuproved the local wellposedness of (1)-(2) under the assumptions that the fuid is irrotational andthere is no self-intersection point on the initial surface E(0). More recently, Lannes[71 consideredthe same problem in the case of finite depth.When curlU≠0 and in 2-D, one may check the local wellposedness of the free boundaryproblem for an incompressible ideal Auid in [8] for the case without the surface tension, [9] forthe case with the surface tension, and [10] for the case of finite depth.The main aim of this paper is trying to remove the irrotational assumption from the well-posedness result in [6] but with the additional stability assumption (3). Let us remark that oneof the key tools used in [6] is the so called Cliford analysis. In particular, the incompressiblefuid velocity U is Cliford analytic if curlU = 0. With this observation, Wu transformed theoriginal full water wave problem to an evolution problem of the free surface, and formulatedan equivalent quasilinear system for the evolution of the free surface. Without the irrotationalassumption on the fuid, one needs to consider not only the evolution of the free surface, 2(t),but also the evolution of the fuid inside the fuid region N(t). Moreover, the velocity field is .not Cliford analytic anymore.However, as pointed out by Wu at the begining of this work that one may still use theframework of the Clifford analysis to formulate an evolution equation for the free interface evenwithout the irrotational assumption on the fAuid. Actually from our calculation in the secondsection, we find that the free surface can be determined by∈(t,a,3) = So(a,B) +u(r,a, B)dr,with u satisfyingUtt + aVnu+anXw= A1 + A2,(4)where n is the outer normal unit vector to the free boundary, A1∈L∞[0, T]; H"(R2)) andA2∈L∞([0, T]; H8- (R2)). Comparing (4) with (5.21) of [6], here we have the additional termA2，which turns out to be a new dificulty due to the loss nf nne half snace derivative. Weovercome this dificulty by the observation that aAYH中国煤化工). Furthermore,since in our case the velocity is not irrotational, we haC N M H Ggy functional asdefined in (4.6) of [11] that enables us to handle the nonzero vorticity and that provides theestimates of lut and |lu|t，On the local wellposedness of 3 D water wave problem with vorticity1067Finally due to the motion of the fuid region, we need to introduce a completely new iterationscheme to construct the approximate solutions sequence, see (4.1- -4.3) of [11], and much moreis involved in the convergence proof to this approximate scheme.Recently under the assumptions of (3) and that the initial domain is diffeomorphic to a ball,Christodoulou and Lindblad[12,13] considered the motion of incompressible inviscid flow in avacuum. In [12], the authors proved the a priori estimates, and in [13], Lindblad proved thelocal existence of smooth solutions to this problem.We should also mention the most recent interesting results in this topic by Coutand andShkoller[14, Shatah and Zeng[15]. Again when No is a smoothly bounded domain of Rd ford = 2, 3, by using the Lagrangian framework, the authors[14] first constructed the approximatesolutions via a subtle mollifcation argument and obtained the uniform estimate to these ap-proximate solutions via the energy estimate, then the existence of solutions was provided bythe weak convergence method while the authors of [15] derived some very interesting a prioriestimates to the free boundary problem of the Euler system by the variational approach. Com-pared with [14], here we set up our problem in an unbounded initial domain and will provethe strong convergence to our approximate scheme (4.1 -4.3) of [11], and we have more detailedinformation on the evolution of free boundary. Furthermore both the energy functional and themethodology used in the present paper are completely different from those employed in [13- 15].Instead, some of the key observations made by Wu in [6] will be very useful in the subsequence.Let us complete this introduction by the notations we are going to use in this paper.Notations. As in [6], we will use the Cifford algebra C(V3) in the subsequence. We denoteR(σ) to be the real part ofσ∈C(V3). We regard pointsξ = (x,y,z) ∈R3 and their corre-sponding Cifford 1-vectorξ = xe1 + ye2 + ze3 as equivalent. This also applies to 1-vectors ofoperators such as D= 8xe1 +8ye2 + 0ze3, of which V = (8x, ay,8z) is the vector counterpart.And we will denote the conjugate ofξ byξ=xE1 + ye2+ ze3, withei= -ei fori=1,2, 3.One may check [16] for the detailed explanations on the Cliford algebra.Assume n is a C2 domain in R3 with boundary E and outer unit normal n. Let f be aC(V})-valued function on 2, we denote the Hilbert transformof f on 2 byH°f(ξ)=二p.0.ξζ-ξ， gism(')f()dS(E)， for ξ∈E,and we use Cf to denote the Cauchy integral of fξ-zCf(<)=4Js g=ajim(°)()dS(")，for z∈n.We denote the double layered potential operator by K, which is defined for the scalar-valuedfunction f on 2 byKf()=二p.o.('-) n(')f[()S()， for E∈s,2πJz Iξ'- ξ引中国煤化工and its adjoint operator K* f is defined asYHCNMHGC*f(ξ)= -元P.0.[s(∈'-ξ):n(F)2f(E')dS(∈)， for ξ∈E.1068Ping ZHANG & Zhi-fei ZHANG2 An equivalent formulation of (1)-(2)As what we have explained in the introduction, the main ideas of this section are motivatedby Section 5 of [6]. The new ingredient here is that the motion and the vorticity of the fAuidinside the fuid region have to be taken into account. For simplicity, we will present part of thedetails here, while the others have been directly cited in [6]Let 6o(a,β) with -∞< a,β <∞be a parametrization of the initial surface 2(0). LetE(t,a,B,) = (x(t, a,B,r),y(t,a,B,y),z(t,a,B,r)) with (a,B,r)∈R子be a parametrizationof the fAuid domain Q(t) by the Lagrangian coordinates (a,B, r), that is, 三(t, a, B, ) satisfies(40), with the initial datum chosen so that Eo(a,,0) = ξo(a,3). We denote ∈(t,a,β) “三(t, a,B,0) to be a parametrization of the interface 8(t).Let N示(N1,N2, N3)= fa x ξβ. Similar calculations as (5.4) of [6] giveNt = (Ea. V)U(,∈(t) Xξβ +ξa x (ξβ. V)U(t,E(t))-Nzw3+N3w2= -N. VεU(t,∈(t)} +Nrw3 - Nzw1=-N. VgU(t,E(t))- N Xw，-N1w2 + N2w1 /where w(t,a,B) = (t,E(t,a,B)) and m ef V x∪= (201, 202 003) denotes the vorticity ofthe fuid inside the fuid region, from which, we obtainNe=-Vnξt-nxw,(5)IN|with n= N being the outward unit normal to E(t).We denoteadet -品(t, ), then thanks to (2), on the interface 2(t), the incompressible Eulerequation can be rewritten asN{tt +e3= an = aN;(6)taking one time derivative to (6), we getaSttt =(NI){N+TNNe.(7)Thanks to (5), we already have an explicit formula for the second term in (7). To calculate(吊)tN, we need to use the fllowing formula (Theorem 3.19 of [16):正一石1x-zF(z)=-4n]:n(x)F(x)dS(x) -4.|x- z;DF(x) dx.When we restrict z to the interface》(t), and denote F(t,a, 3)) by f(t, a, 3), there holdsf(t,a,B) = (f(t,a,B) + Hr(xf(,a,)) -中国煤化工"(x) dx,MHCNM HGfrom which, we deducef(t,a,B) = (Hs(e)f)(t,a,B) - S(DF(,a,), .(8)On the local wellposedness of 3- D water wave problem with vorticity1069whereSG(t,a,3))'JoK(E- E(t,a, B))G(3)dV(t),Js(t)and as in [6]He(ef(t,a,B) e H()F(t,a,))=p.v.. f。K(,a',B) - (,a,)(Ea x 5Ey)f(t,a',β')dc' dβ',JR2with K(E) e去际. In short, we will denote in the subsequenceHz(of=p.o. K(5' -)(E&, x5y)f' da' dB'.Noting by the convention made at the end of the introduction, we haveDU = zj=1eiO(Ujej)= -divU + 231e2e3 + 2W2e3e1 + 203xe1e2,therefore thanks to (8), we obtainξt = Hs(t)ζt - (2)().(9)Taking two time derivatives to (9) givesStt =0 Hz(e)ζt - S(27)()=Hs(c)stt + [8; Hz()]Stt + B,(8; Hz()]Sx) - [(2)(),which ensures(I - H2(x)stt = [8; Hs()Stt + B,()x; Hs(e)S&) - ((2)()].(10)On the other hand, thanks to (5), (6) and (9), one hasNINc= (an.Vg)ξt +anxw .= ((t+e3).Vg)ξt +(Stt +e3)xw= H()(ttxξt + yttOyξt + (att + 1)82ξt) - (Stt + e3) . S(V27)(ξ)+ [xt; Hz(x)8xSe + lyt; Hs(c)]82ξt + [tt; Hz()]8.ξt + (5tt +e3) xw,from which, we deduce(I - Hxc)(-品N)=- (Stt + e3) . S(V21)(ξ) + [xtt; Hs(t)]8rSt + [yt; Hz(o)8yξt+ [zut; Hs(c)]8-ξt + (I - H()((Stt +e3) xw).(11)By summing up (7), (10), and (11), we obtain中国煤化工(I- Hs(e)= (I - Hz(x)Eu + (1-120)八MHCNMHG(12)((),川)=|NI"')= [8; Hz(<)]Stt + 0)(); Hz<()]$t) + [t; Hxc()]8rζt1070Ping ZHANG & Zhifei ZHANG+ lyut; Hs(o)!8,St + [aut; Hs()]82St - a[(23)()]- (5tt +e3). S(V2M)(E) +(1- H()(tt +ez) xw)些f.Using the notation from [6] that HE(t)nHs(t)n, then for any scalar valued function f, weR(He(a)f)= K*(t)f,(13)from which, we obtainR{(1 -- H2()别),1)}(品),|NI+R{Hio (\) w(=+(+()()which together with (12) implies that(品)。|N|=-(I + K(*())-" R(nf),(14)with f being defined in (12).By sunming up (5), (7) and (14), we obtainEttt +aVnξt + anXw= -n(1 + K<*())- 'R(nf).Now let us turn to the calculation of f. Following exactly the same proof of Lemma 3.1 of [6], .we can prove that[8;Hs(e)Jf=p.v.. K(E' -()((Se- 54)x (5gpfo' - Eafy)) da' dB'.(15)p2Thanks to (15), we have[0);Hz()St=p.o./. K('-){(x-S)x 5g)titor- ((&-()x S&)fup }da'dB'，(16)JR2and8t ([8; Hsxc)]St) = p.o.K(∈' - ()({(ut - St) x Sa)Sla- ((tt - 5t) x 5a)ξ{gr} da' dB'+p.o. K(E' -{(){(Ee - $')x ξ1{p)fe- ((e- 5{) x SIar )Stpr } dd' dβ'+ p.o.K(E' -)({(e - ()x 5g)fta- ((e- 5{) x5&)ftpv } da' dβ'+p.0.0,K(' -({(x-f4)x ξu)fla一(St- 5{) x E%,){sr }da' d$',中国煤化工(17)YHCNMHGwhile a similar calculation as (5.17) of [6] gives(ηx ξs)&a - (nx Ea)ξts =ηX {5e(Ga . Vg) - Sa(ξβ. Vz)}ξtOn the local wellposedne88 of 3-D xwater wave problem with vorticity1071=ηX {(Sa xξs)x Vg}ξt=(5a x ξs)(n. Vg)ξt一(n. (ξa x ξρ))Dξt=(Ea x ξa)(n. Vq)ξe- (n. (Ea x ξ))w,(18)from which, we rewrite the last line of (17) asp.o.A,K(∈' -{){(e- ()x 5o)Slor -((5e-5)x 6)lgr } da' d3'= p.0.8rK<(∈' - 5)(° x5})((e -().Vg)ξl dd' dB'- p.0.| arK(∈'-ξ)((561xξ]).(ξt- ))w' do' dB',(19)JR2where we denote w'些20(t, ξ(t, d', B)), and similar notations will be used in the subsequence.However noticing from (3.5) of {6] that-(n. V)K(Ea xξx) + (ξa. V)K(nxξa)+ (ξg.V)K(ξa xη)=0,we get by integration by parts to the first line of (19)8,K(' - ()(Sx x ξ)((e - 52). Ve)ξ{ da' d3'=-p.0.. K(E'-ξ)(6{ -5)x $}r{(6e- 5的).8aVe}ζ{ do' dβ'R2- p.u._K(ξ'-ξ)%ar x (ξl - ξ{){(ξt一5[) . Op,Ve;}5t do' dB'+p.o. L. K(∈'-ξ){(6l-ξi)xξa(Slar°Vg)ξ{ +5ae x (ξ{ - 5o)(5tβr . Vg)5$} da' dβ'- p.o.K<(∈' -)(Etoxξ} + E&' x SHp)((et - 5). Vq)ξi do' dB'.(20)Using (18) once again, we obtain[xu; Hsc)(8xξ) + [yut; Hs(c)](B,5s) + [zt; Hs(<)](B.Sz)= p.o.K(∈'-E) {(Ett -ft) x ξ5g)Sla'- (tt - fH)xξ, )S{gr} do' d3'+ p.u.. K(E'-(){(t- 5t) (Sa x 5}g)}w' de' ds'.(21)Substituting (16)-(17) and (19)-(21) into f, we obtainStt +aVnζt + anxw= -n(I + K())-R {nF(,t, st, w,2MI)},whereF(5,St,5tt,w, 20)= F(ξ5ζ,tt) + Fa(6, St, tt,w) - (8[(2)(5)]+ (Ett +e3) . (221)(中国煤化工e3)Xw)，(22)where similar to (5.20) of [6],MYHCNMHGFr(ξ,e,tSt)1072Ping ZHANG & Zhi-fei ZHANG2p.o.[K(E' -) {(u- 5(r)x 5gx)Sla- (u - ()x S)5lβr }dd' dS'+ p.o.JR2K(ξ' - (){((t -52) x 5lb)Slar- (( -5{) x 5lor)Slβ } da' dβ'+2p.o. L K(E' -){(6e- 5{)x 5)tta- (e-54)x Ex)lpr } da' dB'- p.o.K(E' - ()(5l -Et) x 5{(t- 5{). 8wVg;}S{ da' d3'-p.u.K(∈' - 5)%', x (6l - ξ{){(t -5(). 8p/Vg' }ξ{ do' dβ'K(ξ' -f){(6{ -f)xξ}(Slar°Vg){ +Sarx (E{ - so)(≤Lg, . V;)ζ{} da' d$'一p.u.K(∈' - 5)(Slar xξj, +6, x S{)((t - 6). Vg)ξ{ da' dB',(23)andF2(5,St,5tt,w) =- p.o.8,K(' -()((6e- 5().(E, x 5))w' da' dβ'J+ p.0. .K(∈'-ξ) (Su- ft)(Eoe x5}))w' da' dB'.We denote苗et EaxEforw, then similar argument as (20) gives8K(∈' -()((5t-f)-(Sa x Ey))w' da' dβ'JR= p.v._. 8rK(ξ' -{)(E&r x})((t -f52).(S&/ x()>) da' d3'[8rK((SE -ft)xξg) + 8pvK(E& x(St -&))]((5t- 5)- (S&, x S)))’da' dB', .therefore, we get by integration by partsF(,&,St,w)=p.o. K(∈' -ξ)(s-E)xξρ(∈e→ξ)(E x 5gx)8xn' da'd3'+ p.v.K(E' -()56wx(ξt←ξ)(ξt-ξ2).(E x ξg)8gw da' dβ'- P.v.K(ξ' -ξ)(ξt→ξ4)xξgn5{ax.(S° x ξg)∞' da'dβ'.- p.0.K(E' - ξ)%', x (t - ξ{)SlB，" (Ea x ξ(, )w' da' dB'R2K(∈' -f)(ξe→ξ{)xξβ(ζεc→ξ4).(E'va' xξ}, + S%x Sary)w da'd$'+ p.u.K(∈' -)5&, x(6e- s()(ξε- E).(∈arqrXξ}, +6&/ x 5yx)w' da' dB'中国煤化工K(' - ()(E{orxξxr +E&>MHCN M HGW de'dsK(ξ'-)(st - 5t).(Eo, x 5]xv )w' dc' dB'.(24)On the local wellposedness of 3-D water wave problem with vorticity1073On the other hand, we get by taking curl to (1) thatA2 +U.Vm= m.VU on 2(t),and we can recover U from 00 by-0U=Vx四on S()， and U(t,5(t,c,B)) = Se(t,a,B).An equivalent formulation.Let u"et ζt, and n= u中，we define for the real-valuedfunction gE*(t)g = R{nHs(e(ng)}.(25)Then, thanks to the preceding argument, we reformulate the problem (1)-(2) asutt +aVnu + anxw=-n(I + K<*())-IR{nF(ξ, u, ue,w, 2?)},a20+U.V2= 0.VU on 2(t), t≥0,-0U=Vx2 on' 82(t)， U(,(t,a,3)) = u(t,a,3),(26)u(t= 0,a,B)= uo(a,B)，ut(t = 0,a,B) = u1(a,B),200(t = 0,x,y,z) = 20o(x,y,z) = curlUo(x,y,z), for (x,y,z)∈S2o,wherea= |ut + e3|, Vnu = VnU(t, ξ(t, a, B)), and S(t) is the region under the surface S(t),which is defined by(t,a,β)= So(a,$)+ | u(r,a,B)dr,(27)/ow(,a,B) = 20(t,5(t,a,)), and u =言(u + Hs(x)u - (2)), .F(,ut,w, 20)=方(5,u, u, 2) +方(, u, u,w)- ([{<(27)()] + (u + ea). S(27)())+(I - Hz()((u +e3) x w),(28)with方(5,u, u, 2)=2p.v. .. K(ξ'-){(u -u4)x 5)&e - ((u - u( x 52)1%, } da' dβ'+p.u.. K('-<){(u -u')x up)y%, - ((u -u')x ul)',; } da' dB'JR2+ 2p.0.K(∈'-ξ)({(u-u") x5p)${ar- (u-u') xS&v )hgr } da'd3'一p.v.K(ξ'-f)(u' -u) x ζξr{(u-u).8&Vg}' da' dβ'- p.0.K(E' -<)S x(u' -u){(u-u门). ag,中国煤化工+ p.u.K(ξ'-ξ){(u' -u) x ζ}r(uo, . Veg)lMHCNMH()'} do' d$'一p.o..fK(ξ' -)(uae x ξ', + 5&r x up)((u -门). Ve)u' da' dB',(29)1074Ping ZHANG & Zhi-fei ZHANGandF2(5, u, u,W)= p.0.。K(E' -{)(u-u)xξgr(u-》. (E&x x 5's, )8ov' do'd3'+p.v./ K(E'-ξ)%'r x (u-u')(u-ul). (6o, x 5', )8grW' da' dβ'- p.o.JR2K(ξ' -E)(u-u')x ζr'&.(E&, x 5gv )' dc' d3'K(E' - 5)%ar x (u- u')u'p, .(Ea, x 6臼)0 da' dβ'+ p.u.K(ξ' - ξ)u-u')xξj(u-u').(E&'axξjv +E&r x Sa'pv )~' da' dB'+ p.o.. K(∈'-ξ)S' x(u-u')(u-u)(Sa'p' xξp+ξ&r x 5prp;)' da' dβ'[. K(ξ' - f)(u', x5iv +& xup)(u -心). (Er x 5的,)的da' d3'+ p.v.K(E'-f)(u-ul).(E xζ})w' da' dB'.(30)Remark 1. The reason why we choose to replace u in (29), which is motivated by (5.22)of [6] and (9), is to recover the unique local solution of (1)- (2) from (26), so that in particularthe velocity field U detrmined by (26) satisfies divU = 0 provided that divUo = 0.Finally let us determine the initial condition ofu in (26). We first get by taking Ar to (9)ξtt = [8; Hs(t)]St + Hs(x)ζtt - at[(W)(E)],thennt - nHses5tt = n[8r; He<()]ζt - n8({[(21)()].On the other hand, thanks to (6), we havea-n.e3= -nξtt -n X e3, .noting that a- n.e3 is a real valued function, we arrive at(I +K<*())(a-n.e3) = R{(I + H时(x)(a- n.e3)}= - R{n[8; Hzr()]St - ne([(2)()] + HE((n x e3)},from which, we obtaina=n.e3- (I + K*())-'{R(n[8; Hs(x)5st - n5[(2)(6)] + Hato)(n xe))}_> (31)While notice that divU = 0, let 3(t, a, B, r) be defined in (40), we have8(a,B,r)中国煤化工from which, we deduce thatYHCNMHG8;[()()]= . (El - (&).VK(E' - )0(,')6 dc' d3' dy'On the local wellposedness of 3- D water wave problem twith vorticity1075+ K(E'-)(8 +U. V)20(t,E')J da' dB' dy',whereE' = E(t,a',B',r'). Then thanks to (40) and (26), we obtain8[(20)(ξ)] =Js(t)(U(t,E)- U(t,ξ)). VK(E - )20(t,B)dV(t)+K(E - ()2. VU(t, E)dV().(32)Js()Initial data. Let三(0,a,β,r)= Eo(a,B,r) and U(0,x,y,z)= Uo;ξ(0,a,β)= Eo(a,B,0) and ξ:(0,a, B) = Uo(Eo(a,3)).(33)Thanks to the definition of u and (31), we have .u1=-ez + (no . e3)no - no(I + K8)- 1 {R(mo[8; H2(o)]uo - n([()()]|t=o+ Hz(o)(no x e))}, and uo = U(5o), .(34)where no is the outer unit normal of E(0), Kt is the adjoint operator of the double layeredpotential defined on 2(0). Moreover, thanks to (15) and (32), there holds[0; Hs(o)}uo =p.o.K(E% - 5o) (uo -咖) x (Or568aru - 8a, 60gv u') da' dB';8[(W)(E)]|t=o= | (U(3) - U(Eo)). VK(E - 0)o(3) dV(0)+ K(E- ξo)咖o VU(E) dV(0).JsoWe know no =陆昌, then under the assumption of (3), there holdsao=|u1 +e3|≥2co. .(35)As conventions in the subsequent sections, we will denoteδ= (8a, 8g) or (8a,8g,8),V =(0x, 8y,82), andq> 1 is a positive constant.With the above formulation, we prove the following result in [11]: .Theorem 1. Lets≥ 5 be an integer. Let Eo(a,B,r), (x,β,r)∈ R子, be a parametrizationof the initial domain No, Uo∈H8+1(N2o) with divUo = 0 and there holds (3). We denote .ξo(a,β) eet Eo(a,B,0) to be a parametrization of initial interface 2(0). We assume that thereare positive constants Co,, v> 0 and unit vectors ea, ep, such that1.8aEo-ea, 8gEo-ep, 8Eo-e3∈H*(R年),andeaXeβ=e3;(36)E51-ex，a3o1-ey, Az35l-ez∈H*(S2中国煤化工ea,ep,e3)-1;YHCNMHG2. det(0(a.6,2)> 2v;(37)a(a,B,r)3. |Eo(a,$)-o(a',B})|≥ 2C0((la-a'l+|β-β'|) for V (a,),(a',B)∈R2; .(38)1076Ping ZHANG & Zhi-fei ZHANG4. |a$o(a,B) x 0pξo(a,B)|≥2μ for V(a,B)∈R2,(39)where we denote E5' to be the inverse mapping of Eo. Then there exists a positive time Tsuch that (1)- (2) has a unique solution (u,U,p()), with 8fu∈L∞(0,T];H$+号(R2))∩C([0,T];H*'+号(R2)), fori=0,1,2, ands' < s, the interface 2(t) is determined byE(t):∈(t)=ξo+ | u(r)dr， t∈ [0,T],2(t) is the region below (t), and VU, Vp∈ L∞([0,T]; H*(2(t)) nC(0,T]; H° ()). Pur-thermore, let E(t,a,β, r) solve会(t,a,B,r) =U(t,三(t,a, ,))，(40)E(t = 0,a,β,r) = Eo(a,B,r).Then 8aE-ea, 0gE-ep, a,E-e3∈Lip([0,T]; H*(RA)) and (37)- (39) hold for日with 2v,2Co and 2μ there being replaced by v,Co and μ respectively.Remark 2. The disadvantage of the gravitational field (0,0,-1) in (1) is that we cannotprove∪∈L∞([0, T]; L2(t)) even if Uo∈L2(S2o). Actually under the assumption of (3), wedo not need this term in (1), and it will not arise any further dificulty in the subsequence.However, as (3) can be proved for the irrotational fuid under this gravitational field (see [6]),here we just keep it in (1). .Acknowledgements Wu Sijue pointed out that one may use the Cliford analysis frameworkto study the rotational water wave problem. She presented us an outline of ideas on how tohandle the nonzero vorticity in this framework, and was involved in some initial calculations inthis work. We are grateful for that.A part of this work was done when Zhang Ping was visiting Courant Institute of New YorkUniversity. He would like to thank the hospitality of the Institute.References[1] Nalimov v I. The Cauchy-Poisson problem, Dymarnika Splosh. Sredy (in Russian), 18: 104 -210 (1974)2] Yosihara H. Gravity waves on the free surface of an incompressible perfect fuid of finite depth. Publ ResInst Math Sci, 18: 49 96 (1982)3] Craig W. An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits.Comm Partial Differential Equations, 10: 787- 1003(1985)4] Wu s. Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent Math, 130: 39-72[5] Ambrose D M, Masmoudi N. The zero surface tension limit of two-dimensional water waves, Comm PureAppl Math, 58: 1287-1315 (2005)6] Wu s. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J Amer Math Soc, 12:445- 495 (1999)[7] Lannes D. Well-posednes of the water-waves equations. J Amer Math Soc, 18: 605- 654 (2005)8] Iguchi T, Tanaka N, Tani A. On a free boundary problenAuid in two spacedimensions, Adu Math Sci Appl, 9: 415-472 (1999)中国煤化工9] Ogawa M, Tani A A. Free boundary problem for an incon.TYHC N M H Gface tensin. MathModels Methods Appl Sci, 12: 1725-1740 (2002)[10] Ogawa M, Tani A A. Incompressible perfect fuid motion with frce boundary of finite depth. Adu Math SciAppl, 13: 201-223 (2003)On the local wellposedness of 3-D water wave problem with vorticity1077[11] Zhang P, Zhang Z F. On the free boundary problem of 3-D incompressible Euler equations. preprint 2006[12] Christodoulou D, Lindblad H. On the motion of the free surface of a liquid. Comm Pure Appl Math, 53:1536- 1602 (2000)[13] Lindblad H. Well- posedness for the motion of an incompressible liquid with free surface boundary. Ann ofMath, 162(2): 109 194 (2005)[14] Coutand D, Shkoller s. Well- posedness of the free suface incompressible Euler equations with or withoutsurface tension. J Amer Math Soc, 20: 829 930 (2007)[15] Shatah J, Zeng C C. Geometry and a priori estimates for free boundary problems of the Euler's equation.preprint 2006[16] Gilbert J, Murray M. Clifford Algebras and Dirac Operators in Harmonic Analysis. London: CombridgeUniversity Press, 1991中国煤化工MYHCNMHG