Anisotropic decay and global well-posedness of viscous surface waves without surface tension
aa r X i v : . [ m a t h . A P ] N ov ANISOTROPIC DECAY AND GLOBAL WELL-POSEDNESS OF VISCOUSSURFACE WAVES WITHOUT SURFACE TENSION
YANJIN WANG
Abstract.
We consider a viscous incompressible fluid below the air and above a fixed bottom.The fluid dynamics is governed by the gravity-driven incompressible Navier-Stokes equations,and the effect of surface tension is neglected on the free surface. The global well-posednessand long-time behavior of solutions near equilibrium have been intriguing questions since Beale(
Comm. Pure Appl. Math.
34 (1981), no. 3, 359–392). It had been thought that certainlow frequency assumption of the initial data is needed to derive an integrable decay rate ofthe velocity so that the global solutions in 3 D can be constructed, while the global well-posedness in 2 D was left open. In this paper, by exploiting the anisotropic decay rates of thevelocity, which are even not integrable, we prove the global well-posedness in both 2 D and 3 D ,without any low frequency assumption of the initial data. One of key observations here is acancelation in nonlinear estimates of the viscous stress tensor term in the bulk by using Alinhacgood unknowns, when estimating the energy evolution of the highest order horizontal spatialderivatives of the solution. Introduction
Formulation in Eulerian coordinates.
We consider a viscous, incompressible fluidevolving in a d -dimensional moving domainΩ( t ) = { y ∈ R d | − b < y d < η ( y h , t ) } . (1.1)Here the dimension d = 2 ,
3, and y = ( y h , y d ) for y h = ( y , y d − ) ∈ R d − the horizontalcoordinate and y d the vertical one. The lower boundary of Ω( t ), denoted by Σ b , is assumed tobe rigid and given, but the upper boundary, denoted by Σ( t ), is a free surface that is the graphof the unknown function η : R d − × R + → R . We assume that b > t ≥ u ( · , t ) : Ω( t ) → R d and p ( · , t ) : Ω( t ) → R , respectively.For each t > u, p, η ) satisfy the gravity-driven free-surface incompressibleNavier-Stokes equations: ∂ t u + u · ∇ u + ∇ p − µ ∆ u = 0 in Ω( t )div u = 0 in Ω( t )( pI d − µ D u ) ν = gην on Σ( t ) ∂ t η = u d − u h · Dη on Σ( t ) u = 0 on Σ b . (1.2)Here µ > g > pI d − µ D u isknown as the viscous stress tensor for I d the d × d identity matrix and D u = ∇ u + ( ∇ u ) t thesymmetric gradient of u , and ν = ( − Dη, / q | Dη | is the outward-pointing unit normal onΣ( t ) for Dη for the horizontal gradient of η . The fourth equation in (1.2) is called the kinematicboundary condition which implies that the free surface is advected with the fluid, where u h and u d are the horizontal and vertical components of the velocity, respectively. Note that in Date : November 12, 2019.2000
Mathematics Subject Classification.
Primary 35Q30, 35R35, 76D03; Secondary 35B40, 76E17.
Key words and phrases.
Viscous surface waves; Free boundary problems; Navier-Stokes equations; Globalwell-posedness; Decay.This research was supported by the National Natural Science Foundation of China (No. 11771360, 11531010)and the Natural Science Foundation of Fujian Province of China (No. 2019J02003). (1.2) we have shifted the gravitational forcing from the bulk to the boundary and eliminatedthe constant atmospheric pressure, p atm , in the usual way by adjusting the actual pressure ¯ p according to p = ¯ p + gy d − p atm . Without loss of generality, we may assume that µ = g = 1.To complete the formulation of the problem, we must specify the initial conditions. Wesuppose that the initial surface Σ(0) is given by the graph of the function η (0) = η : R d − → R ,which yields the initial domain Ω(0) on which we specify the initial data for the velocity, u (0) = u : Ω(0) → R d . We will assume that η > − b on R d − and that ( u , η ) satisfy certaincompatibility conditions, which we will describe later.1.2. Reformulation in flattening coordinates.
In order to work in a fixed domain, wewant to flatten the free surface via a coordinate transformation. We will use a flatteningtransformation introduced by Beale in [3]. To this end, we consider the fixed equilibriumdomain Ω := { x ∈ R d | − b < x d < } (1.3)for which we will write the coordinates as x ∈ Ω. We write Σ := { x d = 0 } for the upperboundary of Ω, and we view η as a function on Σ × R + . We define¯ η := P η = harmonic extension of η into the lower half space , (1.4)where P η is defined by (A.1). The harmonic extension ¯ η allows us to flatten the coordinatedomain via the mappingΩ ∋ x ( x h , φ ( x, t )) := Φ( x, t ) = ( y h , y d ) ∈ Ω( t ) , (1.5)where φ ( x, t ) = x d + ϕ ( x, t ) for ϕ ( x, t ) = ˜ b ¯ η ( x, t ) with ˜ b = (1 + x d /b ) . Note that Φ(Σ , t ) = Σ( t )and Φ( · , t ) | Σ b = Id Σ b , i.e. Φ maps Σ to the free surface and keeps the lower surface fixed. Wehave ∇ Φ = (cid:18) I d − Dϕ J (cid:19) and A := ( ∇ Φ − ) T = (cid:18) I d − − DϕK K (cid:19) (1.6)for J = 1 + ∂ d ϕ = 1 + ¯ η/b + ∂ d ¯ η ˜ b and K = J − . (1.7)Here J = det ∇ Φ is the Jacobian of the coordinate transformation.If η is sufficiently small (in an appropriate Sobolev space), then the mapping Φ is a diffeo-morphism. This allows us to transform the problem to one on the fixed spatial domain Ω for t ≥
0. In the new coordinates, the system (1.2) becomes ∂ A t u + u · ∇ A u + ∇ A p − ∆ A u = 0 in Ωdiv A u = 0 in Ω S A ( p, u ) N = η N on Σ ∂ t η = u · N on Σ u = 0 on Σ b ( u, η ) | t =0 = ( u , η ) . (1.8)Here we have written the differential operators ∂ A t , ∇ A , div A , and ∆ A by ∂ A t := ∂ t − K∂ t ϕ∂ d ,( ∇ A ) i := A ij ∂ j , div A := ∇ A · , and ∆ A := div A ∇ A . We have also written N := ( − Dη,
1) for thenon-unit normal to Σ( t ) and S A ( p, u ) = ( pI d − D A u ) for D A u := ∇ A u + ( ∇ A u ) t the symmetric A− gradient of u . Note that if we extend div A to act on symmetric tensors in the natural way,then div A S A ( p, u ) = ∇ A p − ∆ A u for div A u = 0. Recall that A is determined by η through therelation (1.6). This means that all of the differential operators in (1.8) are connected to η , andhence to the geometry of the free surface. ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 3
Previous results.
Free boundary problems in fluid mechanics have been studied by manyauthors in many different contexts. Here we will mention only the work most relevant to ourpresent setting, that is, the viscous surface wave problem, which has attracted the attention ofmany mathematicians since the pioneering work of Beale [2].In [2], Beale proved the local well-posedness of the viscous surface wave problem withoutsurface tension, (1.2), in Lagrangian coordinates: given Ω(0) = Ω and u ∈ H r − (Ω ) for r ∈ (3 , / T > , T ] so that v ∈ L (0 , T ; H r (Ω )) ∩ H r/ (0 , T ; L (Ω )), where v = u ◦ ζ for ζ the Lagrangian flow map satisfying ∂ t ζ = v in Ω and ζ (0) = Id in Ω . Beale [2] also showed that for certain Θ ∈ H r (Ω ) with Θ = 0 on Σ b , therecannot exist a curve of solutions v ε , defined for ε near 0, with ζ ε (0) = Id + ε Θ and v ε (0) = 0,and v ε is of the form v ε = εv (1) + ε v (2) + O ( ε ), such that v ε ∈ L (0 , ∞ ; H r (Ω )) (1.9)and lim t →∞ ζ ε ( t ) | Σ = 0 . (1.10)This would suggest a nondecay theorem that a “reasonable” small-data global well-posednesswith decay of the free surface is false and that any existence theorem for all time would nec-essarily have a more special hypothesis or a weaker conclusion than the assertion which wasshown to be untrue in [2]. Thereafter, the global well-posedness and long-time behavior of solu-tions to (1.2) near equilibrium have been intriguing questions since [2]. Sylvester [13] and Taniand Tanaka [16] studied the existence of small-data global-in-time solutions via the parabolicregularity method as [2], and they make no claims about the decay of the solutions. Sylvester[14] discussed the decay of the solution for the linearized problem around equilibrium in 2 D . Aspointed out by Guo and Tice [6, 7], due to the growth in the highest order spatial derivatives of η as will be seen later, it seems impossible to construct global-in-time solutions to (1.2) withoutalso deriving a decay result, at least by energy methods.For the problem with surface tension, that is, the fourth equation in (1.2) is modified to be( pI d − µ D u ) ν = gην − σHν on Σ( t ) , (1.11)where H = D · (cid:18) Dη/ q | Dη | (cid:19) is the mean curvature of the free surface and σ > u ∈ H r − / (Ω) and η ∈ H r (Σ) for r ∈ (3 , /
2) are sufficiently small. Moreover, Bealeand Nishida [4] showed that for the flat bottom in 3 D if η ∈ L (Σ) is small, then the solutionconstructed in [3] obeys(1 + t ) k u ( t ) k H (Ω) + X j =0 (1 + t ) j +1 (cid:13)(cid:13) D j η ( t ) (cid:13)(cid:13) L (Σ) < ∞ (1.12)and that this decay rate is optimal. Note that if ignoring the different coordinates, the decayof η in (1.12) implies (1.10), but the decay rate of u is not sufficiently rapid to guarantee (1.9),even with surface tension.If the domain is horizontally periodic and assuming that η has the zero average, then thesituation is significantly different. Nishida, Teramoto, and Yoshihara [12] showed that for theproblem with surface tension and with a flat bottom, there exists γ > e γt (cid:16) k u ( t ) k H (Ω) + k η ( t ) k H (Σ) (cid:17) < ∞ . (1.13)Hataya [8] proved that for the problem without surface tension and with a flat bottom, if u ∈ H r − (Ω) and η ∈ H r − / (Σ) for r ∈ (5 , /
2) are sufficiently small, then there exists aunique global solution satisfying Z ∞ (1 + t ) k u ( t ) k H r − (Ω) dt + (1 + t ) k η ( t ) k H r − (Σ) < ∞ . (1.14) YANJIN WANG
Guo and Tice [6] showed that for the problem without surface tension and with a curved bottomif u ∈ H N (Ω) and η ∈ H N +1 / (Σ) for N ≥ t ) N − (cid:16) k u ( t ) k H N +4 (Ω) + k η ( t ) k H N +4 (Σ) (cid:17) < ∞ . (1.15)Tan and Wang [15] established the global-in-time zero surface tension limit of the problem withsurface tension and with a curved bottom for the sufficiently small initial data. We remark thatthe argument in Beale’s nondecay theorem of [2] works in horizontally periodic domains as well,and Guo and Tice [6] showed that the zero average of η prevents the choice of Θ in [2].In light of the decay of u in (1.12) of [3], we may not expect a global well-posedness ofthe problem in horizontally infinite domains without surface tension, (1.2), with the solutionsatisfying (1.9). Hataya and Kawashima [9] announced that for the problem (1.8) with a flatbottom in 3 D , if u ∈ H r − (Ω) and η ∈ H r − / (Σ) for r ∈ (5 , /
2) and η ∈ L (Σ) aresufficiently small, then there exists a unique global solution satisfying(1 + t ) k u ( t ) k H (Ω) + X j =0 (1 + t ) j +1 (cid:13)(cid:13) D j η ( t ) (cid:13)(cid:13) L (Σ) < ∞ , (1.16)but they provides only a terse sketch of their proposed proof and the full details are not availablein the literature to date. Guo and Tice [7] proved that for the problem (1.8) with a flat bottomin 3 D , if u ∈ H (Ω), η ∈ H / (Σ), I λ u ∈ L (Ω) and I λ η ∈ L (Σ) for 0 < λ < I λ is the Riesz potential in the horizontal space) are sufficiently small, then there exists a uniqueglobal solution such that(1 + t ) λ k u ( t ) k H (Ω) + X j =0 (1 + t ) j + λ (cid:13)(cid:13) D j η ( t ) (cid:13)(cid:13) L (Σ) < ∞ . (1.17)We remark that it was pointed out in [7] that the requirement of λ > D wasleft open. The main purpose of this paper is to show the global well-posedness of (1.8) inboth 2 D and 3 D , without any low frequency assumption of the initial data. This gives a closeranswer to the question in the nondecay theorem of Beale [2]. It should be pointed out that theglobal well-posedness of the problem with a curved bottom is still open, and the key point willbe how to deduce the decay of the solution.2. Main results
Statement of the results.
We will work in a high-regularity context, essentially withregularity up to 2 N temporal derivatives for an integer N ≥
5. This requires us to use u and η ,by using the equations (1.8), to construct the initial data ∂ jt u (0) and ∂ jt η (0) for j = 1 , . . . , N and ∂ jt p (0) for j = 0 , . . . , N −
1. These data must then satisfy various conditions, which inturn require u and η to satisfy 2 N compatibility conditions. We refer the reader to [5] for theconstruction of those initial data and the precise description of the 2 N compatibility conditions.We write H k (Ω) with k ≥ H s (Σ) with s ∈ R for the usual Sobolev spaces, with normsdenoted by k·k k and |·| s , respectively. For a vector v ∈ R d for d = 2 ,
3, we write v = ( v h , v d ) for v h the horizontal component of v and v d the vertical component. We write Df for the horizontalgradient of f , while ∇ f denotes the usual full gradient. Let d = 2 , N ≥
5. We define thehigh-order energy as E N := N X j =0 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j + N − X j =0 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N − j − + | η | N − + N X j =1 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N − j (2.1) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 5 and the high-order dissipation rate as D N := k u k N + N X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j +1 + k∇ p k N − + N − X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N − j + | Dη | N − / + | ∂ t η | N − / + N +1 X j =2 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N − j +5 / . (2.2)We define E N := | η | N (2.3)and D N := k u k N +1 + k∇ p k N − + | Dη | N − / + | ∂ t η | N − / . (2.4)We also define F N := | η | N +1 / . (2.5)We define the low-order energy as E N +2 , = k Du h k N +2) − + k u d k N +2) + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j + (cid:13)(cid:13) ∇ p (cid:13)(cid:13) N +2) − + k ∂ d p k N +2) − + N +1 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N +2) − j − + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − + N +2 X j =1 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j . (2.6)Here the subscript “2” basically refers to the “minimal derivative” count 2 of η .Finally, we define G N ( t ) := sup ≤ r ≤ t E N ( r ) + Z t D N ( r ) dr + sup ≤ r ≤ t E N ( r )(1 + r ) ϑ + Z t E N ( r )(1 + r ) ϑ dr + Z t D N ( r )(1 + r ) ϑ + κ d dr + sup ≤ r ≤ t F N ( r )(1 + r ) ϑ + Z t F N ( r )(1 + r ) ϑ dr + sup ≤ r ≤ t (1 + r ) E N +2 , ( r ) , (2.7)where ϑ > d = 3, κ > d = 2, κ = 1 /
2. Then the main result of this paper is stated as follows.
Theorem 2.1.
Let d = 2 , and N ≥ . Suppose the initial data ( u , η ) satisfy the necessarycompatibility conditions of the local well-posedness of (1.8) . There exists an ε > so that if E N (0) + F N (0) ≤ ε , then there exists a unique solution ( u, p, η ) to (1.8) on the interval [0 , ∞ ) that achieves the initial data. The solution obeys the estimate G N ( ∞ ) . E N (0) + F N (0) . (2.8) In particular, we have (1 + t ) − κ d k u h ( t ) k C (¯Ω) + (1 + t ) (cid:16) k Du h ( t ) k C (¯Ω) + | ∂ d u h ( t ) | C (Σ) + k u d ( t ) k C (¯Ω) (cid:17) . E N (0) + F N (0) . (2.9) Remark 2.2.
The boundedness of G N ( ∞ ) in Theorem 2.1, (2.8) , yields the decay estimateof E N +2 , ( t ) . Note that the decay of the velocity is anisotropic, in terms of the horizontalor vertical components, and the horizontal or vertical spatial derivatives (on Ω or Σ ) of thehorizontal component. Such anisotropic decay of the velocity is essential in the derivation of YANJIN WANG the estimate (2.8) . We remark that without the low frequency assumption of the initial data thedecay rate (1 + t ) − of E N +2 , ( t ) , mainly referring to the term (cid:12)(cid:12) D η (cid:12)(cid:12) , is optimal, see [4, 7] . Strategy of the proof.
The proof of Theorem 2.1 is a revised one of Theorem 1.3 in [7],and the main part is to prove a priori (2.8). However, a crucial new observation and severalnew ideas, which will be explained in details below, yield our improvement of Theorem 2.1compared to Theorem 1.3 in [7]; especially, we do not need the low frequency assumption of theinitial data as required in [7], and our results cover both the 3 D and 2 D cases.We first briefly sketch the proof of Theorem 1.3 in [7] for the 3 D case. Note that E N := E N + E N and D N := D N + D N are the high-order energy and dissipation rate used in [7]when λ = 0 therein. Basing on the natural energy structure of (1.8),12 ddt (cid:18)Z Ω J | u | + Z Σ | η | (cid:19) + 12 Z Ω J | D A u | = 0 , (2.10)and making full use of the structure of the problem, [7] proved that for E N is small, E N ( t ) + Z t D N ( r ) dr . E N (0) + Z t K ( r ) F N ( r ) dr, (2.11)where K = k∇ u k C (¯Ω) + | D ∇ u | C (Σ) . The difficulty is then to control the right hand side of (2.11),and the only way to estimate F N is through the fourth transport equation for η in (1.8); [7]derived F N ( t ) . exp (cid:18) C Z t p K ( r ) dr (cid:19) (cid:20) F N (0) + t Z t D N ( r ) dr (cid:21) . (2.12)Hence to close (2.11) and (2.12), we would see twice the necessity of showing K ( t ) . (1 + t ) − − γ for some γ >
0. By assuming that I λ u ∈ L (Ω) and I λ η ∈ L (Σ) for 0 < λ < E N +2 , (see (3.10) for the definition), E N +2 , ( t ) . ( E N (0) + F N (0))(1 + t ) − − λ . (2.13)It follows by the interpolation estimates (similarly as in Lemma 3.4) and (2.13) that K . E κ / N E − κ / N +2 , . (1 + t ) − (2+ λ )(1 − κ / . (2.14)Consequently, [7] can close the estimates (2.11)–(2.12) by requiring λ > D . We remarkthat the energy estimates for the 2 D case can not be closed along the same way as above since(2 + λ )(1 − κ / < / κ = 1 / < λ < / λ < / I λ in dimension 1). Hence, the global well-poseness in 2 D was left open.Our main goals of this paper are to remove the assumption of λ > D and to coverthe 2 D case. This requires us to revise all the estimates (2.11)–(2.13). Our starting point isthat we can improve the decay estimate (2.13) by replacing E N +2 , with the stronger low-orderenergy E N +2 , so that for λ = 0,(1 + t ) E N +2 , ( t ) . E N (0) + F N (0) . (2.15)Such improved decay estimate is essential for our improvement compared to [7]. Secondly, werefine from the derivation of the estimate (2.12) that ∂ t F N . | u | N +1 / p F N + | Du h | C (Σ) F N . (2.16)The subtle points here are that we need only the boundary control of | u | N +1 / rather than D N and that | Du h | C (Σ) . E N +2 , . So if we can prove that Z t | u ( r ) | N +1 / (1 + r ) ϑ dr . E N (0) + F N (0) , ϑ ≥ , (2.17)then a time weighted argument on (2.16) yields F N ( t )(1 + t ) ϑ + Z t F N ( r )(1 + r ) ϑ dr . E N (0) + F N (0) , ϑ ≥ . (2.18) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 7
Apparently, the estimates (2.15) and (2.18) are not sufficient for closing the estimate (2.11),and we need to revise (2.11). We recall from [7] that the term KF N in (2.11) results from thenonlinear estimates of the viscous stress tensor term when estimating the highest order spatialderivatives of the solution. So we will split the estimates (2.11) by singling out E N from E N and D N from D N . In this splitting, we can first prove E N ( t ) + Z t D N ( r ) dr . E N (0) + F N (0) . (2.19)The crucial part is then to control E N and D N . Applying the highest order horizontal spatialderivatives ∂ α for α ∈ N d − with | α | = 4 N to (1.8), we find12 ddt (cid:18)Z Ω J | ∂ α u | (cid:19) + Z Ω J (cid:18) ∂ α ( D A u ) : D A ∂ α u − ∂ α p div A ∂ α u (cid:19) = Z Σ ∂ α ( pI − D A u ) N · ∂ α u + · · · . (2.20)Note that when ∂ α hints A leaded to the appearance of the term KF N in (2.11), and “+ · · · ”denotes the terms that can be controlled other than KF N . Then our idea is to use a crucialcancelation observed by Alinhac [1]. More precisely, the direct computation yields ∂ α ∂ A i u = ∂ A i ( ∂ α u − ∂ A d u∂ α ϕ ) + ∂ A d ∂ A i u∂ α ϕ + · · · , (2.21)which implies that the highest order term of η will be canceled when we use the good unknown U α = ∂ α u − ∂ A d u∂ α ϕ . Thus considering the equations of U α in Ω, instead of (2.20), we get12 ddt (cid:18)Z Ω J | U α | (cid:19) + 12 Z Ω J | D A U α | = Z Σ ( ∂ α pI − D A U α ) N · U α + · · · . (2.22)This means that we have canceled the term KF N in the bulk. By estimating the right handside of (2.22) by using the boundary conditions in (1.8), we obtain12 ddt (cid:18)Z Ω J | U α | + Z Σ | ∂ α η | (cid:19) + 12 Z Ω J | D A U α | . |∇ u | C (Σ) F N + |∇ u | C (Σ) E N + · · · . (2.23)Note that the term |∇ u | C (Σ) F N in (2.23) stems from using the third equation in (1.8), whichcan not be canceled by using Alinhac good unknowns, and |∇ u | C (Σ) E N results from usingthe fourth equation in (1.8). However, the crucial point here is that by using the horizontalcomponent of the third equation in (1.8), we can show that |∇ u | C (Σ) . E N +2 , . Then a timeweighted argument on (2.23), together with (2.10), yields E N ( t )(1 + t ) ϑ + Z t E N ( r )(1 + r ) ϑ dr + Z t (cid:13)(cid:13) U N ( r ) (cid:13)(cid:13) (1 + r ) ϑ dr . E N (0) + F N (0) , ϑ > , (2.24)where we denote (cid:13)(cid:13) U N (cid:13)(cid:13) := k u k + P α ∈ N d − , | α | =4 N k U α k . Another advantage of the goodunknown is that, by the definition of U α , | u | N +1 / . (cid:13)(cid:13) U N (cid:13)(cid:13) + | ∂ d u | C (Σ) F N . (cid:13)(cid:13) U N (cid:13)(cid:13) + E N +2 , F N . (2.25)Thus, (2.24) implies the validity of (2.17) for ϑ > ϑ >
0. Then we seereasonably that not assuming λ > F N in time. Finally, tocontrol D N it still involves KF N , and by the interpolation estimate (3.36) of K in Lemma 3.4, K . (1 + t ) − − κ d / , we can show Z t D N ( r )(1 + r ) ϑ + κ d dr . E N (0) + F N (0) . (2.26)We remark that in the derivation of the estimates (2.15), (2.18), (2.19), (2.24) and (2.26),certain powers of greater than 1 of G N ( t ) need to be added on the right hand sides of theseestimates. Consequently, summing over these estimates, the a priori estimate (2.8) is thenclosed by assuming that E N (0) + F N (0) is sufficiently small. The proof of Theorem 2.1 canbe thus completed by a continuity argument by combining the local existence theory in Guo YANJIN WANG and Tice [5] and our a priori estimates. We remark that although the strategy is carried outunifiedly for both 3 D and 2 D cases, the analysis in 2 D is much more involved.2.3. Notation.
We write N = { , , , . . . } for the collection of non-negative integers. Whenusing space-time differential multi-indices, we write N m = { α = ( α , α , . . . , α m ) } to empha-size that the 0 − index term is related to temporal derivatives. For just spatial derivatives wewrite N m . For α ∈ N m we write ∂ α = ∂ α t ∂ α · · · ∂ α m m . We define the parabolic counting ofsuch multi-indices by writing | α | = 2 α + α + · · · + α m . For α ∈ N d , we write α = ( α h , α d ) . For a given norm k·k and integers k ≥ m ≥
0, we introduce the following notation for sumsof spatial derivatives: (cid:13)(cid:13)(cid:13) D km f (cid:13)(cid:13)(cid:13) := X α ∈ N d − m ≤| α |≤ k k ∂ α f k and (cid:13)(cid:13)(cid:13) ∇ km f (cid:13)(cid:13)(cid:13) := X α ∈ N d − m ≤| α |≤ k k ∂ α f k . (2.27)For space-time derivatives we add bars to our notation: (cid:13)(cid:13)(cid:13) ¯ D km f (cid:13)(cid:13)(cid:13) := X α ∈ N d − m ≤| α |≤ k k ∂ α f k and (cid:13)(cid:13)(cid:13) ¯ ∇ km f (cid:13)(cid:13)(cid:13) := X α ∈ N d m ≤| α |≤ k k ∂ α f k . (2.28)When k = m ≥ (cid:13)(cid:13)(cid:13) D k f (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) D kk f (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) ∇ k f (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ∇ kk f (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) ¯ D k f (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ¯ D kk f (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) ¯ ∇ k f (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ¯ ∇ kk f (cid:13)(cid:13)(cid:13) . (2.29)We employ the Einstein convention of summing over repeated indices for vector and tensoroperations. Throughout the paper we assume that N ≥ C > d = 2 , N, ϑ and κ d , but does notdepend on the data, etc. We refer to such constants as “universal”. They are allowed to changefrom line to line. We employ the notation A . A to mean that A ≤ CA for a universalconstant C >
0. To avoid the constants in various time differential inequalities, we employ thefollowing two conventions: ∂ t A + A . A means ∂ t e A + A . A for any A . e A . A (2.30)and ∂ t ( A + A ) + A . A means ∂ t ( C A + C A ) + A . A for constants C , C > . (2.31)We omit the differential elements of the integrals over Ω and Σ, and also sometimes the timedifferential elements. 3. Preliminaries
We will assume throughout the rest of the paper that the solutions are given on the interval[0 , T ] and obey the a priori assumption G N ( t ) ≤ δ, ∀ t ∈ [0 , T ] (3.1)for an integer N ≥ δ > . This implies in particular that12 ≤ J ≤ , ∀ ( t, x ) ∈ [0 , T ] × ¯Ω . (3.2)(3.1) and (3.2) will be used frequently, without mentioning explicitly. ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 9
Energy functionals.
Below we define the energy functionals used in our analysis. Werecall the definitions of E N , D N , E N , D N , F N , E N +2 , and G N from (2.1)–(2.7) in Section2.1, respectively. We define the low-order dissipation rate by D N +2 , = (cid:13)(cid:13) D u h (cid:13)(cid:13) N +2) − + k Du d k N +2) + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j +1 + (cid:13)(cid:13) ∇ p (cid:13)(cid:13) N +2) − + k D∂ d p k N +2) − + k∇ ∂ t p k N +2) − + N +1 X j =2 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N +2) − j + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / + | D∂ t η | N +2) − / + N +3 X j =2 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j +5 / . (3.3)Recall that we employ the derivative conventions (2.27)–(2.29) from Section 2.3. We define thehigh-order tangential energy by¯ E N := (cid:13)(cid:13)(cid:13) D N − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D N − ∂ t u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) D N − η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t η (cid:12)(cid:12)(cid:12) (3.4)and the corresponding tangential dissipation rate by¯ D N := (cid:13)(cid:13)(cid:13) D N − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D N − ∂ t u (cid:13)(cid:13)(cid:13) . (3.5)The low-order tangential energy is¯ E N +2 , := (cid:13)(cid:13)(cid:13) ¯ D N +2)2 u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N +2)2 η (cid:12)(cid:12)(cid:12) , (3.6)and the corresponding tangential dissipation rate is¯ D N +2 , := (cid:13)(cid:13)(cid:13) ¯ D N +2)2 u (cid:13)(cid:13)(cid:13) . (3.7)We also define two special quantities K := k u k C (¯Ω) + (cid:12)(cid:12) ∇ u (cid:12)(cid:12) C (Σ) (3.8)and ¯ K := |∇ u | C (Σ) . (3.9)Note that ¯ K . K .We have the following lemma that constrains N . Lemma 3.1. If N ≥ , then we have that E N +2 , . E N and D N +2 , . E N . Proof.
The proof follows by simply comparing the definitions of these terms. (cid:3)
For the convenience of comparing our estimates with those of [7], we also recall the energyfunctionals used in [7]. First, E N := E N + E N and D N := D N + D N are the high-orderenergy and dissipation rate used in [7] when λ = 0 therein. Next, [7] used the following low-orderenergy E N +2 , = (cid:13)(cid:13)(cid:13) D N +2)2 u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j + (cid:13)(cid:13) ∇ p (cid:13)(cid:13) N +2) − + N +1 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N +2) − j − + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − + N +2 X j =1 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j (3.10) and the corresponding dissipation rate D N +2 , = (cid:13)(cid:13)(cid:13) D N +2)2 u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) ∇ u (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j +1 + (cid:13)(cid:13) ∇ p (cid:13)(cid:13) N +2) − + k∇ ∂ t p k N +2) − + N +1 X j =2 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N +2) − j + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / + | D∂ t η | N +2) − / + N +3 X j =2 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j +5 / . (3.11)[7] also used F N , but they used K := k u k C (¯Ω) + | D ∇ u | C (Σ) . (3.12)Note that ¯ K . K . K .3.2. Perturbed linear form.
In order to use the linear structure of the equations (1.8), wewill write it as the a perturbation of the linearized equations: ∂ t u + ∇ p − ∆ u = G in Ωdiv u = G in Ω( pI d − D u ) e d = ηe d + G on Σ ∂ t η = u d + G on Σ u = 0 on Σ b . (3.13)Here we have written the vector G = G , + G , + G , + G , + G , for G , i = ( δ ij − A ij ) ∂ j p, (3.14) G , i = u j A jk ∂ k u i , (3.15) G , i = [ K (1 + | Dϕ | ) − ∂ d u i − KDϕ · D∂ d u i , (3.16) G , i = [ − K (1 + | Dϕ | ) ∂ d J + K Dϕ · ( DJ + D∂ d ϕ ) − KD · Dϕ∂ d u i , (3.17) G , i = K∂ t ϕ∂ d u i . (3.18) G is the function G = KDϕ · ∂ d u h + (1 − K ) ∂ d u d , (3.19)and G is the vector defined by that for d = 3, G := ∂ η p − η − ∂ u − AK∂ u ) − ∂ u − ∂ u + BK∂ u + AK∂ u − ∂ u − K∂ u + AK∂ u + ∂ η − ∂ u − ∂ u + BK∂ d u + AK∂ u p − η − ∂ u − BK∂ u ) − ∂ u − K∂ u + BK∂ u + ( K − ∂ u + AK∂ u ( K − ∂ u + BK∂ u K − ∂ u (3.20)and that for d = 2, G := ∂ η (cid:18) p − η − ∂ u − AK∂ u ) − ∂ u − K∂ d u + AK∂ u (cid:19) + (cid:18) ( K − ∂ u + AK∂ u K − ∂ u (cid:19) , (3.21)where A = ∂ η and B = ∂ η . Note that, according to (3.13), p − η = ∂ d u d + G d = − Dη · ( Du d + K∂ d u h − KDη∂ d u d ) . (3.22)Finally, G = − Dη · u h . (3.23) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 11
Interpolation estimates.
The fact that E N +2 , and D N +2 , have a minimal count ofderivatives creates numerous problems when we try to estimate terms with fewer derivatives interms of E N +2 , and D N +2 , . Our way around this is to interpolate between E N +2 , (or D N +2 , )and E N . We will prove various interpolation inequalities of the form k X k . ( E N +2 , ) θ ( E N ) − θ and k X k . ( D N +2 , ) θ ( E N ) − θ , (3.24)where θ ∈ [0 , X is some quantity, and k·k is some norm (usually either L or L ∞ ). In theinterest of brevity, we will record these estimates in tables that only list the value of θ in theestimate. For example, L | E N +2 , DDη / L ∞ | D N +2 , D ∇ ¯ η / | Dη | . ( E N +2 , ) / ( E N ) / in 2 D, and k∇ ¯ η k L ∞ . ( D N +2 , ) / ( E N ) / in 3 D. We record the interpolation estimates in the following lemma, where the norms for ¯ η, u, p, G and G are on Ω and the norms for η, G and G are on Σ. In the below r > Lemma 3.2.
Let u, p, η be the solution of (3.13) and G i be defined in (3.14) – (3.23) . (1) The following tables encode the powers in the L and L ∞ interpolation estimates for thesolution and their derivatives in terms of E N +2 , : L | E N +2 , D Dη, ¯ η Dη, ∇ ¯ η / / Dp / / u h , ∂ d u h , ∂ d u h / / L ∞ | E N +2 , D Dη, ¯ η / / Dη, ∇ ¯ η / / (1 + r ) Dp / / (1 + r ) u h , ∂ d u h , ∂ d u h / / (1 + r ) (3.25) The following tables encode the powers in the L and L ∞ interpolation estimates forthe solution and their derivatives in terms of D N +2 , : L | D N +2 , D Dη, ¯ η Dη, ∇ ¯ η / / D η, ∇ ¯ η, ∂ t η, ∂ t ¯ η / / Dp / / D p, ∂ d p, ∂ d p, ∂ t p / / u h , ∂ d u h , ∂ d u h / / Du h , D∂ d u h , D∂ d u h / / ∂ d u h / u d , ∂ d u d , ∂ d u d , ∂ d u d / / L ∞ | D N +2 , D Dη, ¯ η / / Dη, ∇ ¯ η / / D η, ∇ ¯ η, ∂ t η, ∂ t ¯ η / / (1 + r ) Dp / / D p, ∂ d p, ∂ d p, ∂ t p / / (1 + r ) u h , ∂ d u h , ∂ d u h / / Du h , D∂ d u h , D∂ d u h / / (1 + r ) ∂ d u h u d , ∂ d u d , ∂ d u d , ∂ d u d / / (1 + r ) (3.26)(2) The following tables encode the powers in the L and L ∞ interpolation estimates for thenonlinear terms G i and their derivatives in terms of E N +2 , : L | E N +2 , D DG / ∇ G G d G G G L ∞ | E N +2 , D DG G G G The following tables encode the powers in the L and L ∞ interpolation estimates forthe nonlinear terms G i and their derivatives in terms of D N +2 , : L | D N +2 , D DG / / ∇ G / ∇ G G d / ∇ G d G / ∇ G G / DG G / DG L ∞ | D N +2 , D DG / ∇ G G d G G G Proof.
The interpolation powers θ recorded in the above tables of (3.25)–(3.28) have beendetermined by using the full structure of the linear parts and the nonlinear terms G i in theequations (3.13). We must record estimates for too many choices of X to allow us to writethe full details of each estimate. However, most of the estimates are straightforward, so wewill present only a sketch of how to obtain them, providing details only for the most delicateestimates.The procedure is basically as follows. First, the definition (2.6) of E N +2 , and the definition(3.3) of D N +2 , , Sobolev embeddings and Lemmas A.1–A.4 give some preliminary estimates of u, p, η, ¯ η (and some of their derivatives), which may have smaller powers θ than those recordedin the tables of (3.25)–(3.26). With these estimates of u, p, η, ¯ η , we then estimate the G i terms.The definitions (3.14)–(3.23) of G i show that these terms are linear combinations of products ofone or more terms that can be estimated in either L or L ∞ . For the L ∞ tables of (3.27)–(3.28)we estimate products with k XY k L ∞ ≤ k X k L ∞ k Y k L ∞ ; for the L tables, we estimate productswith both k XY k ≤ k X k k Y k L ∞ and k XY k ≤ k Y k k X k L ∞ , and then take the larger valueof θ produced by these two bounds, where Lemma 3.1 will be used implicitly. These gives somepreliminary estimates of G i (and some of their derivatives), which may have smaller powers θ than those recorded in the tables of (3.27)–(3.28). With these estimates of G i , we then usethe linear structure of (3.13) to improve the estimates of u , ∇ p , etc, which in turn improve theestimates of G i . Such iterative scheme can be carried out repeatly until that we have the desiredpowers θ as recorded in these tables of (3.25)–(3.28). The procedure is mostly straightforward,and below we explain only how to determine the powers θ in a bit more details.Before proceeding further, we may explain the different powers in the tables (3.25)–(3.26),in terms of L or L ∞ , E N +2 , or D N +2 , , and 2 D or 3 D . First, in the L tables, basicallya 1 spatial derivative count contributes a 1 / E N +2 , and a 1 / D N +2 , ;this results mainly the differences between the first tables of (3.25) and (3.26), and also thedifferences between the first tables of (3.27) and (3.28). Second, by Sobolev embeddings, an L ∞ norm contributes an L norm of a 1 / D and of a 1 horizontalderivative count in 3 D (or 1 / − and 1 − due to the L ∞ limiting embedding cases); this resultsmainly the differences between the two tables of (3.25) and (3.26), respectively. Note that thedifferent powers in the tables of (3.25)–(3.26) reflects the ones in the tables in (3.27)–(3.28).Now we start our estimates. First, the powers of η, Dη , D η , ¯ η, D ¯ η , D ¯ η and Dp , D p asrecorded in these tables of (3.25)–(3.26) follows by Sobolev embeddings and Lemmas A.1–A.4,and they can not be improved.Next, we estimate u h . We use the horizontal component of the first equation in (3.13) to find ∂ d u h ∼ Dp + D u h + ∂ t u h + G , h + G , h + D ¯ ηD∂ d u h + G , h + G , h . (3.29)It is then straightforward to check that the the powers of ∂ d u h in these tables of (3.25)–(3.26)are determined by those of Dp . On the other hand, we use the horizontal component of the ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 13 third equation in (3.13) to find ∂ d u h ∼ Du d + Dη ∇ u on Σ . (3.30)Since u h = 0 on Σ b , this together with Poincar´e’s inequality of Lemmas A.5–A.6 and thetrace theory allows us to check that the the powers of u h , ∂ d u h in these tables of (3.25)–(3.26)are determined by those of ∂ d u h and hence Dp . Note that in particular, the proof of (3.25)is completed. Similarly, the powers of Du h , D∂ d u h , D∂ d u h in the two tables of (3.26) aredetermined by those of D p .We then estimate u d . We use the second equation in (3.13) to find ∂ d u d ∼ Du h + D ¯ η∂ d u h . (3.31)Since u d = 0 on Σ b , this together with Poincar´e’s inequality of Lemma A.6 implies that thepowers of u d , ∂ d u d , ∂ d u d , ∂ d u d in the two tables of (3.26) are determined by those of Du h , D∂ d u h , D∂ d u h .We now estimate ∂ d p . We use the vertical component of the first equation in (3.13) to find ∂ d p ∼ ∇ u d + ∂ t u d + G , d + G , d + G , d + G , d . (3.32)We can check that the the powers of ∂ d p, ∂ d p in the two tables of (3.26) are determined by thoseof ∂ d u d , ∂ d u d .Now we estimate ∂ d u h . We use the horizontal component of the first equation in (3.13) tofind ∂ d u h = ∂ d D u d + ∂ d Dp + ∂ d ∂ t u d + G , d + G , d + G , d + G , d + G , d . (3.33)We can check that the the powers of ∂ d u h in the two tables of (3.26) are determined by those of G , d ∼ ¯ η∂ d p , which are the sum of power of ∂ d p in the same table and power of ¯ η in the secondtable of (3.26).Finally, we estimate ∂ t η, ∂ t ¯ η and ∂ t p . We use the fourth equation in (3.13) to find ∂ t η ∼ u d + G on Σ . (3.34)Then the trace theory Lemma A.1 imply that the the powers of ∂ t η, ∂ t ¯ η in the two tables of(3.26) are determined by those of u d . On the other hand, we use the vertical component of thethird equation in (3.13) to find ∂ t p ∼ ∂ t η + 2 ∂ t ∂ d u d + ∂ t G d on Σ . (3.35)This together with Poincar´e’s inequality of Lemma A.5 yields that the the powers of ∂ t p inthe two tables of (3.26) are determined by those of ∂ t η . Note that the proof of (3.26) is thuscompleted.With the estimates (3.25)–(3.26) in hand, it is then fairly routine to prove these tables of(3.27)–(3.28). (cid:3) Lemma 3.3.
Note that most of the interpolation powers θ in D of Lemma 3.2 improve those ofLemma 3.1–Proposition 3.16 in [7] . This is due to that our low-order energy E N +2 , is strongerthan E N +2 , , which results also that our analysis is much more simple and direct than those in [7] . Now we record the interpolation estimates for K , ¯ K and K , as defined by (3.8), (3.9) and(3.12), respectively. Lemma 3.4.
It holds that K . K . E κ d / N E − κ d / N +2 , , d = 2 , and ¯ K . E N +2 , . (3.37) Proof. (3.36) follows directly from the definition (2.6) of E N +2 , and the L ∞ table in (3.25) withthe choice r = κ / (2 − κ ) when d = 3. On the other hand, we use the horizontal componentof the third equation in (3.13) to find ∂ d u h = − Du d − G h on Σ . (3.38) (3.37) then follows from the definition (2.6) of E N +2 , and the L table in (3.27). (cid:3) Nonlinear estimates.
We now present the estimates of the nonlinear terms G i . We firstrecord the estimates at the N + 2 level. Lemma 3.5.
Let G i be defined by (3.14) – (3.23) . Then (1) It holds that (cid:13)(cid:13)(cid:13) ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N +2) − G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) ¯ D N +2) − G (cid:12)(cid:12)(cid:12) / . E N E N +2 , . (3.39)(2) It holds that (cid:13)(cid:13)(cid:13) ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N +2) − G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) ¯ D N +2) − G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) ¯ D N +2) − ∂ t G (cid:12)(cid:12)(cid:12) / . E N D N +2 , . (3.40) Proof.
The estimates of these nonlinearities are fairly routine to derive: we note that all G i termsare quadratic or of higher order; then we apply the differential operator and expand using theLeibniz rule; each term in the resulting sum is also at least quadratic, and we estimate one termin H k ( k = 0 or 1 / G i ) and the other term in L ∞ or H m for m depending on k ,using Sobolev embeddings, trace theory, and Lemmas A.1–A.4 and A.9.Note that the derivative count in the differential operators is chosen in order to allow estima-tion by E N +2 , in (3.39) and by D N +2 , in (3.40). Because E N +2 , and D N +2 , involve minimalderivative counts, there may be terms in the sum ∂ α G i that cannot be directly estimated, andwe must appeal to the interpolation results in the L tables of (3.27) and (3.28) in Lemma 3.2.This yields directly the estimate (3.39) by using the L table of (3.27) as the derivative countinvolved also does not exceed those in E N +2 , . For the estimate (3.40), by using the L table of(3.28) it suffices to estimate three exceptions whose derivative count involved exceed those in D N +2 , : (cid:13)(cid:13) ∇ N +2)+1 ¯ η ∇ u (cid:13)(cid:13) when estimating G and G , (cid:12)(cid:12) D N +2) − ∇ ¯ η ∇ u (cid:12)(cid:12) / when estimating G and (cid:12)(cid:12) D N +2) ηu (cid:12)(cid:12) / when estimating G . To control them, we use the Sobolev interpolationto have (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) / ≤ (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / (cid:17) (4 N − / (4 N − (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N − (cid:17) / (4 N − ≤ D (4 N − / (4 N − N +2 , E / (4 N − N . (3.41)On the other hand, the L ∞ table in (3.26) implies in particular that k u k C (¯Ω) . D / N +2 , E / N . (3.42)Hence, by (3.41)–(3.42) and Lemma A.10, we obtain that for N ≥ (cid:12)(cid:12)(cid:12) D N +2) ηu (cid:12)(cid:12)(cid:12) / . (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) / | u | C (Σ) . D (4 N − / (4 N − N +2 , E / (4 N − N D / N +2 , E / N . E N D N +2 , . (3.43)Similarly, by using additionally the trace theory and Lemma A.1, we have (cid:12)(cid:12)(cid:12) D N +2) − ∇ ¯ η ∇ u (cid:12)(cid:12)(cid:12) / . (cid:12)(cid:12)(cid:12) D N +2) − ∇ ¯ η (cid:12)(cid:12)(cid:12) / |∇ u | C (Σ) . (cid:13)(cid:13)(cid:13) D N +2) − ∇ ¯ η (cid:13)(cid:13)(cid:13) |∇ u | C (Σ) . (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) / |∇ u | C (Σ) . E N D N +2 , (3.44) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 15 and (cid:13)(cid:13)(cid:13) ∇ N +2)+1 ¯ η ∇ u (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13)(cid:13) ∇ N +2)+1 ¯ η (cid:13)(cid:13)(cid:13) k∇ u k C (¯Ω) . (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) / k∇ u k C (¯Ω) . E N D N +2 , . (3.45)We then conclude (3.40). (cid:3) Now we record the estimates at the 2 N level. Lemma 3.6.
Let G i be defined by (3.14) – (3.23) . Then (1) It holds that (cid:13)(cid:13)(cid:13) ∇ N − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∇ N − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) D N − G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) D N − G (cid:12)(cid:12)(cid:12) / (3.46)+ (cid:13)(cid:13)(cid:13) ¯ ∇ N − ∂ t G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ ∇ N − ∂ t G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t G (cid:12)(cid:12)(cid:12) / . ( E N ) , and (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:12)(cid:12) ∇ N − G (cid:12)(cid:12) / + (cid:12)(cid:12) ∇ N − G (cid:12)(cid:12) / . ( E N ) + K E N . (3.47)(2) It holds that (cid:13)(cid:13)(cid:13) ∇ N − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∇ N − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) D N − G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) D N − G (cid:12)(cid:12)(cid:12) / (3.48)+ (cid:13)(cid:13)(cid:13) ¯ ∇ N − ∂ t G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ ∇ N − ∂ t G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t G (cid:12)(cid:12)(cid:12) / + (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t G (cid:12)(cid:12)(cid:12) / . E N D N , (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:12)(cid:12) D N − G (cid:12)(cid:12) / + (cid:12)(cid:12) D N − G (cid:12)(cid:12) / . E N D N + K D N , (3.49) and (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:13)(cid:13) ∇ N − G (cid:13)(cid:13) + (cid:12)(cid:12) D N − G (cid:12)(cid:12) / + (cid:12)(cid:12) D N − G (cid:12)(cid:12) / . E N D N + K F N , (3.50) Proof.
The proof is similar to Lemma 3.5. The estimate (3.46) is straightforward since E N hasno minimal derivative restrictions, and there is no problem for the estimate (3.48) even η and¯ η appearing in G i are not included in D N , since we can always control them by E N and otherterms multiplying them by D N .We now turn to the derivation of the estimate (3.50). With three exceptions, we may argueas in the derivation of (3.48) to estimate the desired norms of all terms by E N D N . The excep-tional terms are (cid:13)(cid:13) ∇ N +1 ¯ η ∇ u (cid:13)(cid:13) when estimating G and G , (cid:12)(cid:12) D N − ∇ ¯ η ∇ u (cid:12)(cid:12) / when estimating G and (cid:12)(cid:12) D N ηu (cid:12)(cid:12) / when estimating G . We will now show how to estimate the exceptionalterms with K F N . Indeed, by Lemma A.10, we have (cid:12)(cid:12) D N ηu (cid:12)(cid:12) / . (cid:12)(cid:12) D N η (cid:12)(cid:12) / | u | C (Σ) . F N K . (3.51)Similarly, by using additionally the trace theory and Lemma A.1, we have (cid:12)(cid:12) D N − ∇ ¯ η ∇ u (cid:12)(cid:12) / . (cid:12)(cid:12) D N − ∇ ¯ η (cid:12)(cid:12) / |∇ u | C (Σ) . k ¯ η k N +1 K . | η | N +1 / K . F N K (3.52)and (cid:13)(cid:13) ∇ N +1 ¯ η ∇ u (cid:13)(cid:13) . (cid:13)(cid:13) ∇ N +1 ¯ η (cid:13)(cid:13) k∇ u k C (¯Ω) . | η | N +1 / K . F N K . (3.53)We then conclude (3.50).Finally, the estimates (3.47) and (3.49) follow similarly as (3.50) by bounding | η | N − / by E N in (3.47) and by D N in (3.49). (cid:3) Energy evolution
Energy evolution in perturbed linear form.
To derive the energy evolution of themixed time-horizontal spatial derivatives of the solution to (1.8), that is, excluding the highestorder time derivative and the highest order horizontal spatial derivatives, we shall use theperturbed linear formulation (3.13). Recall that we employ the derivative conventions (2.27)–(2.29) from Section 2.3.We first record the estimates for the evolution of the energy at the 2 N level. Proposition 4.1.
It holds that ddt (cid:18)(cid:13)(cid:13)(cid:13) ¯ D N − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) ¯ D N − D∂ t u (cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N − η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) ¯ D N − D∂ t η (cid:12)(cid:12) (cid:19) + (cid:13)(cid:13)(cid:13) ¯ D N − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) ¯ D N − D∂ t u (cid:13)(cid:13) . p E N D N + p D N K D N . (4.1) Proof.
Let α ∈ N d − be so that 1 ≤ | α | ≤ N and 1 ≤ | α h | ≤ N −
1. Applying ∂ α to thefirst equation of (3.13) and then taking the dot product with ∂ α u , using the other equations of(3.13) as in Proposition 6.2 of [7], we find that12 ddt (cid:18)Z Ω | ∂ α u | + Z Σ | ∂ α η | (cid:19) + 12 Z Ω | D ∂ α u | = Z Ω ∂ α u · ( ∂ α G − ∇ ∂ α G ) + Z Ω ∂ α p∂ α G + Z Σ − ∂ α u · ∂ α G + ∂ α η∂ α G . (4.2)We now estimate the right hand side of (4.2). Since 1 ≤ | α h | ≤ N −
1, we may write α = β + ( α − β ) for some β ∈ N d − with | β | = 1. Hence | α − β | ≤ N − | α h − β h | ≤ N − (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∂ α u · ( ∂ α G − ∇ ∂ α G ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∂ α + β u · ( ∂ α − β G − ∇ ∂ α − β G ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ∂ α u k (cid:16)(cid:13)(cid:13)(cid:13) ∂ α − β G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∂ α − β G (cid:13)(cid:13)(cid:13) (cid:17) . p D N p E N D N + K D N . (4.3)For the G term we do not need to integrate by parts: by (3.48)–(3.49), (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∂ α p∂ α G (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ∂ α p k (cid:13)(cid:13)(cid:13) ∂ α − β ∂ β G (cid:13)(cid:13)(cid:13) ≤ k ∂ α p k (cid:13)(cid:13)(cid:13) ∂ α − β G (cid:13)(cid:13)(cid:13) . p D N p E N D N + K D N . (4.4)For the G term we integrate by parts and use the trace estimate to see that, by (3.48)–(3.49), (cid:12)(cid:12)(cid:12)(cid:12)Z Σ ∂ α u · ∂ α G (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Σ ∂ α + β u · ∂ α − β G (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ∂ α + β u (cid:12)(cid:12)(cid:12) − / (cid:12)(cid:12)(cid:12) ∂ α − β G (cid:12)(cid:12)(cid:12) / ≤ | ∂ α u | / (cid:12)(cid:12)(cid:12) ∂ α − β G (cid:12)(cid:12)(cid:12) / ≤ k ∂ α u k (cid:12)(cid:12)(cid:12) ∂ α − β G (cid:12)(cid:12)(cid:12) / . p D N p E N D N + K D N . (4.5)For the G term we must split to two cases: α ≥ α = 0. In the former case, there is atleast one temporal derivative in ∂ α , so by (3.48) we have (cid:12)(cid:12)(cid:12)(cid:12)Z Σ ∂ α η∂ α G (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t η (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ¯ D N − ∂ t G (cid:12)(cid:12)(cid:12) . p D N p E N D N . (4.6)In the latter case, there involves only spatial derivatives, that is, 1 ≤ | α | = | α h | ≤ N −
1, andwe write ∂ α G = ∂ α ( Dη · u h ) = − D∂ α η · u h − [ ∂ α , u h ] · Dη. (4.7)
ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 17
We use the integration by parts to see that, by Lemma A.10, − Z Σ ∂ α ηD∂ α η · u h = − Z Σ D | ∂ α η | · u h = 12 Z Σ | ∂ α η | D · u h . | ∂ α η | − / | ∂ α ηD · u h | / . | ∂ α η | − / | ∂ α η | / | Du h | C (Σ) . p D N D N K . (4.8)On the other hand, similarly as the derivation of (3.49) in Lemma 3.6, we have | [ ∂ α , u ] Dη | / . E N D N + K D N (4.9)and hence Z Σ ∂ α η [ ∂ α , u ] Dη . | ∂ α η | − / | [ ∂ α , u ] Dη | / . p D N p E N D N + K D N . (4.10)In light of (4.8) and (4.10), we conclude that (cid:12)(cid:12)(cid:12)(cid:12)Z Σ ∂ α η · ∂ α G (cid:12)(cid:12)(cid:12)(cid:12) . p D N p E N D N + K D N . (4.11)Now, by the estimates (4.3)–(4.6) and (4.11), we deduce from (4.2) that for all α ∈ N d − with 1 ≤ | α | ≤ N and 1 ≤ | α h | ≤ N − ddt (cid:18)Z Ω | ∂ α u | + Z Σ | ∂ α η | (cid:19) + 12 Z Ω | D ∂ α u | . p D N p E N D N + K D N . p E N D N + p D N K D N . (4.12)The estimate (4.1) then follows from (4.12) by summing over such α and (2.10), using Korn’sinequality of Lemma A.8 since ∂ α u = 0 for α ∈ N d − . (cid:3) We then record the estimates for the evolution of the energy at the N + 2 level. Proposition 4.2.
It holds that ddt (cid:18)(cid:13)(cid:13)(cid:13) ¯ D N +2) − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D N +2) − Du (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D N +2) − η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ¯ D N +2) − Dη (cid:12)(cid:12)(cid:12) (cid:19) + (cid:13)(cid:13)(cid:13) ¯ D N +2) − u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D N +2) − Du (cid:13)(cid:13)(cid:13) . p E N D N +2 , . (4.13) Proof.
This is just a restatement of Proposition 6.4 of [7] when m = 2. Let α ∈ N d − beso that 2 ≤ | α | ≤ N + 2) and α ≤ N + 1. Note that (4.2) holds, and the right hand side of(4.2) can be estimated similarly as in Proposition 4.1 by using (3.40) in place of (3.48)–(3.49)so that they can be bounded by √ E N D N +2 , , except the following terms: Z Σ ∂ α η∂ α G when | α | = 2( N + 2) and α = 0 (4.14)and Z Σ ∂ α η∂ α G and Z Ω ∂ α p∂ α G when | α | = 2 . (4.15)If | α | = 2( N + 2) and α = 0, then we have Z Σ ∂ α η∂ α G . (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) D N +2) G (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) (cid:16)(cid:12)(cid:12)(cid:12) D N +2) Dη (cid:12)(cid:12)(cid:12) | u h | L ∞ (Σ) + | Dη | L ∞ (Σ) (cid:12)(cid:12)(cid:12) D N +2) u h (cid:12)(cid:12)(cid:12) (cid:17) . (4.16) To control the right hand side of the above, we use the Sobolev interpolation to have (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) ≤ (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / (cid:17) (4 N − / (4 N − (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N − (cid:17) / (4 N − ≤ D (4 N − / (4 N − N +2 , E / (4 N − N (4.17)and (cid:12)(cid:12)(cid:12) D N +2)+1 η (cid:12)(cid:12)(cid:12) ≤ (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / (cid:17) (4 N − / (4 N − (cid:16)(cid:12)(cid:12) D η (cid:12)(cid:12) N − (cid:17) / (4 N − ≤ D (4 N − / (4 N − N +2 , E / (4 N − N . (4.18)On the other hand, the L ∞ table in (3.26) implies in particular that | u h | C (Σ) . D / N +2 , E / N and | Dη | C (Σ) . D / N +2 , E / N . (4.19)Hence, by (4.17)–(4.19), we obtain from (4.16) that for N ≥ Z Σ ∂ α η∂ α G . q D (4 N − / (4 N − N +2 , E / (4 N − N q D / N +2 , E / N q D (4 N − / (4 N − N +2 , E / (4 N − N + D N +2 , . p E N D N +2 , . (4.20)Now for | α | = 2, if α = 0, then by integrating by parts and the L table in (3.28), we have Z Σ ∂ α η∂ α G + Z Ω ∂ α p∂ α G . (cid:12)(cid:12) D η (cid:12)(cid:12) (cid:12)(cid:12) DG (cid:12)(cid:12) + (cid:13)(cid:13) D p (cid:13)(cid:13) (cid:13)(cid:13) DG (cid:13)(cid:13) . p D N +2 , p E N D N +2 , . p E N D N +2 , . (4.21)If α = 1, then by the two tables in (3.26), we have Z Σ ∂ t η∂ t G . | ∂ t η | (cid:12)(cid:12) ∂ t G (cid:12)(cid:12) . | ∂ t η | (cid:16) | ∂ t Dη | | u h | C (Σ) + | Dη | C (Σ) | ∂ t u h | (cid:17) . q D / N +2 , E / N p D N +2 , q D / N +2 , E / N . p E N D N +2 , . (4.22)For the G term, we need to use the structure of G when d = 2; indeed, we can treat this termunifiedly for d = 2 ,
3. Recall the Piola identity ∂ j ( J A ij ) = 0. Then by the second equation in(1.8), we have ∂ j ( J A ij u i ) = 0 , which implies that G = div u = ∂ j (( δ ij − J A ij ) u i ) = − D · (( J − u h ) + ∂ d ( Dϕ · u h ) . (4.23)Hence, by integrating by parts in horizontal variable and the two tables in (3.26), we obtain Z Ω ∂ t p∂ t G = Z Ω D∂ t p · ∂ t (( J − u h ) + Z Ω ∂ t p∂ t ∂ d ( Dϕ · u h ) . k D∂ t p k (cid:16) k ∂ t J k k u h k C (¯Ω) + k J − k C (¯Ω) k ∂ t u h k (cid:17) + k ∂ t p k (cid:16) k ∂ t ∂ d Dϕ k k u h k C (¯Ω) + k ∂ d Dϕ k C (¯Ω) k ∂ t u h k (cid:17) + k ∂ t p k (cid:16) k ∂ t Dϕ k k ∂ d u h k C (¯Ω) + k Dϕ k C (¯Ω) k ∂ t ∂ d u h k (cid:17) . p D N +2 , (cid:18)q D / N +2 , E / N q D / N +2 , E / N + p E N p D N +2 , (cid:19) + q D / N +2 , E / N (cid:18)p D N +2 , q D / N +2 , E / N + q D / N +2 , E / N p D N +2 , (cid:19) + q D / N +2 , E / N (cid:18)p D N +2 , q D / N +2 , E / N + q D / N +2 , E / N p D N +2 , (cid:19) . p E N D N +2 , . (4.24)Consequently, by (4.20), (4.21), (4.22) and (4.24), we know that the exceptional terms (4.14)and (4.15) are all bounded by √ E N D N +2 , , and hence that the right hand side of (4.2) are ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 19 bounded by √ E N D N +2 , for all α ∈ N d − with 2 ≤ | α | ≤ N + 2) and α ≤ N + 1. Theestimate (4.13) thus follows by summing over such α . (cid:3) Energy evolution using geometric form.
To derive the energy evolution of the highestorder temporal derivatives of the solution to (1.8), we will not use the perturbed linear formu-lation (3.13). As well explained in [7], if we did this we would be unable to control the G , G and G terms in the right hand side of (4.2). Motivated by [7], we shall use the followinggeometric formulation. Applying the differential operator ∂ α = ∂ α t to (1.8), we find that ∂ A t ( ∂ α u ) + u · ∇ A ( ∂ α u ) + div A S A ( ∂ α p, ∂ α u ) = F in Ω ∇ A · ( ∂ α u ) = F in Ω S A ( ∂ α p, ∂ α u ) N = ∂ α η N + F on Σ ∂ t ( ∂ α η ) = ∂ α u · N + F on Σ ∂ α u = 0 on Σ b , (4.25)where F ,αi = X <β ≤ α C βα n ∂ β ( K∂ t ϕ ) ∂ α − β ∂ d u i + A jk ∂ k ( ∂ β A iℓ ∂ α − β ∂ ℓ u j + ∂ β A jℓ ∂ α − β ∂ ℓ u i ) (4.26)+ ∂ β A jℓ ∂ α − β ∂ ℓ ( A im ∂ m u j + A jm ∂ m u i ) − ∂ β ( u j A jk ) ∂ α − β ∂ k u i − ∂ β A ik ∂ α − β ∂ k p o ,F ,α = − X <β ≤ α C βα ∂ β A ij ∂ α − β ∂ j u i , (4.27) F ,α = X <β ≤ α C βα n ∂ β Dη∂ α − β ( η − p ) + ( ∂ β ( N j A im ) ∂ α − β ∂ m u j + ∂ β ( N j A jm ) ∂ α − β ∂ m u i ) o (4.28)and F ,α = X <β ≤ α C βα ∂ β Dη · ∂ α − β u h . (4.29)We now present the estimates of these nonlinear terms F i at both 2 N and N + 2 levels. Lemma 4.3.
Let F i,α be defined by (4.26) – (4.29) , then the following estimates hold. (1) Let F i, N be F i,α when ∂ αt = ∂ Nt , it holds that (cid:13)(cid:13) F , N (cid:13)(cid:13) + (cid:13)(cid:13) ∂ t ( J F , N ) (cid:13)(cid:13) + (cid:12)(cid:12) F , N (cid:12)(cid:12) + (cid:12)(cid:12) F , N (cid:12)(cid:12) . E N D N (4.30) and (cid:13)(cid:13) F , N (cid:13)(cid:13) . ( E N ) . (4.31)(2) Let F i,N +2 be F i,α when ∂ αt = ∂ N +2 t , it holds that (cid:13)(cid:13) F ,N +2 (cid:13)(cid:13) + (cid:13)(cid:13) ∂ t ( J F ,N +2 ) (cid:13)(cid:13) + (cid:12)(cid:12) F ,N +2 (cid:12)(cid:12) + (cid:12)(cid:12) F ,N +2 (cid:12)(cid:12) . E N D N +2 , (4.32) and (cid:13)(cid:13) F ,N +2 (cid:13)(cid:13) . E N E N +2 , . (4.33) Proof.
All these estimates, with the trivial replacement of E N by E N , etc., are recorded inTheorems 5.1–5.2 of [7]. Indeed, the proof of (4.32)–(4.33) will be simpler than those of Theorem5.2 of [7] as our low-order energy and dissipation are stronger than those in [7]. For example, (cid:13)(cid:13) F ,N +2 (cid:13)(cid:13) . X <ℓ ≤ N +1 (cid:13)(cid:13)(cid:13) ∂ ℓt A ij (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ∂ N +2 − ℓt ∂ j u i (cid:13)(cid:13)(cid:13) C (¯Ω) + (cid:13)(cid:13)(cid:13) ∂ N +2 t A ij (cid:13)(cid:13)(cid:13) k ∂ j u i k C (¯Ω) . E N E N +2 , + E N +2 , E N . E N E N +2 , . (4.34)This proves (4.33). (cid:3) We now present the evolution estimate for 2 N temporal derivatives. Proposition 4.4.
It holds that ddt (cid:18)(cid:13)(cid:13) ∂ Nt u (cid:13)(cid:13) + (cid:12)(cid:12) ∂ Nt η (cid:12)(cid:12) + 2 Z Ω J ∂ N − t pF , N (cid:19) + (cid:13)(cid:13) ∂ Nt u (cid:13)(cid:13) . p E N D N (4.35) and Z Ω J ∂ N − t pF , N . ( E N ) / . (4.36) Proof.
This is just a restatement of Proposition 5.3 of [7] in the time-differential form, which isproved by employing the geometric formulation (4.25) and using the estimates (4.30)–(4.31) ofLemma 4.3. Note that an integration by parts in time is used for the pressure term as there isone more time derivative on p than can be controlled. We refer to [7] for more details. (cid:3) We then record a similar result for N + 2 temporal derivatives. Proposition 4.5.
It holds that ddt (cid:18)(cid:13)(cid:13)(cid:13) ∂ N +2 t u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ∂ N +2 t η (cid:12)(cid:12)(cid:12) + 2 Z Ω J ∂ N +1 t pF ,N +2 (cid:19) + (cid:13)(cid:13)(cid:13) ∂ N +2 t u (cid:13)(cid:13)(cid:13) . p E N D N +2 , (4.37) and Z Ω J ∂ N +1 t pF ,N +2 . p E N E N +2 , . (4.38) Proof.
This is just a restatement of Proposition 5.4 of [7] when m = 2 therein. The proof issimilar to Proposition 4.4 except using (4.32)–(4.33) in place of (4.30)–(4.31). (cid:3) Energy evolution of Alinhac good unknowns.
To derive the energy evolution of thehighest order horizontal spatial derivatives of the solution to (1.8), as explained in Section 2.2,even using the geometric formulation (4.25) will lead to the appearance of KF N . To avoidthis, we will appeal to the reformulation by using Alinhac good unknowns. Recall that whenapplying the differential operator ∂ α for α ∈ N d − to (1.8), we need to commute ∂ α with eachdifferential operator ∂ A in (1.8). It is thus useful to establish the following general expressions.For i = 1 , , d , set ∂ α ∂ A i f = ∂ A i ∂ α f − ∂ A d f ∂ A i ∂ α ϕ + C αi ( f ) , (4.39)where the commutator C αi ( f ) is given for i = d by C αi ( f ) = C αi, ( f ) + C αi, ( f ) (4.40)with C αi, = − (cid:20) ∂ α , ∂ i ϕ∂ d φ , ∂ d f (cid:21) , (4.41) C αi, = − ∂ d f (cid:20) ∂ α , ∂ i ϕ, ∂ d φ (cid:21) − ∂ i ϕ∂ z f (cid:20) ∂ α − α ′ , ∂ d φ ) (cid:21) ∂ α ′ ∂ d η (4.42)for any α ′ ≤ α with | α ′ | = 1. Note that for i = 1 , d − ∂ i φ = ∂ i ϕ and that for α = 0, ∂ α ∂ d φ = ∂ α ∂ d ϕ . For i = d , similar decomposition for the commutator holds (basically, itsuffices to replace ∂ i ϕ by 1 in the above expressions). Since ∂ A i and ∂ A j commute, it holds that ∂ α ∂ A i f = ∂ A i ( ∂ α f − ∂ A d f ∂ α ϕ ) + ∂ A d ∂ A i f ∂ α ϕ + C αi ( f ) . (4.43)It was first observed by Alinhac [1] that the highest order term of ϕ and hence η will be canceledwhen we use the good unknown ∂ α f − ∂ A d f ∂ α ϕ .We shall now derive the equations satisfied by the good unknowns U α := ∂ α u − ∂ A d u ∂ α η and P α := ∂ α p − ∂ A d p ∂ α η for α ∈ N d − with | α | = 4 N. (4.44) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 21
Lemma 4.6.
For | α | = 4 N , it holds that ∂ A t U α + u · ∇ A U α + div A S A ( P α , U α ) = Q ,α in Ω ∇ A · U α = Q ,α in Ω S A ( P α , U α ) N = ∂ α η N + Q ,α on Σ ∂ t ( ∂ α η ) = U α · N + Q ,α on Σ U α = 0 on Σ b , (4.45) where Q i,α , i = 1 , , , are defined by (4.53) , (4.56) , (4.58) and (4.60) , respectively.Proof. We first prove the first equation in (4.6). Note that ∂ A t + u · ∇ A = ∂ t + u h · Du + U d ∂ d , (4.46)where U d := ∂ d φ ( u · N − ∂ t ϕ ) with N extended to Ω by N = ( − Dϕ, ∂ α (cid:0) ∂ A t + u · ∇ A (cid:1) u = ( ∂ t + u h · D + U d ∂ d ) ∂ α u + (cid:0) u · ∂ α N − ∂ t ∂ α ϕ (cid:1) ∂ A d u (4.47) − ∂ A d ∂ α ϕ (cid:0) u · N − ∂ t ϕ (cid:1) ∂ A d u + C α ( T )= (cid:0) ∂ A t + u · ∇ A (cid:1) ∂ α u − ∂ A d u (cid:0) ∂ A t + u · ∇ A ) ∂ α ϕ + C α ( T )= (cid:0) ∂ A t + u · ∇ A (cid:1) U α + ∂ A d (cid:0) ∂ A t + u · ∇ A (cid:1) u∂ α ϕ − ∂ A d u · ∇ A u∂ α ϕ + C α ( T ) , where the commutator C α ( T ) is defined by C α ( T ) = [ ∂ α , u h ] · Du + [ ∂ α , U d , ∂ d u ] + (cid:20) ∂ α , U d , ∂ d φ (cid:21) ∂ d u + 1 ∂ d ϕ [ ∂ α , u ] · N ∂ d u, + ∂ d ϕU d ∂ d u (cid:20) ∂ α − α ′ , ∂ d φ ) (cid:21) ∂ α ′ ∂ d ϕ (4.48)for any α ′ ≤ α with | α ′ | = 1. On the other hand, (4.43) implies that ∂ α ∇ A p = ∇ A P α + ∂ ϕd ∇ A p∂ α η + C α ( p ) . (4.49)It remains to compute ∂ α ∆ A u = ∂ α ∇ A · ( D A u ) . Note that ∂ α ∇ A · ( D A u ) = ∇ A · (cid:0) ∂ α D A u (cid:1) − (cid:0) ∂ A d D A u (cid:1) ∇ A ( ∂ α ϕ ) + D α (cid:0) D A u (cid:1) (4.50)with D α (cid:0) D A u (cid:1) i = C αj (cid:0) D A u ) ij , and ∂ α (cid:0) D A u (cid:1) = D A (cid:0) ∂ α u (cid:1) − ∂ A d u ⊗ ∇ A ∂ α ϕ − ∇ A ∂ α ϕ ⊗ ∂ A d u + E α ( u ) (4.51)with E α ( u ) ij = C αi ( u j ) + C αj ( u i ) . Then we deduce that ∂ α ∆ A u = ∇ A · D A ( ∂ α u ) − ∇ A · (cid:16) ∂ A d v ⊗ ∇ A ∂ α ϕ − ∇ A ∂ α ϕ ⊗ ∂ A d u (cid:17) − (cid:0) ∂ A d D A u (cid:1) ∇ A ( ∂ α ϕ ) + D α (cid:0) D A u (cid:1) + ∇ A · E α ( u )= ∆ A U α − ∂ A d ∆ A u∂ α ϕ + D α (cid:0) D A u (cid:1) + ∇ A · E α ( u ) . (4.52)Hence, (4.47), (4.49), (4.52) and the first equation in (1.8) imply the first equation in (4.6) with Q ,α defined by Q ,α = e Q ,α + ∇ A · E α ( u ) + D α (cid:0) D A u (cid:1) , (4.53)where e Q ,α = ∂ A d u · ∇ A u∂ α ϕ − C α ( T ) − C α ( p ) . (4.54)Next, (4.43) yields that ∂ α ∇ A · u = ∇ A · U α + ∂ ϕd ∇ A · u∂ α ϕ − Q ,α , (4.55)where Q ,α := − X i =1 C αi ( u i ) . (4.56)(4.55) and the second equation in (1.8) imply the second equation in (4.6). Now applying ∂ α to the third equation in (1.8) and using (4.51), we get (cid:0) D A ( ∂ α u ) − ∂ A d u ⊗ ∇ A ∂ α ϕ − ∇ A ∂ α u ⊗ ∂ A d u + E α ( u ) (cid:1) N − ( ∂ α p − ∂ α η ) N = − ( D A u − ( p − η ) I d ) ∂ α N − [ ∂ α , D A u − ( p − η ) I d , N ] . = − ( D A u − D A u N · N I d ) ∂ α N − [ ∂ α , D A u − D A u N · N I d , N ] (4.57) ≡ − D A u Π ∂ α N − [ ∂ α , D A u Π , N ] , where Π = I d − N ⊗ N . This yields the third equation in (4.6) with Q ,α defined by Q ,α := − ∂ A d p ∂ α η − D A u Π ∂ α N − ∂ α η∂ A d (cid:0) D A u (cid:1) N − [ ∂ α , D A u Π , N ] − E α ( u ) N . (4.58)We then apply ∂ α to the fourth equation in (1.8) to find ∂ t ∂ α η + u h · D∂ α η − ∂ α u · N = [ ∂ α , u, Dη ] . (4.59)This yields the fourth equation in (4.6) with Q ,α defined by Q ,α := − u h · D∂ α η − ∂ A d u · N ∂ α η + e Q ,α , (4.60)where e Q ,α = [ ∂ α , u h · , Dη ] . (4.61)Finally, the fifth equation in (1.8) follows directly since ϕ = 0 on Σ b . (cid:3) We shall present the estimates of some of these nonlinear terms in (4.45).
Lemma 4.7.
It holds that (cid:13)(cid:13)(cid:13) e Q ,α (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) Q ,α (cid:13)(cid:13) . E N +2 , D N , (4.62) kE α ( u ) k + |E α ( u ) | − / . E N +2 , D N , (4.63) (cid:12)(cid:12) Q ,α (cid:12)(cid:12) − / . ( E N +2 , + K ) D N + ¯ K F N (4.64) and (cid:12)(cid:12)(cid:12) e Q ,α (cid:12)(cid:12)(cid:12) . E N +2 , D N + ¯ K E N . (4.65) Proof.
The estimates (4.62)–(4.63) follow similarly as Lemma 3.5. For the estimate (4.64), theadditional terms result from the followings: | D A u Π ∂ α N | − / . |∇ u | C (Σ) | ∂ α Dη | − / . ¯ K | ∂ α η | / . ¯ K F N (4.66)and (cid:12)(cid:12) p α η∂ A d (cid:0) D A u (cid:1) N (cid:12)(cid:12) − / . | ∂ α η | − / (cid:16) |∇ u | C (Σ) + |∇ u | C (Σ) (cid:17) . D N K . (4.67)While for the estimate (4.65), the additional term is due to that for | α ′ | = 1: (cid:12)(cid:12)(cid:12) ∂ α ′ u h · ∂ α − α ′ Dη (cid:12)(cid:12)(cid:12) . | Du h | C (Σ) (cid:12)(cid:12) ∂ N η (cid:12)(cid:12) . ¯ K E N . (4.68)The estimates (4.64)–(4.65) then follow. (cid:3) We now present the energy evolution for 4 N horizontal spatial derivatives. We compactlywrite (cid:13)(cid:13) U N (cid:13)(cid:13) := X α ∈ N d − , | α | =4 N k U α k + k u k . (4.69) Proposition 4.8.
It holds that ∂ t (cid:16)(cid:13)(cid:13) U N (cid:13)(cid:13) + E N (cid:17) + (cid:13)(cid:13) U N (cid:13)(cid:13) . q E N +2 , + ¯ K E N + ( p E N +2 , + K ) D N + ¯ K F N . (4.70) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 23
Proof.
Let α ∈ N with | α | = 4 N . Taking the dot product of the first equation in (4.45) with U α and then integrating by parts, using the other equations in (4.45) as Proposition 4.1, weobtain 12 ddt (cid:18)Z Ω J | U α | + Z Σ | ∂ α η | (cid:19) + 12 Z Ω J | D A U α | = Z Ω J (cid:0) U α · Q ,α + P α Q ,α (cid:1) + Z Σ − U α · Q ,α + ∂ α ηQ ,α . (4.71)We now estimate the right hand side of (4.71). For the Q ,α term, by (4.62), we have Z Ω J U α · e Q ,α . k U α k (cid:13)(cid:13)(cid:13) e Q ,α (cid:13)(cid:13)(cid:13) . k U α k p E N +2 , D N . (4.72)It follows from the integration by parts and the trace theory that, by (4.63), Z Ω J U α · div A E α ( u ) = Z Σ U α · E α ( u ) N − Z Ω J ∇ A U α : E α ( u ) . | U α | / |E α ( u ) | − / + k∇ U α k kE α ( u ) k . k U α k p E N +2 , D N . (4.73)By the definition (4.40), Z Ω J U α · D α (cid:0) D A u (cid:1) = Z Ω J C αj, ( D A u ) ij U αj + Z Ω J C αj, ( D A u ) ij U αj . (4.74)For the first term, by expanding, it suffices to estimate terms like Z Ω J U αj ∂ β (cid:0) ∂ j ϕ∂ d φ (cid:1)(cid:0) ∂ γ ∂ d ( D A v ) ij (cid:1) , (4.75)where β = 0 , γ = 0 and | β | + | γ | = 4 N. If | γ | = 4 N − | β | = 1, we integrate by partsto have Z Ω J U αj ∂ β (cid:0) ∂ j ϕ∂ d φ (cid:1)(cid:0) ∂ γ ∂ d ( D A v ) ij (cid:1) = − Z Ω ∂ γ ′ (cid:18) J U αj ∂ β (cid:0) ∂ j ϕ∂ d φ (cid:1)(cid:19) (cid:0) ∂ γ − γ ′ ∂ d ( D A v ) ij (cid:1) . k U α k p E N +2 , p D N . (4.76)For | γ | ≤ N − | β | ≥ Z Ω J U αj ∂ β (cid:0) ∂ j ϕ∂ d φ (cid:1)(cid:0) ∂ γ ∂ d ( D A v ) ij (cid:1) . k U α k p E N +2 , D N . (4.77)Similarly, we have Z Ω J C αj, ( D A u ) ij U αj . k U α k p E N +2 , D N . (4.78)It then follows from (4.74)–(4.78) that Z Ω J U α · D α (cid:0) D A u (cid:1) . k U α k p E N +2 , D N . (4.79)Hence, by (4.72), (4.73) and (4.79), we get Z Ω J U α · Q ,α . k U α k p E N +2 , D N . (4.80)For the Q ,α term, by (4.62), we have Z Ω J P α Q ,α . k P α k (cid:13)(cid:13) Q ,α (cid:13)(cid:13) . p D N p E N +2 , D N . (4.81)For the Q ,α term, by (4.64) and the trace theory, we obtain Z Σ − U α · Q ,α ≤ | U α | / (cid:12)(cid:12) Q ,α (cid:12)(cid:12) − / . k U α k q ( E N +2 , + K ) D N + ¯ K F N . (4.82) For the Q ,α term, the integration by parts gives Z Σ ∂ α η (cid:0) u h · D∂ α η + ∂ A d u · N ∂ α η (cid:1) = Z Σ (cid:18) − D · u h + ∂ A d u · N (cid:19) | ∂ α η | . |∇ u | C (Σ) | ∂ α η | . p ¯ K E N . (4.83)By (4.65), we have Z Σ ∂ α η e Q ,α . | ∂ α η | (cid:12)(cid:12)(cid:12) e Q ,α (cid:12)(cid:12)(cid:12) . p E N q E N +2 , D N + ¯ K E N . (4.84)Finally, we may follow the estimates (5-22)–(5-25) of [7] to have Z Ω J | D A U α | ≥ k D U α k . (4.85)Summing (4.80)–(4.85), we deduce (4.70) from (4.71) and (2.10) by using Cauchy’s and Korn’sinequalities. (cid:3) Comparison results
In this section we show that, up to some errors, the full energies and dissipations are compa-rable to those tangential ones at both 2 N and N + 2 levels.5.1. Energy comparison.
We begin with the result for the instantaneous energies. Recall thedefinitions of E N , E N +2 , , ¯ E N , ¯ E N +2 , , E N and K from (2.1), (2.6), (3.4), (3.6), (2.3) and(3.8), respectively. Theorem 5.1.
It holds that E N . ¯ E N + ( E N ) + K E N (5.1) and E N +2 , . ¯ E N +2 , + E N E N +2 , . (5.2) Proof.
We first prove (5.1). Note that the definition (3.4) of ¯ E N guarantees that (cid:13)(cid:13) ∂ Nt u (cid:13)(cid:13) + | η | N − + N X j =1 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N − j . ¯ E N . (5.3)We let j = 0 , . . . , N − ∂ jt to the equations in (3.13) to find − ∆ ∂ jt u + ∇ ∂ jt p = − ∂ j +1 t u + ∂ jt G in Ωdiv ∂ jt u = ∂ jt G in Ω( ∂ jt pI d − D ( ∂ jt u )) e d = ∂ jt ηe d + ∂ jt G on Σ ∂ jt u = 0 on Σ b . (5.4)Applying the elliptic estimates of Lemma A.12 with r = 4 N − j ≥ j = 1 , . . . , N − (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j + (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N − j − . (cid:13)(cid:13)(cid:13) ∂ j +1 t u (cid:13)(cid:13)(cid:13) N − j − + (cid:13)(cid:13)(cid:13) ∂ jt G (cid:13)(cid:13)(cid:13) N − j − + (cid:13)(cid:13)(cid:13) ∂ jt G (cid:13)(cid:13)(cid:13) N − j − + (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N − j − / + (cid:12)(cid:12)(cid:12) ∂ jt G (cid:12)(cid:12)(cid:12) N − j − / . (cid:13)(cid:13)(cid:13) ∂ j +1 t u (cid:13)(cid:13)(cid:13) N − j +1) + ¯ E N + ( E N ) . (5.5)A simple induction on (5.5) yields, by (5.3) again, N X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j + N − X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N − j − . (cid:13)(cid:13) ∂ Nt u (cid:13)(cid:13) + ¯ E N + ( E N ) . ¯ E N + ( E N ) . (5.6) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 25
On the other hand, applying the elliptic estimates of Lemma A.12 with r = 4 N − j = 0 and using (5.6), (5.3) and (3.47), we have k u k N + k p k N − . k ∂ t u k N − + (cid:13)(cid:13) G (cid:13)(cid:13) N − + (cid:13)(cid:13) G (cid:13)(cid:13) N − + | η | N − / + (cid:12)(cid:12) G (cid:12)(cid:12) N − / . ¯ E N + ( E N ) + K E N . (5.7)Consequently, by the definition (2.1) of E N , summing (5.6) and (5.7) gives (5.1).Now we prove (5.2). Note that the definition (3.6) of ¯ E N +2 , guarantees that (cid:13)(cid:13)(cid:13) ∂ N +2 t u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − + N +2 X j =1 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j . ¯ E N +2 , . (5.8)It follows similarly as the derivation of (5.6) and (5.7), except using (5.8) in place of (5.3) and(3.39) in place of (3.46)–(3.47), that (cid:13)(cid:13) D u (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) D p (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j + N +1 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N − j − . ¯ E N +2 , + (cid:13)(cid:13)(cid:13) ¯ D ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ¯ D ¯ D N +2) − G (cid:12)(cid:12)(cid:12) / . ¯ E N +2 , + E N E N +2 , . (5.9)By an inspection of the definition (2.6) of E N +2 , and the estimate (5.9), it remains to improvethe estimates of terms without temporal derivatives in (5.9). First, by Poincar´e’s inequality ofLemma A.7 and the second equation in (3.13), using (5.9) and (3.39), we have k Du d k N +2) − . (cid:13)(cid:13) D u d (cid:13)(cid:13) N +2) − + k ∂ d Du d k N +2) − . (cid:13)(cid:13) D u (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) DG (cid:13)(cid:13) N +2) − . ¯ E N +2 , + E N E N +2 , . (5.10)Then by the vertical component of the first equation in (3.13), using (5.10), (5.9) and (3.39),we obtain k D∂ d p k N +2) − . k Du d k N +2) − + k D∂ t u d k N +2) − + (cid:13)(cid:13) DG d (cid:13)(cid:13) N +2) − . ¯ E N +2 , + E N E N +2 , . (5.11)Next, by the horizontal component of the first equation in (3.13), using (5.9), (5.11) and (3.39),we get (cid:13)(cid:13) ∇ ∂ d u h (cid:13)(cid:13) N +2) − . (cid:13)(cid:13) ∇ D u h (cid:13)(cid:13) N +2) − + k∇ ∂ t u h k N +2) − + k∇ Dp k N +2) − + (cid:13)(cid:13) ∇ G h (cid:13)(cid:13) N +2) − . ¯ E N +2 , + E N E N +2 , . (5.12)By Poincar´e’s inequality of Lemma A.5 and the horizontal component of the fourth equation in(3.13), using (5.9), (5.12), the trace theory and (3.39), we have k ∂ d Du h k N +2) − . | D∂ d u h | + k∇ ∂ d Du h k N +2) − . (cid:12)(cid:12) D u d (cid:12)(cid:12) + (cid:12)(cid:12) DG (cid:12)(cid:12) + k∇ ∂ d Du h k N +2) − . (cid:13)(cid:13) D u d (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) ∂ d Du h (cid:13)(cid:13) N +2) − + (cid:12)(cid:12) DG (cid:12)(cid:12) . ¯ E N +2 , + E N E N +2 , . (5.13)Then by Poincar´e’s inequality of Lemma A.7, using (5.9) and (5.13), we obtain k Du h k N +2) − . k∇ Du h k N +2) − . ¯ E N +2 , + E N E N +2 , . (5.14) Now, by the second equation in (3.13) again, using (5.14), (5.10) and (3.39), we have k u d k N +2) . k Du d k N +2) − + k ∂ d u d k N +2) − . k Du k N +2) − + (cid:13)(cid:13) G (cid:13)(cid:13) N +2) − . ¯ E N +2 , + E N E N +2 , . (5.15)By the vertical component of the first equation in (3.13), using (5.9), (5.15) and (3.39), weobtain k ∂ d p k N +2) − . k ∂ t u d k N +2) − + k u d k N +2) + (cid:13)(cid:13) G d (cid:13)(cid:13) N +2) − . ¯ E N +2 , + E N E N +2 , . (5.16)Consequently, by the definition (2.6) of E N +2 , , summing (5.9), (5.12) and (5.14)–(5.16) gives(5.2). (cid:3) Dissipation comparison.
Now we consider a similar result for the dissipations. Recallthe definitions of D N , D N +2 , , ¯ D N , ¯ D N +2 , , D N and F N from (2.2), (3.3), (3.5), (3.7),(2.4) and (2.5), respectively. We also recall the notation (4.69) of (cid:13)(cid:13) U N (cid:13)(cid:13) for Alinhac goodunknowns. Theorem 5.2.
It holds that D N . ¯ D N + E N D N + K D N , (5.17) D N . ¯ D N + (cid:13)(cid:13) U N (cid:13)(cid:13) + E N D N + K F N (5.18) and D N +2 , . ¯ D N +2 , + E N D N +2 , . (5.19) Proof.
We first prove (5.17). Notice that we have not yet derived an estimate of η in terms ofthe dissipation, so we can not apply the elliptic estimates of Lemma A.12 as in Theorem 5.1. Itis crucial to observe that we can get higher regularity estimates of u on the boundary Σ from¯ D N . Indeed, since Σ is flat, we may use the definition of Sobolev norm on Σ and the tracetheorem to obtain, by the definition (3.5) of ¯ D N , that for j = 1 , . . . , N , (cid:12)(cid:12)(cid:12) ∂ jt u (cid:12)(cid:12)(cid:12) N − j +1 / . (cid:12)(cid:12)(cid:12) ∂ jt u (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D N − j ∂ jt u (cid:12)(cid:12)(cid:12) / . (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D N − j ∂ jt u (cid:13)(cid:13)(cid:13) . ¯ D N (5.20)and | u | N − / . | u | + (cid:12)(cid:12) D N − u (cid:12)(cid:12) / . k u k + (cid:13)(cid:13) D N − u (cid:13)(cid:13) . ¯ D N (5.21)This motivates us to use the elliptic estimates of Lemma A.13.Now we let j = 0 , . . . , N − ∂ jt to the equations in (3.13) to find − ∆ ∂ jt u + ∇ ∂ jt p = − ∂ j +1 t u + ∂ jt G in Ωdiv ∂ jt u = ∂ jt G in Ω ∂ jt u = ∂ jt u on Σ ∂ jt u = 0 on Σ b . (5.22)Applying the elliptic estimates of Lemma A.13 with r = 4 N − j + 1 ≥ j = 1 , . . . , N − (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j +1 + (cid:13)(cid:13)(cid:13) ∇ ∂ jt p (cid:13)(cid:13)(cid:13) N − j − . (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∂ j +1 t u (cid:13)(cid:13)(cid:13) N − j − + (cid:13)(cid:13)(cid:13) ∂ jt G (cid:13)(cid:13)(cid:13) N − j − + (cid:13)(cid:13)(cid:13) ∂ jt G (cid:13)(cid:13)(cid:13) N − j + (cid:12)(cid:12)(cid:12) ∂ jt u (cid:12)(cid:12)(cid:12) N − j +1 / . ¯ D N + (cid:13)(cid:13)(cid:13) ∂ j +1 t u (cid:13)(cid:13)(cid:13) N − j +1)+1 + E N D N . (5.23) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 27
A simple induction on (5.23) yields N X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N − j +1 + N − X j =1 (cid:13)(cid:13)(cid:13) ∇ ∂ jt p (cid:13)(cid:13)(cid:13) N − j − . ¯ D N + (cid:13)(cid:13) ∂ Nt u (cid:13)(cid:13) + E N D N . ¯ D N + E N D N . (5.24)On the other hand, applying the elliptic estimates of Lemma A.13 with r = 4 N to the problem(5.22) for j = 0 and using (5.21), (5.24) and (3.49), we have k u k N + k∇ p k N − . k u k + k ∂ t u k N − + (cid:13)(cid:13) G (cid:13)(cid:13) N − + (cid:13)(cid:13) G (cid:13)(cid:13) N − + | u | N − / . ¯ D N + E N D N + K D N . (5.25)Now we estimate η , and we turn to the boundary conditions in (3.13). For the η term, i.e.without temporal derivatives, we use the vertical component of the third equation in (3.13) η = p − ∂ d u d − G d on Σ . (5.26)Notice that at this point we do not have any bound of p on the boundary Σ, but we havebounded ∇ p in Ω. Applying D to (5.26), by the trace theory, (5.25) and (3.49), we obtain | Dη | N − / . | Dp | N − / + | D∂ d u d | N − / + (cid:12)(cid:12) DG d (cid:12)(cid:12) N − / . ¯ D N + E N D N + K D N . (5.27)For the term ∂ jt η for j ≥
1, we use instead the fourth equation in (3.13) to gain the regularity: ∂ t η = u d + G on Σ . (5.28)Indeed, for ∂ t η , we use (5.28), (5.21) and (3.49) to find | ∂ t η | N − / . | u d | N − / + (cid:12)(cid:12) G (cid:12)(cid:12) N − / . ¯ D N + E N D N + K D N . (5.29)For j = 2 , . . . , N + 1 we apply ∂ j − t to (5.28) to see, by (5.20) and (3.48), that (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N − j +5 / ≤ (cid:12)(cid:12)(cid:12) ∂ j − t u d (cid:12)(cid:12)(cid:12) N − j +5 / + (cid:12)(cid:12)(cid:12) ∂ j − t G (cid:12)(cid:12)(cid:12) N − j +5 / . (cid:12)(cid:12)(cid:12) ∂ j − t u d (cid:12)(cid:12)(cid:12) N − j − / + (cid:12)(cid:12)(cid:12) ∂ j − t G (cid:12)(cid:12)(cid:12) N − j − / . ¯ D N + E N D N . (5.30)The ∂ jt η estimates allows us to further bound (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) for j = 1 , . . . , N −
1. Indeed, applying ∂ jt for j = 1 , . . . , N − (cid:12)(cid:12)(cid:12) ∂ jt p (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ d ∂ jt u d (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ jt G d (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12) ∂ jt G (cid:12)(cid:12)(cid:12) . ¯ D N + E N D N + K D N . (5.31)By Poincar´e’s inequality of Lemma A.5, by (5.24) and (5.31), we have (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) . (cid:12)(cid:12)(cid:12) ∂ jt p (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) ∂ d ∂ jt p (cid:13)(cid:13)(cid:13) . ¯ D N + E N D N + K D N . (5.32)Consequently, by the definition (2.2) of D N , summing (5.24), (5.25), (5.27), (5.29), (5.30) and(5.32) gives (5.17).We next prove (5.18). We must resort to Alinhac good unknowns U α for | α | = 4 N . Indeed,by the definition of U α in (4.44), the trace theory and Lemma A.10, we have | u | N +1 / . | u | + (cid:12)(cid:12) D N u (cid:12)(cid:12) / . | u | + (cid:12)(cid:12) U N (cid:12)(cid:12) / + (cid:12)(cid:12) ∂ A d uD N η (cid:12)(cid:12) / . (cid:13)(cid:13) U N (cid:13)(cid:13) + | ∂ d u | C (Σ) (cid:12)(cid:12) D N η (cid:12)(cid:12) / . (cid:13)(cid:13) U N (cid:13)(cid:13) + ¯ K F N . (5.33) Applying the elliptic estimates of Lemma A.13 with r = 4 N + 1 to the problem (5.22) for j = 0and using (5.33), (5.24) and (3.50), we obtain k u k N +1 + k∇ p k N − . k u k + k ∂ t u k N − + (cid:13)(cid:13) G (cid:13)(cid:13) N − + (cid:13)(cid:13) G (cid:13)(cid:13) N + | u | N +1 / . ¯ D N + (cid:13)(cid:13) U N (cid:13)(cid:13) + E N D N + K F N . (5.34)Applying D to (5.26), by the trace theory, (5.34) and (3.50), we have | Dη | N − / . | Dp | N − / + | D∂ d u d | N − / + (cid:12)(cid:12) DG d (cid:12)(cid:12) N − / . ¯ D N + (cid:13)(cid:13) U N (cid:13)(cid:13) + E N D N + K F N . (5.35)On the other hand, we use (5.28), (5.33) and (3.50) to find | ∂ t η | N − / . | u d | N − / + (cid:12)(cid:12) G (cid:12)(cid:12) N − / . (cid:13)(cid:13) U N (cid:13)(cid:13) + E N D N + K F N . (5.36)Consequently, by the definition (2.4) of D N , summing (5.34)–(5.36) gives (5.18).Now we prove (5.19). Note that the definition (3.7) of ¯ D N +2 , and the trace theory, similarlyas (5.20)–(5.21), guarantee that N +2 X j =0 (cid:12)(cid:12)(cid:12) ∂ jt u (cid:12)(cid:12)(cid:12) N − j +1 / . ¯ D N +2 , (5.37)It follows similarly as the derivation of (5.24), (5.25), (5.27), (5.30) and (5.32), except using(5.37) in place of (5.20)–(5.21) and (3.40) in place of (3.48)–(3.49), that (cid:13)(cid:13) D u (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) ∇ D p (cid:13)(cid:13) N +2) − + N +2 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt u (cid:13)(cid:13)(cid:13) N +2) − j +1 + k∇ ∂ t p k N +2) − + N +1 X j =1 (cid:13)(cid:13)(cid:13) ∂ jt p (cid:13)(cid:13)(cid:13) N +2) − j + (cid:12)(cid:12) D η (cid:12)(cid:12) N +2) − / + N +3 X j =2 (cid:12)(cid:12)(cid:12) ∂ jt η (cid:12)(cid:12)(cid:12) N +2) − j +5 / . ¯ D N +2 , + (cid:13)(cid:13)(cid:13) ¯ D ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ¯ D ¯ ∇ N +2) − G (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12) D G d (cid:12)(cid:12) N +2) − / + (cid:12)(cid:12)(cid:12) ¯ ∇ N +2) − ∂ t G (cid:12)(cid:12)(cid:12) . ¯ D N +2 , + E N D N +2 , . (5.38)By an inspection of the definition (3.3) of D N +2 , and the estimate (5.38), it remains to improvethe estimates of u and p terms without temporal derivatives in (5.38) and derive the estimateof ∂ t η . First, by the second equation in (3.13), using (5.38) and (3.40), we have k Du d k N +2) . (cid:13)(cid:13) D u d (cid:13)(cid:13) N +2) − + k D∂ d u d k N +2) − . (cid:13)(cid:13) D u (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) DG (cid:13)(cid:13) N +2) − . ¯ D N +2 , + E N D N +2 , . (5.39)We then use (5.28), (5.39), the trace theory and (3.40) to find | D∂ t η | N +2) − / . | Du d | N +2) − / + (cid:12)(cid:12) DG (cid:12)(cid:12) N +2) − / . ¯ D N +2 + E N D N +2 , . (5.40)By the vertical component of the first equation in (3.13), using (5.39), (5.38) and (3.40), weobtain k D∂ d p k N +2) − . k Du d k N +2) + k D∂ t u d k N +2) − + (cid:13)(cid:13) DG d (cid:13)(cid:13) N +2) − . ¯ D N +2 , + E N D N +2 , . (5.41) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 29
By the horizontal component of the first equation in (3.13), using (5.38), (5.41) and (3.40), wehave (cid:13)(cid:13) ∇ ∂ d u h (cid:13)(cid:13) N +2) − . (cid:13)(cid:13) ∇ ∂ d D u h (cid:13)(cid:13) N +2) − + k∇ ∂ d ∂ t u h k N +2) − + k∇ ∂ d Dp k N +2) − + (cid:13)(cid:13) ∇ ∂ d G h (cid:13)(cid:13) N +2) − . ¯ D N +2 , + E N D N +2 , . (5.42)Next, by the second equation in (3.13), using (5.42) and (3.40), we obtain (cid:13)(cid:13) ∂ d u d (cid:13)(cid:13) N +2) − . (cid:13)(cid:13) ∂ d Du h (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) ∂ d G (cid:13)(cid:13) N +2) − . ¯ D N +2 , + E N D N +2 , . (5.43)Finally, by the vertical component of the first equation in (3.13), using (5.38), (5.43) and (3.40),we have (cid:13)(cid:13) ∂ d p (cid:13)(cid:13) N +2) − . (cid:13)(cid:13) ∂ d ∆ u d (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) ∂ d ∂ t u d (cid:13)(cid:13) N +2) − + (cid:13)(cid:13) ∂ d G d (cid:13)(cid:13) N +2) − . ¯ D N +2 , + E N D N +2 , . (5.44)Consequently, by the definition (3.3) of D N +2 , , summing (5.38)–(5.44) gives (5.19). (cid:3) A priori estimates
Bounded estimates of E N and D N . Now we show the boundedness of E N ( t )+ R t D N . Proposition 6.1.
There exists δ > so that if G N ( T ) ≤ δ , then E N ( t ) + Z t D N ( r ) dr . E N (0) + ( G N ( t )) . (6.1) Proof.
Note first that since E N ( t ) ≤ G N ( T ) ≤ δ , by taking δ small, we obtain from (5.1) ofTheorem 5.1 and (5.17) of Theorem 5.2 that¯ E N . E N . ¯ E N + K E N , and ¯ D N . D N . ¯ D N + K D N . (6.2)Now we integrate (4.1) of Proposition 4.1 and (4.35) of Proposition 4.4 in time and then addup the two resulting estimates to conclude that, by the definition (3.4) of ¯ E N , the definition(3.5) of ¯ D N and (4.36),¯ E N ( t ) + Z t ¯ D N . E N (0) + ( E N ( t )) / + Z t (cid:16)p E N D N + p D N K D N (cid:17) . (6.3)By (6.2), we may improve (6.3) to be, since δ is small and by Cauchy’s inequality, E N ( t ) + Z t D N . E N (0) + K ( t ) E N ( t ) + Z t K D N . (6.4)By the estimate (3.36) of Lemma 3.4 and the definition (2.7) of G N ( t ), we obtain K ( t ) . E N ( t ) κ d / E N +2 , ( t ) − κ d / . G N ( t )(1 + t ) − κ d , (6.5)and hence we deduce from (6.4) that E N ( t ) + Z t D N ( r ) dr . E N (0) + G N ( t ) E N ( t )(1 + t ) − κ d + Z t G N ( r ) D N ( r )(1 + r ) − κ d dr . E N (0) + ( G N ( t )) (6.6)for 2 − κ d ≥ ϑ + κ d > ϑ >
0. This gives (6.1). (cid:3)
Growth estimates of E N and D N . We first derive the estimates of E N by a time-weighted argument. Note that a time integral of the time weighted estimates of Alinhac goodunknowns U α for | α | = 4 N will be obtained, which is crucial for closing the estimate for F N . Proposition 6.2.
There exists δ > so that if G N ( T ) ≤ δ , then E N ( t )(1 + t ) ϑ + Z t E N ( r )(1 + r ) ϑ dr + Z t (cid:13)(cid:13) U N ( r ) (cid:13)(cid:13) (1 + r ) ϑ dr . G N (0) + ( G N ( t )) / . (6.7) Proof.
Multiplying (4.70) of Proposition 4.8 by (1 + t ) − ϑ with ϑ >
0, we find ∂ t (cid:13)(cid:13) U N (cid:13)(cid:13) + E N (1 + t ) ϑ ! + ϑ (cid:13)(cid:13) U N (cid:13)(cid:13) + E N (1 + t ) ϑ + (cid:13)(cid:13) U N (cid:13)(cid:13) (1 + t ) ϑ . q E N +2 , + ¯ K E N + ( p E N +2 , + K ) D N + ¯ K F N (1 + t ) ϑ . (6.8)By the estimate (3.37) of Lemma 3.4 and the definition (2.7) of G N ( t ), we obtain¯ K ( t ) . E N +2 , ( t ) . G N ( t )(1 + t ) − . (6.9)By (6.9) and (6.5), we deduce from (6.8) that ∂ t (cid:13)(cid:13) U N (cid:13)(cid:13) + E N (1 + t ) ϑ ! + ϑ E N (1 + t ) ϑ + (cid:13)(cid:13) U N (cid:13)(cid:13) (1 + t ) ϑ . p G N E N (1 + t ) ϑ + p G N D N (1 + t ) ϑ + G N F N (1 + t ) ϑ (6.10)for 2 − κ d ≥
1. Integrating (6.16) directly in time, since δ > E N ( t )(1 + t ) ϑ + Z t E N ( r )(1 + r ) ϑ dr + Z t (cid:13)(cid:13) U N ( r ) (cid:13)(cid:13) (1 + r ) ϑ dr . G N (0) + p G N ( t ) Z t (cid:18) D N ( r )(1 + r ) ϑ + F N ( r )(1 + r ) ϑ (cid:19) dr . G N (0) + ( G N ( t )) / (6.11)for 1 + ϑ ≥ ϑ + κ d . This yields (6.7). (cid:3) Now we show a time weighted integral-in-time estimate of D N . Proposition 6.3.
There exists δ > so that if G N ( T ) ≤ δ , then Z t D N ( r )(1 + r ) ϑ + κ d dr . E N (0) + ( G N ( t )) / . (6.12) Proof.
Multiplying (5.18) by (1 + t ) − ϑ − κ d of Theorem 5.2 and then integrating in time, by (6.7),(6.1), (6.5) and and the definition (2.7) of G N ( t ), we have Z t D N ( r )(1 + r ) ϑ + κ d dr . Z t (cid:13)(cid:13) U N (cid:13)(cid:13) + D N + K F N (1 + r ) ϑ + κ d dr . G N (0) + ( G N ( t )) / + Z t G N ( r ) F N ( r )(1 + r ) ϑ dr . G N (0) + ( G N ( t )) / . (6.13)This gives (6.12). (cid:3) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 31
Growth estimate of F N . To estimate F N , it is crucial that we will use a differentargument from [7]. Applying J N +1 / to the fourth equation in (1.8), we have ∂ t J N +1 / η + u h · D J N +1 / η = J N +1 / u d − h J N +1 / , u h i · Dη. (6.14)We now derive the estimate of F N by a time-weighted argument. Proposition 6.4.
There exists δ > so that if G N ( T ) ≤ δ , then F N ( t )(1 + t ) ϑ + Z t F N ( r )(1 + r ) ϑ dr . E N (0) + ( G N ( t )) / . (6.15) Proof.
Taking the dot product of (6.14) with J N +1 / η , and then integrate by parts; using thecommutator estimate of Lemma A.11 with s = 4 N + 1 /
2, by (5.33), we find that12 ddt F N = 12 ddt Z Σ |J N +1 / η | = Z Σ J N +1 / u d J N +1 / η − Z Σ u h · D |J N +1 / η | − Z Σ h J N +1 / , u h i · Dη J N +1 / η = Z Σ J N +1 / u d J N +1 / η + 12 Z Σ D · u h |J N +1 / η | − Z Σ h J N +1 / , u h i · Dη J N +1 / η . | u d | N +1 / | η | N +1 / + | Du h | L ∞ (Σ) | η | N +1 / + (cid:16) | Du h | L ∞ (Σ) | η | N +1 / + | u h | N +1 / | Dη | L ∞ (Σ) (cid:17) | η | N +1 / . | u | N +1 / p F N + | Du | L ∞ (Σ) F N . (cid:13)(cid:13) U N (cid:13)(cid:13) p F N + p ¯ K F N (6.16)Multiplying (6.16) by (1 + t ) − − ϑ , by (6.9) and the definition of G N ( t ), we have ∂ t (cid:18) F N (1 + t ) ϑ (cid:19) + (1 + ϑ ) F N (1 + t ) ϑ . (cid:13)(cid:13) U N (cid:13)(cid:13) √F N + √ ¯ K F N (1 + t ) ϑ . (cid:13)(cid:13) U N (cid:13)(cid:13) (1 + t ) ϑ/ √F N (1 + t ) ϑ/ + p G N F N (1 + t ) ϑ . (6.17)Integrating (6.17) directly in time, since δ > F N ( t )(1 + t ) ϑ + Z t F N ( r )(1 + r ) ϑ dr . G N (0) + Z t (cid:13)(cid:13) U N ( r ) (cid:13)(cid:13) (1 + r ) ϑ dr . G N (0) + ( G N ( t )) / . (6.18)This yields (6.15). (cid:3) Decay estimate of E N +2 , . It remains to show the decay estimate of E N +2 , . Proposition 6.5.
There exists δ > so that if G N ( T ) ≤ δ , then (1 + t ) E N +2 , ( t ) . E N (0) + ( G N ( T )) for all ≤ t ≤ T. (6.19) Proof.
Since E N ( t ) ≤ G N ( T ) ≤ δ , by taking δ small, we obtain from (5.2) of Theorem 5.1 and(5.19) of Theorem 5.2 that E N +2 , . ¯ E N +2 , . E N +2 , and D N +2 , . ¯ D N +2 , . D N +2 , . (6.20)Now we combine the estimate (4.13) of Proposition 4.2 and the estimate (4.37) of Proposition4.5 to conclude that, by the definition (3.6) of ¯ E N +2 , and the definition (3.7) of ¯ D N +2 , , ddt (cid:18) ¯ E N +2 , + 2 Z Ω J ∂ N +1 t pF ,N +2 (cid:19) + ¯ D N +2 , . p E N D N +2 , (6.21) By (4.36), (6.20) and the smallness of δ , we deduce that there exists an instantaneous energy,which is equivalent to ¯ E N +2 , and E N +2 , , but for simplicity is still denoted by ¯ E N +2 , , suchthat ddt ¯ E N +2 , + D N +2 , ≤ . (6.22)In order to get decay from (6.22), we shall now estimate ¯ E N +2 , in terms of D N +2 , . Noticethat D N +2 , can control every term in ¯ E N +2 , except (cid:12)(cid:12) D N +2) η (cid:12)(cid:12) , (cid:12)(cid:12) D η (cid:12)(cid:12) and | ∂ t η | . The keypoint is to use the Sobolev interpolation as in [7]. Indeed, we first have that (cid:12)(cid:12)(cid:12) D N +2) η (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) D η (cid:12)(cid:12) θ N +2) − / | η | − θ )4 N − . D θN +2 , E − θ N , where θ = 4 N − N − . (6.23)and (cid:12)(cid:12) D η (cid:12)(cid:12) ≤ | η | / (cid:12)(cid:12) D η (cid:12)(cid:12) / . E / N D / N +2 , . (6.24)By the L table in (3.26), | ∂ t η | . E / N D / N +2 , . (6.25)Hence, in light of (6.23)–(6.25), since N ≥ N − / (4 N − ≥ /
3, we deduce E N +2 , . E / N D / N +2 , . (6.26)Denote by M := sup [0 ,T ] E N . Hence by (6.22) and (6.26), there exists some constant C > ddt E N +2 , + C M / E / N +2 , ≤ . (6.27)Solving this differential inequality directly, we obtain E N +2 , ( t ) ≤ M ( M / + C ( E N +2 , (0)) / t ) E N +2 , (0) . (6.28)Using that E N +2 , (0) . M , we obtain from (6.28) that E N +2 , ( t ) . M t . (6.29)This together with (6.1) directly implies (6.19). (cid:3) Proof of main theorem
Now we can arrive at our ultimate energy estimates for G N . Theorem 7.1.
There exists a universal < δ < so that if G N ( T ) ≤ δ , then G N ( T ) . G N (0) . (7.1) Proof.
It follows directly from the definition of G N and Propositions 6.1–6.5 that G N ( t ) . G N (0) + ( G N ( T )) / for all 0 ≤ t ≤ T. (7.2)This proves (7.1) since δ > (cid:3) Now we can present the
Proof of Theorem 2.1.
By the a priori estimates (7.1) of Theorem 7.1, we may employ a conti-nuity argument as in Section 11 of [7] to prove that there exists a universal ε > E N (0) + F N (0) ≤ ε , then the unique local solution constructed in [5] is indeed a globalsolution to (1.8) on [0 , ∞ ). This completes the proof of Theorem 2.1. (cid:3) ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 33
Appendix A. Analytic tools
A.1.
Poisson integral.
For a function f , defined on Σ = R d − , the Poisson integral in R d − × ( −∞ ,
0) is defined by P f ( x ′ , x d ) = Z R d − ˆ f ( ξ ) e π | ξ | x d e πix ′ · ξ dξ. (A.1)Although P f is defined in all of R d − × ( −∞ , R d − × ( − b, P f on the set R d − × ( −∞ , Lemma A.1.
Let P f be the Poisson integral of a function f that is either in ˙ H q (Σ) or ˙ H q − / (Σ) for q ∈ N , where ˙ H s is the usual homogeneous Sobolev space of order s . Then k∇ q P f k . k f k H q − / (Σ) and k∇ q P f k . k f k H q (Σ) . (A.2) Proof.
It follows from the definition (A.1). We refer to Lemma A.7 of [5]. (cid:3)
A.2.
Interpolation estimates.
Assume that Σ = R d − and Ω = Σ × ( − b, Lemma A.2.
Let P f be the Poisson integral of f , defined on Σ . Let q, s ∈ N , and r ≥ . Thenthe following estimates hold. (1) Let θ = sq + s and − θ = qq + s . (A.3) Then k∇ q P f k . (cid:16) | f | (cid:17) θ (cid:16)(cid:12)(cid:12) D q + s f (cid:12)(cid:12) (cid:17) − θ . (A.4)(2) Let r + s > ( d − / , θ = r + s − q + s + r , and − θ = q + 1 q + s + r . (A.5) Then k∇ q P f k L ∞ . (cid:16) | f | (cid:17) θ (cid:16)(cid:12)(cid:12) D q + s f (cid:12)(cid:12) r (cid:17) − θ . (A.6)(3) Let s > ( d − / . Then k∇ q P f k L ∞ . | D q f | s . (A.7) Proof.
We refer to Lemma A.6 of [7]. (cid:3)
The next result is a similar interpolation result for functions defined only on Σ.
Lemma A.3.
Let f be defined on Σ . Then the following estimates hold. (1) Let q, s ∈ (0 , ∞ ) and θ = sq + s and − θ = qq + s . (A.8) Then | D q f | . (cid:16) | f | (cid:17) θ (cid:16)(cid:12)(cid:12) D q + s f (cid:12)(cid:12) (cid:17) − θ . (A.9)(2) Let q, s ∈ N , r ≥ , r + s > ( d − / , θ = r + s − q + s + r , and − θ = q + 1 q + s + r . (A.10) Then | D q f | L ∞ . (cid:16) | f | (cid:17) θ (cid:16)(cid:12)(cid:12) D q + s f (cid:12)(cid:12) r (cid:17) − θ . (A.11) Proof.
See Lemma A.7 of [7]. (cid:3)
Now we record a similar result for functions defined on Ω that are not Poisson integrals. Theresult follows from estimates on fixed horizontal slices.
Lemma A.4.
Let f be a function on Ω . Let q, s ∈ N , and r ≥ . Then the following estimateshold. (1) Let θ = sq + s and − θ = qq + s . (A.12) Then k D q f k . (cid:16) k f k (cid:17) θ (cid:16)(cid:13)(cid:13) D q + s f (cid:13)(cid:13) (cid:17) − θ . (A.13)(2) Let r + s > ( d − / , θ = r + s − q + s + r , and − θ = q + 1 q + s + r . (A.14) Then k D q f k L ∞ . (cid:16) k f k (cid:17) θ (cid:16)(cid:13)(cid:13) D q + s f (cid:13)(cid:13) r +1 (cid:17) − θ (A.15) and k D q f k L ∞ (Σ) . (cid:16) k f k (cid:17) θ (cid:16)(cid:13)(cid:13) D q + s f (cid:13)(cid:13) r +1 (cid:17) − θ (A.16) Proof.
We refer to Lemma A.8 of [7]. (cid:3)
A.3.
Poincar´e-type inequalities.
We have the following Poincar´e-type inequalities.
Lemma A.5.
It holds that k f k . | f | + k ∂ d f k and k f k L ∞ . | f | L ∞ + k ∂ d f k L ∞ . (A.17) Lemma A.6.
For f = 0 on Σ b , it holds that k f k . k ∂ d f k and k f k L ∞ . k ∂ d f k L ∞ . (A.18) Lemma A.7.
For f = 0 on Σ b , it holds that k f k . k f k . k∇ f k and k f k L ∞ . k f k W , ∞ . k∇ f k L ∞ . (A.19)We will need a version of Korn’s inequality. Lemma A.8.
It holds that k u k . k D u k for all u ∈ H (Ω; R d ) so that u = 0 on Σ b .Proof. We refer to Lemma 2.7 of [2]. (cid:3)
A.4.
Product estimates.
We will need some estimates of the product of functions in Sobolevspaces.
Lemma A.9.
Let the domain of function spaces be either Σ or Ω . Let ≤ r ≤ s ≤ s be suchthat s > r + n/ , n = d − or d , then k f g k H r . k f k H s k g k H s . (A.20) Proof.
We refer to Lemma A.1 of [7]. (cid:3)
We will also need the following variant.
Lemma A.10.
It holds that | f g | / . | f | C (Σ) | g | / and | f g | − / . | f | C (Σ) | g | − / . (A.21) Proof.
We refer to Lemma A.2 of [7]. (cid:3)
A.5.
Commutator estimate.
Let J = (1 − ∆) / with ∆ the Laplace operator on R n .We recall the following commutator estimate: Lemma A.11. k [ J s , f ] g k L ≤ k∇ f k L ∞ kJ s − g k L + kJ s f k L k g k L ∞ . (A.22) Proof.
We refer to Lemma X1 of [11]. (cid:3)
ISCOUS SURFACE WAVES WITHOUT SURFACE TENSION 35
A.6.
Stokes elliptic estimates.
We recall the elliptic estimates for two classical Stokes prob-lems with different boundary conditions.
Lemma A.12.
Let r ≥ . Suppose that f ∈ H r − (Ω) , h ∈ H r − (Ω) and ψ ∈ H r − / (Σ) . Thenthere exists unique u ∈ H r (Ω) , p ∈ H r − (Ω) solving the problem − ∆ u + ∇ p = f in Ωdiv u = h in Ω( pI d − D u ) e d = ψ on Σ u = 0 on Σ b . (A.23) Moreover, k u k r + k p k r − . k f k r − + k h k r − + | ψ | r − / . (A.24) Proof.
We refer to the proof in Lemma 3.3 of [2]. (cid:3)
Lemma A.13.
Let r ≥ . Suppose that f ∈ H r − (Ω) , h ∈ H r − (Ω) , ϕ ∈ H r − / (Σ) and that u ∈ H r (Ω) , p ∈ H r − (Ω) (up to constants) solving the problem − ∆ u + ∇ p = f in Ωdiv u = h in Ω u = ϕ on Σ u = 0 on Σ b . (A.25) Then k u k r + k∇ p k r − . k u k + k f k r − + k h k r − + | ϕ | r − / . (A.26) Proof.
We refer to the proof of (3.7) in [10]. (cid:3)
References [1] S. Alinhac. Existence d’ondes de rar´efaction pour des syst`emes quasi-lin´eaires hyperboliques multidimension-nels.(French. English summary) [Existence of rarefaction waves for multidimensional hyperbolic quasilinearsystems]
Comm. Partial Differential Equations (1989), no. 2, 173–230.[2] J. Beale. The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl.Math. (1981), no. 3, 359–392.[3] J. Beale. Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. (1983/84), no. 4,307–352.[4] J. Beale, T. Nishida. Large-time behavior of viscous surface waves. Recent topics in nonlinear PDE, II (Sendai, 1984), 1–14, North-Holland Math. Stud., 128, North-Holland, Amsterdam, 1985.[5] Y. Guo, I. Tice. Local well-posedness of the viscous surface wave problem without surface tension.
Anal.PDE (2013), no. 2, 287–369.[6] Y. Guo, I. Tice. Almost exponential decay of periodic viscous surface waves without surface tension. Arch.Ration. Mech. Anal. (2013), no. 2, 459–531.[7] Y. Guo, I. Tice. Decay of viscous surface waves without surface tension in horizontally infinite domains.
Anal. PDE (2013), no. 6, 1429–1533.[8] Y. Hataya. Decaying solution of a Navier-Stokes flow without surface tension. J. Math. Kyoto Univ. Nonlinear Anal. (2009), no. 12, e2535–e2539.[10] Y. Kagei. Large time behavior of solutions to the compressible Navier-Stokes equation in an infinite layer. Hiroshima Math. J. (2008), no. 1, 95–124.[11] T. Kato, G. Ponce. Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl.Math. (1988), no. 7, 891–907.[12] T. Nishida, Y. Teramoto, H. Yoshihara. Global in time behavior of viscous surface waves: horizontallyperiodic motion. J. Math. Kyoto Univ. Comm.Partial Differential Equations (1990), no. 6, 823–903.[14] D. Sylvester. Decay rates for a two-dimensional viscous ocean of finite depth. J. Math. Anal. Appl. (1996), no. 2, 659–666.[15] Z. Tan, Y. J. Wang. Zero surface tension limit of viscous surface waves.
Comm. Math. Phys. (2014), no.2, 733–807. [16] A. Tani, N. Tanaka. Large-time existence of surface waves in incompressible viscous fluids with or withoutsurface tension.
Arch. Rational Mech. Anal. (1995), no. 4, 303–314.
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
E-mail address , Y. J. Wang:, Y. J. Wang: