Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus
aa r X i v : . [ m a t h . A P ] F e b LOCAL WELL-POSEDNESS AND PARABOLIC SMOOTHING OFSOLUTIONS OF FULLY NONLINEAR THIRD-ORDEREQUATIONS ON THE TORUS
TRISTAN ROY
Abstract.
In this paper we study the initial value problem of fully nonlinearthird-order dispersive equations on the torus, that is ∂ t u = F (cid:0) ∂ x u, ∂ x u, ∂ x u, u, x, t (cid:1) with F a smooth function depending on the space variable x , the time variable t , the first three derivatives of u with respect to x , and u . In particular we findconditions on F and u (0) for which one can construct a local and unique solu-tion u . In particular if F and u (0) satisfy some conditions then the equationbehaves like a diffusive one and it has a parabolic smoothing property: the so-lution is infinitely smooth in one direction of time and the problem is ill-posedin the other direction of time. If F and u (0) satisfy some other conditionsthen the equation behaves like a dispersive one. We also prove continuousdependence with respect to the data. The proof relies upon energy estimatescombined with a gauge transform [7, 8] and the Bona-Smith argument [1]. Contents
1. Introduction 12. Notation 63. Preliminary lemmas 84. Regularized problem: local well-posedness 165. Gauged energy 206. Gauged energy estimates 226.1. General estimates 226.2. Gauged energy estimates: statement 257. Gauged energy estimates for the difference of two solutions 287.1. General estimates 287.2. Gauged energy estimates for difference of solutions: statement 328. Proof of Theorem 359. Proof of Corollary 1.2 3710. Proof of Lemmas 3810.1. Proof of Lemma 6.4 3810.2. Proof of Lemma 7.4 4211. Appendix 52References 531.
Introduction
We consider the third-order nonlinear Cauchy problem on the torus T := R / π Z (1.1) ∂ t u = F ( ~u, t ) · Here F : R × T × R → R , ω := ( ω , .., ω − ) → F ( ω ) denotes a C ∞ function, ~f := (cid:0) ∂ x f, ∂ x f, ∂ x f, f, x (cid:1) with f a function defined on T that lies in an appropriatespace, x ∈ T the space variable, and t the time variable.Equations of the form (1.1) have been extensively studied in the literature, inparticular on the real line (i.e x ∈ R ) and for equations(1.2) ∂ t u + ∂ x u = P ( ∂ x u, ∂ x u, u ) · Here P is a polynomial that satisfies some conditions. We mention some results onthe real line. In this case, it is well-known that the linear part of these equationscreates a strong dispersive effect, which should overcome the effect of the nonlin-earity at least locally and be a plus for the construction of local solutions. In [11],local well-posedness (L.W.P) on a small interval [ − T, T ] (with
T >
0) was provedfor small data in [11] and for large data in [10] in weighted Sobolev spaces withregularity exponent high enough for equations (1.2) by using, among other things,the dispersive effect of these equations. We also refer to [12] who studied L.W.P forgeneralizations of (1.2) to systems. In [13], L.W.P was proved in weighted Besovspaces for small data. In [5], L.W.P was studied in some translation-invariant sub-spaces of Sobolev spaces. In [4], under some additional assumptions on P , L.W.Pwas proved in (standard) Sobolev spaces H k . Recently, in [2], the authors consid-ered the full nonlinear problem (1.1) and found that under some assumptions on F and the data, the problem is locally well-posed and that the dispersive effect ofthe equation dominates.In this paper we aim at constructing local strong solutions of the fully nonlin-ear problem (1.3) on the torus T . By strong solutions of (1.1) we mean solutions u ∈ C (cid:0) I, H k (cid:1) of the following integral equation on a interval I containing 0:(1.3) t ∈ I : u ( t ) = u (0) + R t F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17) dt ′ , with u (0) (the initial data) a given function that lies in H k . Here H k denotesthe usual Sobolev space with index k , i.e the closure of smooth functions f withrespect to the norm k f k H k := k{h n i k ˆ f ( n ) } n ∈ Z k l that are defined on the torus.Here h n i := (1 + n ) and ˆ f ( n ) denotes the n th − Fourier coefficient of f definedon the torus, that is ˆ f ( n ) := π R π f ( x ) e − inx dx .To this end we define some quantities that we use throughout this paper and thatappear in the statement of your theorem. Let h f i denote the average of a function f , that is h f i := π R T f ( x ′ ) dx ′ . Let t ∈ R . We define δ ( ~f , t ) := inf x ∈ T (cid:12)(cid:12)(cid:12) ∂ ω F ( ~f , t )( x ) (cid:12)(cid:12)(cid:12) , and ˜ δ ( ~f ) := δ ( ~f , · It is well-known that ˜ δ (cid:16) −−→ u (0) (cid:17) measures the strength of the dispersion of (1.3).Intuitively, dispersion should facilitate the rule-out of blow-up and the constructionof local solutions: see e.g [2] for discussions related to this number. Therefore in ONLINEAR THIRD-ORDER EQUATIONS 3 the sequel we only consider solutions of (1.1) with initial data u (0) that satisfy thewell-known non-degeneracy dispersion property, that is ˜ δ (cid:16) −−→ u (0) (cid:17) >
0. Let P ( f, t ) := 2 P p =0 ∂ ω ω p F ( ~f , t ) ∂ p +1 x f + ∂ ω ω − F ( ~f , t ) + ∂ ω F ( ~f , t ) ,Q ( f, t ) := − ∂ ω F ( ~f , t ) (cid:28) P ( f,t ) ∂ω F ( ~f,t ) (cid:29) ,δ ′ ( f, t ) := inf x ∈ T | Q ( f, t )( x ) | , and ˜ δ ′ ( ~f ) := δ ′ ( ~f , · We define now the notion of parabolic resonance for (1.3), by analogy with that in[14] in the framework of fifth-order (semilinear) dispersive equations. We say that(1.3) is of non-parabolic resonance type if D P ( f,t ) ∂ ω F ( ~f,t ) E = 0 for all t ∈ R and for all f ∈ E := n h ∈ C ∞ ( T ) : δ ( ~h, t ) > t ∈ R o . If not, we say that (1.3) is of parabolic resonance type .We shall see that L.W.P holds on a small forward-in-time interval [0 , T ] (resp.a small backward-in-time interval [ − T, x ∈ T Q ( u (0) , x ) > x ∈ T Q ( u (0) , x ) < Q ( u (0) , P + ,k and P − ,k : P + ,k := n f ∈ H k : ˜ δ ( ~f ) > x ∈ T Q ( f, x ) > o P − ,k := n f ∈ H k : ˜ δ ( ~f ) > x ∈ T Q ( f, x ) < o · We shall also see that L.W.P holds on a small interval [ − T, T ] if we assume that(1.1) is of non-parabolic resonance type, and that the data has enough regularity.We define P k := n f ∈ H k : ˜ δ ( ~f ) > o We now state the first theorem of this paper:
Theorem 1.1.
Assume that (1.3) is of parabolic resonance type. Let k ≥ k > .Then the following properties hold:(1) ( Local existence and P arabolic smoothing ) Let φ ∈ P + ,k (resp. φ ∈ P − ,k ). Then one can find T := T (cid:16) k φ k H k , ˜ δ ( ~φ ) , ˜ δ ′ ( φ ) (cid:17) > for which thereexists a solution u ∈ C (cid:0) [0 , T ] , H k (cid:1) (resp. u ∈ C (cid:0) [ − T, , H k (cid:1) ) of (1.3)with u (0) := φ such that u ( t ) ∈ P + ,k for t ∈ [0 , T ] (resp. u ( t ) ∈ P − ,k for all t ∈ [ − T, ). Moreover a parabolic smoothing effect holds, thatis u ∈ C ∞ ((0 , T ] × T ) (resp. u ∈ C ∞ ([ − T, × T ) ) if φ ∈ P + ,k (resp. φ ∈ P − ,k ).(2) ( U niqueness ) Assume that for some ˘ T > there exist u , u ∈ C (cid:16) [0 , ˘ T ] , H k (cid:17) (resp. u , u ∈ C (cid:16) [ − ˘ T , , H k (cid:17) ) solutions of (1.3) with u (0) = u (0) suchthat u q ( t ) ∈ P + ,k (resp. u q ( t ) ∈ P − ,k ) holds for q ∈ { , } and for t ∈ [0 , ˘ T ] (resp. t ∈ [ − ˘ T , ). Then u ( t ) = u ( t ) on h , ˘ T i (resp. h − ˘ T , i ). TRISTAN ROY (3) ( Continuous dependence on initial data ) Let φ ∞ ∈ P + ,k (resp. φ ∞ ∈ P − ,k ). Let φ n ∈ H k be such that φ n → φ ∞ in H k as n → ∞ . Then there exists N ∈ N such that for ∞ ≥ n ≥ N one can find ˘ T := ˘ T (cid:16) k φ ∞ k H k , ˜ δ ( −→ φ ∞ ) , ˜ δ ′ ( φ ∞ ) (cid:17) > for which there are unique solutions u n ∈ C (cid:16) [0 , ˘ T ] , H k (cid:17) (resp. u n ∈C (cid:16) [ − ˘ T , , H k (cid:17) ) of (1.3) with data u n (0) := φ n that satisfy u n ( t ) ∈ P + ,k for t ∈ [0 , ˘ T ] (resp. u n ( t ) ∈ P − ,k for t ∈ h − ˘ T , i ). Moreover sup t ∈ [0 , ˘ T ] k u n ( t ) − u ∞ ( t ) k H k → (resp. sup t ∈ [ − ˘ T, k u n ( t ) − u ∞ ( t ) k H k → ) as n → ∞ .Remark . The proof of the local existence part of the theorem shows that T canbe chosen as a continuous function that decreases as ˜ δ ( ~φ ) decreases, that decreasesas ˜ δ ′ ( φ ) decreases, and that decreases as k φ k H k increases. Remark . Roughly speaking, the first part of the theorem says that if inf x ∈ T Q ( u (0) ,
0) ( x ) > x ∈ T Q ( u (0) ,
0) ( x ) < φ ∈ P + ,k or φ ∈ P − ,k . Observe that ∂ ω F ∈ C , the spaceof continuous functions: this follows from Lemma 3.3 and the well-known Sobolevembedding H m ֒ → C , with m > . Hence the function ∂ ω F ( ~φ,
0) has a constantsign. Hence Q ( φ ) = ˜ δ ( ~φ )¯ δ ′ ( φ ) with¯ δ ′ ( f ) := − (cid:28) P ( f, | ∂ ω F ( ~f, | (cid:29) · Hence if ¯ δ ′ ( φ ) > δ ′ ( φ ) <
0) then ¯ δ ′ ( φ ) (resp. − ¯ δ ′ ( φ ))) measures thestrength of the diffusion in the forward (resp. backward) direction, holding ˜ δ ( ~φ )constant: see e.g [3] for discussions related to this number, for particular functions F of (1.1). Remark . Assume that φ ∈ P + ,k or φ ∈ P − ,k . Recall from (1.2) that thefunction ∂ ω F ( ~φ ) has a constant sign. This implies that there are two and only twoscenarios: either inf x ∈ T ∂ ω F ( ~φ )(0 , x ) > x ∈ T ∂ ω F ( ~φ )(0 , x ) <
0. Observealso that φ ∈ P + ,k ⇔ (cid:16) inf x ∈ T ∂ ω F (cid:16) −→ φ , (cid:17) ( x ) > x ∈ T ∂ ω F (cid:16) −→ φ , (cid:17) ( x ) < (cid:17) and R T P ( φ, dx < φ ∈ P − ,k ⇔ (cid:16) inf x ∈ T ∂ ω F (cid:16) −→ φ , (cid:17) ( x ) > x ∈ T ∂ ω F (cid:16) −→ φ , (cid:17) ( x ) < (cid:17) and R T P ( φ, dx > · Hence, in view of these observations and the theorem above we can draw con-clusions regarding the local existence, parabolic smoothing, uniqueness, and con-tinuous dependence on initial data of the solutions of (1.1) from the positivityof inf x ∈ T ∂ ω F ( ~φ, x ), the negativity of sup x ∈ T ∂ ω F ( ~φ, x ) , and the sign of R T P ( φ, dx . Remark . Consider φ n and φ ∞ that are defined in the continuous dependenceon initial data part of the theorem. We see from Section 11 with ( f, g ) := ( φ n , φ ∞ )that ˜ δ ( −→ φ n ) → ˜ δ ( −→ φ ∞ ), that ˜ δ ′ ( φ n ) → ˜ δ ′ ( φ ∞ ) as n → ∞ and that φ n ∈ P + ,k (resp. ONLINEAR THIRD-ORDER EQUATIONS 5 φ n ∈ P − ,k ) if φ ∞ ∈ P + ,k (resp. φ ∞ ∈ P − ,k ) for n large. Hence we see fromthe local existence part of the theorem, the uniqueness part of the theorem, andRemark 1.1 that the first statement of the continuous dependence on initial datapart of the theorem holds. It remains to prove the second statement: see Section8. Remark . The proof of the local existence part of the theorem also shows thatif φ ∈ P + ,k (resp. φ ∈ P − ,k ) holds , then u ( t ) ∈ P + ,k (resp. u ( t ) ∈ P − ,k )holds for t ∈ [0 , T ] (resp. t ∈ [ − T,
0] ), but also that δ (cid:16) −−→ u ( t ) , t (cid:17) & ˜ δ ( ~φ ) and that δ ′ ( u ( t ) , t ) & ˜ δ ′ ( φ ) for t ∈ [0 , T ] (resp. t ∈ [ − T, Corollary 1.2.
Assume that (1.3) is of parabolic resonance type. Let k > .Assume that φ ∈ P + ,k (resp. φ ∈ P − ,k ) and that φ / ∈ c C ∞ ( T ) . Then for any T > ,there does not exist any solution u ∈ C (cid:0) [ − T, , H k (cid:1) (resp. u ∈ C (cid:0) [0 , T ] , H k (cid:1) ) of(1.3) with u (0) := φ .Remark . Corollary 1.2 can be viewed as an ill-posedness result for equations(1.3) that are of parabolic resonance type. It shows that these equations are bynature parabolic, and not dispersive.We now state the second theorem of this paper:
Theorem 1.3.
Assume that (1.3) is of non-parabolic resonance type. Let k ≥ k > . Then the following properties hold:(1) ( Local existence ) Let φ ∈ P k . Then one can find T := T (cid:16) k φ k H k , ˜ δ ( ~φ ) (cid:17) > for which there exists a solution u ∈ C (cid:0) [ − T, T ] , H k (cid:1) of (1.3) with u (0) := φ such that u ( t ) ∈ P k for t ∈ [ − T, T ] .(2) ( U niqueness ) Assume that for some ˘ T > there exist u , u ∈ C (cid:16) [ − ˘ T , ˘ T ] , H k (cid:17) solutions of (1.3) with u (0) = u (0) such that u q ( t ) ∈ P k holds for q ∈{ , } and for t ∈ [ − ˘ T , ˘ T ] . Then u ( t ) = u ( t ) on h − ˘ T , ˘ T i .(3) ( Continuous dependence on initial data ) Let φ ∞ ∈ P k . Let φ n ∈ H k besuch that φ n → φ ∞ in H k as n → ∞ . Then there exists N ∈ N such thatfor ∞ ≥ n ≥ N one can find ˘ T := ˘ T (cid:16) k φ ∞ k H k , ˜ δ ( −→ φ ∞ ) (cid:17) > for which thereare unique solutions u n ∈ C (cid:16) [ − ˘ T , ˘ T ] , H k (cid:17) (resp. u n ∈ C (cid:16) [ − ˘ T , ˘ T ] , H k (cid:17) )of (1.3) with data u n (0) := φ n that satisfy u n ( t ) ∈ P k for t ∈ [ − ˘ T , ˘ T ] .Moreover sup t ∈ [ − ˘ T , ˘ T ] k u n ( t ) − u ∞ ( t ) k H k → as n → ∞ .Remark . The proof of the local existence part of the theorem shows that T can be chosen as a continuous function that decreases as ˜ δ ( ~φ ) decreases, and thatdecreases as k φ k H k increases. Remark . Roughly speaking, the first part of the theorem says that the dispersivepart of the equation dominates. Moreover the strength of the dispersion increasesas the amplitude of ˜ δ ( ~φ ) increases. TRISTAN ROY
Remark . Consider φ n and φ ∞ that are defined in the continuous dependenceon initial data part of the theorem. We see from Section 11 with ( f, g ) := ( φ n , φ ∞ )that ˜ δ (cid:16) −→ φ n (cid:17) → ˜ δ (cid:16) −→ φ ∞ (cid:17) as n → ∞ . Hence we see from the local existence part ofthe theorem, the uniqueness part of the theorem, and Remark 1.7 that the firststatement of the continuous dependence on initial data part of the theorem holds.It remains to prove the second statement of the theorem. Remark . The proof of the local existence part of the theorem also shows thatif φ ∈ P k , then u ( t ) ∈ P k holds for t ∈ [ − T, T ], but also that δ (cid:16) −−→ u ( t ) , t (cid:17) & ˜ δ ( ~φ ) for t ∈ [ − T, T ].Assumption: Observe that if v ( t ) := u ( − t ) then it satisfies (1.3), replacing F with − F . Therefore we will assume WLOG throughout the proof of Theorem 1.1that φ ∈ P + ,k . We can also WLOG restrict our analysis to positive times in theproof of Theorem 1.3. Actually the reader can check the proof of Theorem 1.3 is astraightforward modification of that of Theorem 1.1: therefore it is omitted.We now explain how this paper is organized. In Section 2 we write down somenotation that we use in this paper: in particular we introduce the gauge function.In Section 3, we prove some preliminary lemmas. In Section 4, we study the localwell-posedness of a parabolic regularization of (1.1). In Section 5, we introducethe gauged energy functionals. In Section 6, we prove gauged energy estimates:these estimates will allow, among other things, to find a time of existence of thesolution of the regularized problem that is bounded from below by a positive time(namely T ′ ) that does not depend on the parameter of regularization and that thesolution satisfies similar properties as the data. In Section 7, we prove gauged en-ergy estimates for the difference of two solutions of the regularized problem withsmoothed out data that is also regularized by operators of the form J ǫ,s (this isthe Bona-Smith approximation [1]). In Section 8, we prove Theorem 1.1 by usinga limit process. Throughout this paper, one has to perform a delicate analysis tomake sure that the solution of the regularized problem and the solution of (1.3)belongs to the same sets as the data. Acknowledgments : The author would like to thank Kotaro Tsugawa for in-teresting discussions related to this problem.2.
Notation
In this section we introduce some notation that we use throughout this paper.In this paper, unless otherwise specified, we do not mention for sake of notationsimplicity the spaces to which some functions (or more generally distributions) be-long in some propositions or lemmas: this exercise is left to the reader.Let f be a function on T . If n ∈ N then ∂ nx f denotes the n th derivative of f . Let m ≥ n ∈ Z , and ˆ f ( n ) be the n th − Fourier coefficient of f . Let ˜ D m bethe operator such that [ ˜ D m f ( n ) := ( in ) m ˆ f ( n ). More generally if p ∈ N and q ∈ R such that q + p ≥ D q be the operator such that ˆ g ( n ) := ( in ) q + p ˆ f ( n ) Notation convention: ∂ x f := f . ONLINEAR THIRD-ORDER EQUATIONS 7 with g := ˜ D q ( ∂ px f ). It is well-known that if m ∈ N then ˜ D m = ∂ mx . We define k f k ˙ H m := k{ ( in ) m ˆ f ( n ) } n ∈ Z k l ( Z ) .In this paper we use a gauge transform at a time t , in the spirit of [7, 8]. Let k ′ ∈ R . If δ ( ~f ) > k ′ ( f, t )( x ) := (cid:12)(cid:12)(cid:12) ∂ ω F ( ~f , t )( x ) (cid:12)(cid:12)(cid:12) k ′− e R x P ( f,t ) ∂ω F ( ~f,t ) −h P ( f,t ) ∂ω F ( ~f,t ) i ! dx ′ · Observe that the term h P ( f,t ) ∂ω F ( ~f,t ) i in (2.1) makes the function Φ k ′ ( f, t ) periodic and,consequently, it is defined on T . Observe also that(2.2) (cid:0) − k ′ (cid:1) ∂ x ∂ ω F ( ~f , t ) − (cid:16) k ′ ( f, t ) ∂ x Φ − k ′ ( f, t ) ∂ ω F ( ~f , t ) + P ( f, t ) (cid:17) = Q ( f, t ) · This is a crucial observation that we will often use throughout this paper. We willoften work with the function Φ k ′ ( u ( t ) , t ) ∂ x u ( t ) and prove estimates that involvethis function.We write C := C ( α , ..., α n ) if C is a constant depending on the variables α ,..., α n .If C := C ( α , α ) is a constant depending on two variables α and α , then we maytake the liberty to write only C if we do not want to emphasize the dependanceon α and α , or to write only C := C ( α ) if we do not want to emphasize thedependance on α , in order to simplify the notation. This convention naturallyextends to a constant depending on several variables. We write a . b if thereexists a constant C > a ≤ Cb . We write a . α ,...,α n b if here exists C := C ( α , ..., α n ) such that a ≤ Cb . If a . α ,α b then we may take he libertyto write only a . b if we do not want to emphasize the dependance on α and α or to write only a . α b if we do not want to emphasize the dependance on α .This convention naturally extends to a . α ,...,αn b . We write a ≪ b if there existsa small constant c > a ≤ cb . Similarly we write a ≪ α ,...,α n b if thereexists c := c ( α , ..., α n ) such that a ≤ cb . We use the same conventions for c as for C .Let x + = x + ǫ and x − = x − ǫ with 0 < ǫ ≪
1. Let p ∈ N and q ∈ R . Wedefine (cid:18) qp (cid:19) := q ( q − ... ( q − ( p − p ! for p = 0 and (cid:18) q (cid:19) := 1.We recall some lemmas. The first lemma shows that we can control the H k ′ − norm of a product of two functions f and g Lemma 2.1.
Let k ′ > . Let f and g be two functions. Then k f g k H k ′ . k f k H k ′ k g k H
12 + + k f k H
12 + k g k H k ′ For a proof see e.g [9] and references therein. Given ǫ ∈ (0 ,
1] and s ≥
0, let J ǫ,s be the operator defined by [ J ǫ,s f ( n ) := e − ǫ h n i s ˆ f ( n ) for n ∈ Z . The second lemmashows that this operator have some smoothing properties: TRISTAN ROY
Lemma 2.2.
Let ≤ j ≤ s and l ≥ . The following holds: k J ǫ,s f − f k H s − j . ǫ js k f k H s , and k J ǫ,s f k H s + l . ǫ − ls k f k H s · For a proof see e.g [9]. It will be often used in the propositions where we imple-ment the Bona-Smith approximation argument ([1]) .3. Preliminary lemmas
In this section we prove some lemmas.In the first lemma, we estimate an expression that involves two functions f and g .Thw expression is written as a sum of two terms: the main term and the remainder.Observe that in the remainder the excess of m derivatives of g is transferred on f . Lemma 3.1.
Let f and g be two functions. Let k ′ ≥ . Let m be an integer suchthat m ≥ . The following holds: (3.1) (cid:13)(cid:13)(cid:13)(cid:13) ˜ D k ′ ( f ∂ mx g ) − m − P l =0 (cid:0) k ′ l (cid:1) ˜ D l f ˜ D k ′ − l ∂ mx g (cid:13)(cid:13)(cid:13)(cid:13) L . min (cid:16) k f k H k ′ + m k g k H
12 + , k f k H k ′ k g k H ( m + 12 ) + (cid:17) + k f k H ( m + 12 ) + k g k H k ′ · Proof.
We can write
L.H.S of (3 .
1) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P n ∈ Z X ( n, n ) ˆ f ( n − n )ˆ g ( n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l with X ( n, n ) := ( in ) k ′ ( in ) m − m − P l =0 (cid:0) k ′ l (cid:1) ( i ( n − n )) l ( in ) k ′ − l + m · We estimate X ( n, n ). If | n − n | & | n | then clearly | X ( n, n ) | . | n − n | k ′ + m and | X ( n, n ) | . | n | m | n − n | k ′ . If | n − n | ≪ | n | then we factor out n m and weapply the Taylor formula for n = 0 to get | X ( n, n ) | . | n | m | n − n | m | n | k ′ − m . | n − n | m | n | k ′ . If n = 0 then the bound found for X ( n, n ) clearly holds. TheYoung inequality and the Cauchy-Schwarz inequality yield L.H.S of (3 . . min ( k f k H k ′ + m k ˆ g ( n ) k l , k f k H k ′ k n m ˆ g ( n ) k l )+ k n m ˆ f ( n ) k l k g k H k ′ . min (cid:16) k f k H k ′ + m k g k H
12 + , k f k H k ′ k g k H ( m + 12 ) + (cid:17) + k f k H ( m + 12 ) + k g k H k ′ · (cid:3) In the next two lemmas we prove some estimates involving derivatives of powersof functions, quotient of functions, exponential of functions, differences of exponen-tials of functions. We also prove nonlinear estimates. These estimates will be usedto prove some estimates that involve the gauge function.
Lemma 3.2.
Let β ∈ R . Let ¯ δ > and ( k ′ , K ) ∈ ( R + ) . Then the following holds: see also [9] for an exposition of the ideas of this argument ONLINEAR THIRD-ORDER EQUATIONS 9 • Let f be a function. Assume that k f k H
32 + ≤ K and that inf x ∈ T | f ( x ) | ≥ ¯ δ .Then there exists C := C (¯ δ, K ) > such that (3.2) k f β k H k ′ ≤ C (1 + k f k H k ′ ) • Let f and g be two functions. Assume that k f k H
12 + ≤ K , k g k H
32 + ≤ K ,and inf x ∈ T | g ( x ) | ≥ ¯ δ . Then there exists C := C (¯ δ, K ) > such that (3.3) (cid:13)(cid:13)(cid:13) fg (cid:13)(cid:13)(cid:13) H k ′ ≤ C (1 + k f k H k ′ + k g k H k ′ ) There exists C := C (¯ δ, K ) > such that (3.4) k g k H max ( k ′ ,
32 + ) ≤ K = ⇒ (cid:13)(cid:13)(cid:13) fg (cid:13)(cid:13)(cid:13) H k ′ ≤ C k f k H max ( k ′ ,
12 + ) • Let f be a function. Assume that k f k H
12 + ≤ K . Then there exists C := C ( K ) > such that (3.5) k ′ ∈ [0 ,
1] : (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′ ≤ C, and k ′ ≥ (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′ ≤ C (1 + k f k H k ′− ) ·• Let f and g be two functions. Assume that k f k H
12 + ≤ K and that k g k H
12 + ≤ K . Assume also that k f k H k ′− ≤ K and that k g k H k ′− ≤ K if k ′ ≥ . Thenthere exists C := C ( K ) such that (3.6) k ′ ∈ [0 ,
1] : (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ − e R x g ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′ ≤ C k f − g k L , and k ′ ≥ (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ − e R x g ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′ ≤ C k f − g k H k ′− · Remark . The proof shows that the constants C depending only on K can bechosen as continuous functions of K that increase as K increases. It also showsthat the constants C depending on ¯ δ and K can be chosen as continuous functionsof (¯ δ, K ) that increase as K increases and increase as ¯ δ decreases. Proof.
We prove (3.2) by an induction process.Let 0 ≤ k ′ ≤
1. The H¨older inequality, the Sobolev embedding H + ֒ → L ∞ appliedto the cases β < k f β − k L ∞ . β ≥ k f β k L . hk f k H
12 + i β − k f k L and k ∂ x f β k L . hk f ki β − H
12 + k ∂ x f k L . k ∂ x f k L · Hence (3.2) holds for this range of k ′ . Assume now that (3.2) holds for all 0 ≤ k ′ ≤ r with r ∈ N ∗ . We see from Lemma 2.1 thats (cid:13)(cid:13) ∂ x f β (cid:13)(cid:13) H k ′− . hk f β − k H
12 + ik ∂ x f k H k ′− + hk f β − k H k ′− ik ∂ x f k H
12 + ≤ C (1 + k f k H k ′ ) · Hence (3.2) holds for r ≤ k ′ ≤ r + 1.Next we prove (3.3) and (3.4). We see from (3.2) that (cid:13)(cid:13)(cid:13) fg (cid:13)(cid:13)(cid:13) H k ′ . k f k H k ′ k g − k H
12 + + k g − k H k ′ k f k H
12 + . (cid:16) k g k H
12 + (cid:17) k f k H k ′ + (1 + k g k H k ′ ) k f k H
12 + · Hence (3.3) and (3.4) hold.Next we also prove (3.5). Let 0 ≤ k ′ ≤
1. There exists C ′ > (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) L ∞ . e C ′ k f k L . · We also have(3.8) (cid:13)(cid:13)(cid:13) ∂ x e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) L . k f k L (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) L ∞ . · Hence (3.5) holds.Assume that (3.5) holds for all 0 ≤ k ′ ≤ r with r ∈ N ∗ . If r ≤ k ′ ≤ r + 1 then(3.9) (cid:13)(cid:13)(cid:13) ∂ x e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′− . (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H
12 + k f k H k ′− + (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′− k f k H
12 + . k f k H k ′− · Hence (3.5) holds for r ≤ k ′ ≤ r + 1.Next we prove (3.6). Let h ( x ′ ) := f ( x ′ ) − g ( x ′ ), X := e R x h ( x ′ ) dx ′ −
1, and Y := e R x g ( x ′ ) dx ′ .First we estimate k X k H m . We claim that(3.10) m ∈ [0 ,
1] : k X k H m . k h k L m ≥ k X k H m . k h k H m − · Indeed, define ¯ m as follows: let ¯ m := 0 if m ∈ [0 ,
1] and let ¯ m := m − m ≥ k h k H ¯ m & k k H m .
1. If k h k H ¯ m ≪ m ∈ [0 , (cid:12)(cid:12)R x h ( x ′ ) dx ′ (cid:12)(cid:12) . k h k L ≪
1; since e y − ≈ y for | y | ≪ k X k L . k h k L ;proceeding as in (3.8) we get k ∂ x X k L . k h k L and consequently (3.10) holds for¯ m ∈ [0 , ≤ m ≤ r , we proceed in a similarway as (3.9) in order to estimate k X k H m for r ≤ m ≤ r + 1 and we find that (3.10)holds.Hence using also (3.5) we get (cid:13)(cid:13)(cid:13) e R x f ( x ′ ) dx ′ − e R x g ( x ′ ) dx ′ (cid:13)(cid:13)(cid:13) H k ′ . k X k H k ′ k Y k H
12 + + k X k H
12 + k Y k H k ′ . R.H.S of (3 . · (cid:3) ONLINEAR THIRD-ORDER EQUATIONS 11
Lemma 3.3.
Let G : R × T × R −→ R be a smooth function. Let I ⊂ R be aninterval such that | I | ≤ . Let ( k ′ , K ) ∈ ( R + ) . The following holds: • Let f be a function. Assume that k f k H
92 + ≤ K . Then there exists C := C ( K ) > such that for all t ∈ I (3.11) (cid:13)(cid:13)(cid:13) G ( ~f , t ) (cid:13)(cid:13)(cid:13) H k ′ ≤ C (1 + k f k H k ′ +3 ) Let β ∈ R . Assume that k f k H
92 + ≤ K , and that there exists ¯ δ > such thatfor all t ∈ I inf x ∈ T | G ( f, t )( x ) | ≥ ¯ δ . Then there exists C := C (¯ δ, K ) > such that for t ∈ I (3.12) (cid:13)(cid:13)(cid:13) G β ( ~f , t ) (cid:13)(cid:13)(cid:13) H k ′ ≤ C (1 + k f k H k ′ +3 ) • Let f and g be two functions. Assume that k f k H
92 + ≤ K , k g k H
92 + ≤ K , k f k H k ′ +3 ≤ K , and k g k H k ′ +3 ≤ K . Then there exists C := C ( K ) > suchthat for all t , t ∈ I (3.13) (cid:13)(cid:13)(cid:13) G ( ~f , t ) − G ( ~g, t ) (cid:13)(cid:13)(cid:13) H k ′ ≤ C (cid:16) k f − g k H max ( k ′ +3 ,
72 + ) + | t − t | (cid:17) · Let β ∈ R . Assume that k f k H
92 + ≤ K , k g k H
92 + ≤ K , k f k H k ′ +3 ≤ K , k g k H k ′ +3 ≤ K , and that there exists ¯ δ > such that for all t ∈ I and forall θ ∈ [0 , , inf x ∈ T (cid:12)(cid:12)(cid:12) G (cid:16) −→ h θ , t (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≥ ¯ δ , with h θ := θf + (1 − θ ) g . Thenthere exists C := C (¯ δ, K ) > such that for all t , t ∈ I (3.14) (cid:13)(cid:13)(cid:13) G β ( ~f , t ) − G β ( ~g, t ) (cid:13)(cid:13)(cid:13) H k ′ ≤ C (cid:16) k f − g k H max ( k ′ +3 ,
72 + ) + | t − t | (cid:17) · Remark . Regarding the constants in Lemma 3.3, we refer to Remark 3.1.
Proof.
We prove Lemma 3.3 by an induction process.First we prove (3.11).Let 0 ≤ k ′ ≤
1. The H¨older inequality, the Sobolev embedding H + ֒ → L ∞ , andthe composition rule imply that(3.15) (cid:13)(cid:13)(cid:13) G ( ~f , t ) (cid:13)(cid:13)(cid:13) L . sup | ( ~ω,t ) | . h K i | G ( ~ω, t ) | . , and (cid:13)(cid:13)(cid:13) ∂ x (cid:16) G ( ~f , t ) (cid:17)(cid:13)(cid:13)(cid:13) L . sup γ ∈ N : | γ |≤ | ( ~ω,t ) | . h K i | ∂ γ G ( ~ω, t ) | hk f k H i . · Hence (3.11) holds.Let r ′ ∈ N ∗ . Assume that (3.11) holds for all 0 ≤ k ′ ≤ r and for all smoothfunctions G . The induction assumption yields and Lemma 2.1 yield (3.16) (cid:13)(cid:13)(cid:13) ∂ x (cid:16) G ( ~f , t ) (cid:17)(cid:13)(cid:13)(cid:13) H k ′− . sup γ ∈ N : | γ |≤ | ( ~ω,t ) | . h K i k ∂ γ G ( ~ω, t ) k H
12 + k f k H k ′ +3 + sup γ ∈ N : | γ |≤ | ( ~ω,t ) | . h K i k ∂ γ G ( ~ω, t ) k H k ′− hk f k H
92 + i≤ C (1 + k f k H k ′ +3 )Hence (3.11) holds for r ≤ k ′ ≤ r + 1.The proof of (3.12) is very similar to that (3.11). Since inf x ∈ T (cid:12)(cid:12)(cid:12) G ( ~f )( x ) (cid:12)(cid:12)(cid:12) ≥ ¯ δ this implies that sup | ~ω | . h K i (cid:12)(cid:12) G β ( ~ω ) (cid:12)(cid:12) .
1. Hence (3.12) holds for 0 ≤ k ′ ≤ r ≤ k ′ ≤ r +1 by slightly modifying (3.16). G β and G β − are smooth, this implies that G β is smooth so we can apply (3.11)by replacing G with G β . We get (3.12).Next we prove (3.13). Let t θ := θt + (1 − θ ) t . Write G ( ~f , t ) − G ( ~g, t ) = R P i =0 ∂ ω i G (cid:16) −→ h θ , t θ (cid:17) ( ∂ x i f − ∂ x i g )+ ∂ t G (cid:16) −→ h θ , t θ (cid:17) ( t − t ) dθ We get from (3.11) and Lemma 2.1 (cid:13)(cid:13)(cid:13) G ( ~f , t ) − G ( ~g, t ) (cid:13)(cid:13)(cid:13) H k ′ . sup θ ∈ [0 , (cid:13)(cid:13)(cid:13) ∂ t G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ | t − t | + sup i ∈{ ,.., } θ ∈ [0 , (cid:13)(cid:13)(cid:13) ∂ ω i G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ k f − g k H
72 + + sup i ∈{ ,.., } θ ∈ [0 , (cid:13)(cid:13)(cid:13) ∂ ω i G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + k f − g k H k ′ +3 . (1 + k f k H k ′ +3 + k g k H k ′ +3 ) (cid:16) k f − g k H
72 + + | t − t | (cid:17) +(1 + k f k H
72 + + k g k H
72 + ) k f − g k H k ′ +3 ≤ C (cid:16) k f − g k H max ( k ′ +3 ,
72 + ) + | t − t | (cid:17) Finally we prove (3.14). We have
ONLINEAR THIRD-ORDER EQUATIONS 13 (cid:13)(cid:13)(cid:13) G β ( ~f , t ) − G β ( ~g, t ) (cid:13)(cid:13)(cid:13) H k ′ . sup θ ∈ [0 , (cid:16) (cid:13)(cid:13)(cid:13) G β − (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + (cid:13)(cid:13)(cid:13) ∂ t G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ + (cid:13)(cid:13)(cid:13) G β − (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ (cid:13)(cid:13)(cid:13) ∂ t G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + (cid:17) | t − t | + sup i ∈{− ,.., } θ ∈ [0 , (cid:13)(cid:13)(cid:13) G β − (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + (cid:13)(cid:13)(cid:13) ∂ ω i G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + k f − g k H k ′ +3 + sup i ∈{− ,.., } θ ∈ [0 , (cid:13)(cid:13)(cid:13) G β − (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ (cid:13)(cid:13)(cid:13) ∂ ω i G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + k f − g k H
72 + + sup i ∈{− ,.., } θ ∈ [0 , (cid:13)(cid:13)(cid:13) G β − (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H
12 + (cid:13)(cid:13)(cid:13) ∂ ω i G (cid:16) −→ h θ , t θ (cid:17)(cid:13)(cid:13)(cid:13) H k ′ k f − g k H
72 + . (cid:16) k f k H max ( k ′ +3 ,
72 + ) + k g k H max ( k ′ +3 ,
72 + ) (cid:17) (cid:16) k f − g k H max ( k ′ +3 ,
72 + ) + | t − t | (cid:17) . k f − g k H max ( k ′ +3 ,
72 + ) + | t − t |· (cid:3) The next lemma below allows to prove estimates involving the derivatives ofΦ k ′ ( f, t ). More precisely: Lemma 3.4.
Let I ⊂ R be an interval such that | I | ≤ . Let ( k ′ , K ) ∈ R + × R + .The following holds:(1) Let f be a function. Assume that k f k H
92 + ≤ K and that for all t ∈ I inf x ∈ T | ∂ ω F ( −→ f , t )( x ) | ≥ ¯ δ . Then there exists C := C (¯ δ, K ) > such thatfor all t ∈ I (3.17) k Φ k ′ ( f, t ) k H k ′ ≤ C (1 + k f k H k ′ +3 ) , and k Φ − k ′ ( f, t ) k H k ′ ≤ C (1 + k f k H k ′ +3 ) · (2) Define k ′ to be the following number: k ′ ≥ k ′ := k ′ + 31 ≤ k ′ < k ′ := max (cid:0) + , k ′ + 3 (cid:1) ≤ k ′ < k ′ := + Let f and g be two functions. Assume that k f k H k ′ ≤ K and k g k H k ′ ≤ K .Assume also that there exists ¯ δ > such that for all t ∈ I and for all θ ∈ [0 ,
1] inf x ∈ T (cid:12)(cid:12)(cid:12) ∂ ω F (cid:16) −→ h θ , t (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≥ ¯ δ with h θ := θf + (1 − θ ) g . Then thereexists C := C (¯ δ, K ) such that for all t , t ∈ I the following holds: (3.18) k Φ k ′ ( f, t ) − Φ k ′ ( g, t ) k H k ′ ≤ C (cid:0) k f − g k H k ′ + | t − t | (cid:1) , and (cid:13)(cid:13) Φ − k ′ ( f, t ) − Φ − k ′ ( g, t ) (cid:13)(cid:13) H k ′ ≤ C (cid:0) k f − g k H k ′ + | t − t | (cid:1) · Remark . Regarding the constants in Lemma 3.4, we refer to Remark 3.1.
Proof.
We only prove the first estimate of (3.17): the proof of the second estimateis similar and therefore omitted. Let m ≥
0. We have (cid:13)(cid:13)(cid:13) h P ( f,t ) ∂ ω F ( ~f,t ) i (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13)(cid:13) h P ( f,t ) ∂ ω F ( ~f,t ) i (cid:13)(cid:13)(cid:13) L ∞ . (cid:13)(cid:13)(cid:13) P ( f,t ) ∂ ω F ( ~f,t ) (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13)(cid:13) P ( f,t ) ∂ ω F ( ~f,t ) (cid:13)(cid:13)(cid:13) L Hence in view of the Sobolev embedding H + ֒ → L ∞ we have (cid:13)(cid:13)(cid:13) h P ( f,t ) ∂ ω F ( ~f,t ) i (cid:13)(cid:13)(cid:13) L . sup γ ∈ N : | γ |≤ | ( ~ω,t ) | . h K i | ∂ γ F ( ~ω, t ) | k f k H . ∂ nx h P ( f,t ) ∂ ω F ( ~f,t ) i = 0 for n ∈ N ∗ , we get(3.19) (cid:13)(cid:13)(cid:13) h P ( f,t ) ∂ ω F ( ~f,t ) i (cid:13)(cid:13)(cid:13) H m . . Lemma 2.1 and Lemma 3.3 yield(3.20) k P ( f, t ) k H m . sup γ ∈ N : | γ |≤ (cid:13)(cid:13)(cid:13) ∂ γ F ( ~f , t ) (cid:13)(cid:13)(cid:13) H m hk f k H
92 + i + sup γ ∈ N : | γ |≤ (cid:13)(cid:13)(cid:13) ∂ γ F ( ~f , t ) (cid:13)(cid:13)(cid:13) H
12 + k f k H m +4 . k f k H m +4 · We get from Lemma 3.2(3.21) (cid:13)(cid:13)(cid:13) P ( f,t ) ∂ ω F ( ~f,t ) (cid:13)(cid:13)(cid:13) H m . k P ( f, t ) k H m + k ∂ ω F ( ~f , t ) k H m . k f k H m +4 · Let X := e R x (cid:18) P ( f,t ) ∂ω F ( ~f,t ) −h P ( f,t ) ∂ω F ( ~f,t ) i (cid:19) dx ′ . We have k Φ k ′ ( f, t ) k H k ′ . (cid:13)(cid:13)(cid:13) ( ∂ ω F ) k ′− ( ~f , t ) (cid:13)(cid:13)(cid:13) H k ′ k X k H
12 + + k X k H k ′ (cid:13)(cid:13)(cid:13) ( ∂ ω F ) k ′− ( ~f , t ) (cid:13)(cid:13)(cid:13) H
12 + . (1 + k f k H k ′ +3 ) k X k H
12 + + k X k H k ′ . k f k H k ′ +3 · Hence (3.17) holds.Next we prove the first estimate of (3.18): indeed the proof of the second esti-mate is similar and therefore left to the reader. We have k Φ k ′ ( f, t ) − Φ k ′ ( g, t ) k H k ′ . k△ Y ( t , t ) k H k ′ k Z ( ~f , t ) k H
12 + + k△ Y ( t , t ) k H
12 + k Z ( ~f , t ) k H k ′ + k Y ( ~g, t ) k H k ′ k△ Z ( t , t ) k H
12 + + k Y ( ~g, t ) k H
12 + k△ Z ( t ) k H k ′ , with Y ( ~f , t ) := ( ∂ ω F ) k − ( ~f , t ) , △ Y ( t , t ) := Y ( ~f , t ) − Y ( ~g, t ), Z ( ~f , t ) defined by Z ( ~f , t )( x ) := e R x (cid:18) P ( f,t ) ∂ω F ( ~f,t ) −h P ( f,t ) ∂ω F ( ~f,t ) i (cid:19) dx ′ , and △ Z ( t , t ) := Z ( ~f , t ) − Z ( ~g, t ). Let m ∈ (cid:8) , k ′ (cid:9) . Lemma 3.3 shows that k△ Y ( t , t ) k H k ′ . | t − t | + k f − g k H max ( k ′ +3 ,
72 + ) and k△ Y ( t , t ) k H
12 + . | t − t | + k f − g k H
72 + . Wealso have k Y ( ~g, t ) k H k ′ . k g k H k ′ +3 and k Y ( ~g, t ) k H
12 + . k g k H
72 + . We have (cid:13)(cid:13)(cid:13) Z ( ~f , t ) (cid:13)(cid:13)(cid:13) H
12 + . (cid:13)(cid:13)(cid:13) Z ( ~f , t ) (cid:13)(cid:13)(cid:13) H k ′ . (cid:13)(cid:13)(cid:13) Z ( ~f , t ) (cid:13)(cid:13)(cid:13) H k ′ . k f k H k ′ +3 ) if k ′ ≤ k ′ ≥ k△ Z ( t , t ) k H m for m ∈ (cid:8) , k ′ (cid:9) . ONLINEAR THIRD-ORDER EQUATIONS 15
To this end we prove the following claim:Claim: Let p ≥
0. Assume that k f k H max (
92 + ,p +4 ) ≤ K and k g k H max (
92 + ,p +3 ) ≤ K .Then(3.22) k P ( f, t ) − P ( g, t ) k H p . k f − g k H max ( p +4 ,
92 + ) + | t − t | Proof.
We have k P ( f, t ) − P ( g, t ) k H p ≤ A + B with A := sup γ ∈ N : | γ |≤ (cid:13)(cid:13)(cid:13) ∂ γ F ( ~f , t ) − ∂ γ F ( ~g, t ) (cid:13)(cid:13)(cid:13) H p sup ≤ j ≤ k ∂ jx f k H
12 + + sup γ ∈ N : | γ |≤ (cid:13)(cid:13)(cid:13) ∂ γ F ( ~f , t ) − ∂ γ F ( ~g, t ) (cid:13)(cid:13)(cid:13) H p sup ≤ j ≤ k ∂ jx f k H p ,B := sup ≤ j ≤ k ∂ jx ( f − g ) k H p sup γ ∈ N : | γ |≤ k ∂ γ F ( ~g, t ) k H
12 + + sup ≤ j ≤ k ∂ jx ( f − g ) k H
12 + sup γ ∈ N : | γ |≤ k ∂ γ F ( ~g, t ) k H p Lemma 3.3 implies that A . k f − g k H max ( p +3 ,
72 + ) k f k H max (
92 + ,p ) + | t − t | . k f − g k H max ( p +3 ,
72 + ) + | t − t |· We also have B . k f − g k H p +4 (cid:16) k g k H
72 + (cid:17) + k f − g k H
92 + (1 + k g k H p +3 ) . k f − g k H max ( p +4 ,
92 + ) · Hence (3.22) holds. (cid:3)
Let m ≥
1. Lemma 3.2 shows that k△ Z ( t , t ) k H m . (cid:13)(cid:13)(cid:13) P ( f,t ) ∂ ω F ( ~f,t ) − P ( g,t ) ∂ ω F ( ~g,t ) (cid:13)(cid:13)(cid:13) H m − + (cid:13)(cid:13)(cid:13) h P ( f,t ) ∂ ω F ( ~f,t ) − P ( g,t ) ∂ ω F ( ~g,t ) i (cid:13)(cid:13)(cid:13) H m − . k△ Z ( t , t ) k H m − + k△ Z ( t , t ) k H m − , assuming that k△ Z ( t , t ) k H max ( m − ,
12 + ) ≤ K and k△ Z ( t , t ) k H max ( m − ,
12 + ) ≤ K : this will be proved shortly. Observe from (3.21) and (3.19) that k△ Z ( t , t ) k H
12 + . K and that k△ Z ( t , t ) k H
12 + . K . We first estimate k△ Z ( t , t ) k H m − . Let A := P ( f,t ) − P ( g,t ) ∂ ω F ( ~f,t ) and B := P ( g,t ) ( ∂ ω F ( ~g,t ) − ∂ ω F ( ~f,t ) ) ∂ ω F ( ~f,t ) ∂ ω F ( ~g,t ) . We have (see proof of(3.19))(3.23) k△ Z ( t , t ) k H m − . (cid:13)(cid:13)(cid:13) P ( f,t ) ∂ ω F ( ~f,t ) − P ( g,t ) ∂ ω F ( ~g,t ) (cid:13)(cid:13)(cid:13) L . k A k L + k B k L · The above claim implies(3.24) k A k L . k P ( f, t ) − P ( g, t ) k L . k f − g k H
92 + + | t − t |· We see from (3.20) and Lemma 3.3 (3.25) k B k L . k P ( g, t ) k L ∞ k ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) k L . k P ( g, t ) k H
12 + (cid:16) k f − g k H
72 + + | t − t | (cid:17) . k f − g k H
72 + + | t − t |· Hence k△ Z ( t , t ) k H m − . k f − g k H k ′ . We now estimate k△ Z ( t , t ) k H m − . Wehave k△ Z ( t , t ) k H m − . k A k H m − + k B k H m − . Assume that m ≥
3. We useLemma 2.1, Lemma 3.2, Lemma 3.3, (3.20), and (3.22). We have k A k H m − . k P ( f, t ) − P ( g, t ) k H m − . k f − g k H k ′ + | t − t | . We have k B k H m − . (cid:13)(cid:13)(cid:13) P ( g, t ) (cid:16) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:17)(cid:13)(cid:13)(cid:13) H m − . k P ( g, t ) k H m − (cid:13)(cid:13)(cid:13) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:13)(cid:13)(cid:13) H
12 + + k P ( g, t ) k H
12 + (cid:13)(cid:13)(cid:13) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:13)(cid:13)(cid:13) H m − . k f − g k H k ′ · Hence k△ Z ( t , t ) k H m − . k f − g k H k ′ . Assume now that 1 ≤ m <
3. We have(3.26) k A k H m − . k P ( f, t ) − P ( g, t ) k H max ( m − ,
12 + ) . k f − g k H k ′ + | t − t |· We also have(3.27) k B k H m − . (cid:13)(cid:13)(cid:13) P ( g, t ) (cid:16) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:17)(cid:13)(cid:13)(cid:13) H max ( m − ,
12 + ) . k P ( g, t ) k H max ( m − ,
12 + ) (cid:13)(cid:13)(cid:13) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:13)(cid:13)(cid:13) H
12 + + k P ( g, t ) k H
12 + (cid:13)(cid:13)(cid:13) ∂ ω F ( ~g, t ) − ∂ ω F ( ~f , t ) (cid:13)(cid:13)(cid:13) H max ( m − ,
12 + ) . k f − g k H k ′ + | t − t |· Hence k△ Z ( t , t ) k H m . k f − g k H k ′ + | t − t | .Assume now that 0 ≤ m ≤
1. Lemma 3.2 shows that k△ Z ( t , t ) k H m . k△ Z ( t , t ) k L + k△ Z ( t , t ) k L , assuming that k△ Z ( t , t ) k H
12 + . K and k△ Z ( t , t ) k H
12 + . K : this will beproved shortly. From (3.23), (3.24), and (3.25), we see that k△ Z ( t , t ) k H
12 + . k f − g k H k ′ . A similar scheme used in (3.26) and (3.27) shows that k△ Z ( t , t ) k H
12 + . k f − g k H k ′ . Hence the assumptions above are clearly satisfied and moreover k△ Z ( t , t ) k H m . k f − g k H k ′ + | t − t |· (cid:3) Regularized problem: local well-posedness
We consider the following regularized problem (with ǫ ∈ (0 , ∂ t u + ǫ∂ x u = F ( ~u, t ) · Let A ∈ R . Let e − A∂ x f be the operator such that \ e − A∂ x f ( n ) := e − An ˆ f ( n ) for n ∈ Z . The following proposition holds: ONLINEAR THIRD-ORDER EQUATIONS 17
Proposition 4.1.
Let k ′ ≥ k ′ > . Let ˜ φ ∈ H k ′ . Then there exists T ǫ := T ǫ ( ˜ φ ) ∈ (0 , ∞ ] and a unique solution u ∈ C (cid:16) [0 , T ǫ ) , H k ′ (cid:17) such that u satisfies the integralequation for (4.1) with initial data u (0) := ˜ φ on [0 , T ǫ ) , i.e (4.2) t ∈ [0 , T ǫ ) : u ( t ) = T ( u ( t )) := e − ǫt∂ x ˜ φ + R t e − ǫ ( t − t ′ ) ∂ x F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17) dt ′ , and such that either ( i ) : lim inf t → T ǫ k u ( t ) k H k ′ = ∞ or ( ii ) : T ǫ = ∞ holds.Moreover let ˜ φ n ∈ H k ′ and ˜ φ ∞ ∈ H k ′ be such that k ˜ φ n − ˜ φ ∞ k H k ′ → as n →∞ . Let u n ∈ C (cid:16)h , T ǫ ( ˜ φ n ) (cid:17) , H k ′ (cid:17) ( resp. u ∞ ∈ C (cid:16)h , T ǫ ( ˜ φ ∞ ) (cid:17) , H k ′ (cid:17) be thesolution of (4.2), replacing ˜ φ with ˜ φ n (resp. ˜ φ ∞ ). Let T ∈ h , T ǫ ( ˜ φ ∞ ) (cid:17) . Then sup t ∈ [0 ,T ] k u n ( t ) − u ∞ ( t ) k L ∞ t H k ′ ([0 ,T ]) → as n → ∞ .Remark . The conclusion above implies that T ǫ (cid:16) ˜ φ n (cid:17) > T for n large. Proof.
The proof uses standard arguments. In the sequel let C ( x ) be a nonnegativenumber depending on x of which the value is allowed to change from one line tothe other one and such that all the statements below are true.Let T ′ ǫ := T ′ ǫ (cid:16) k ˜ φ k H k ′ (cid:17) be a positive time that is small enough such that all thestatements below are true. Let X be the following Banach space X := n v ∈ C (cid:16) [0 , T ′ ǫ ) , H k ′ (cid:17) : k v k L ∞ t H k ′ ([0 ,T ′ ǫ )) ≤ k ˜ φ k H k ′ ) o · Clearly X , endowed with the norm k v k X := k v k L ∞ t H k ′ ([0 ,T ′ ǫ )) , is a Banach space.Let t ∈ [0 , T ′ ǫ ). Let A v ( t ) := F (cid:16) −−→ v ( t ′ ) , t (cid:17) .We first claim that T ( X ) ⊂ X . The Minkowski inequality shows that kT ( v ( t )) k H k ′ ≤ k ˜ φ k H k ′ + R t (cid:13)(cid:13)(cid:13) h n i e − ǫ ( t − t ′ ) n h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l dt ′ From (3.11) we see that(4.3) (cid:13)(cid:13)(cid:13) h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l ≤ C (cid:16) k ˜ φ k H k ′ (cid:17) (1 + k v ( t ′ ) k H k ′ )Elementary considerations show that(4.4) sup n ∈ Z h n i e − ǫ ( t − t ′ ) n . ǫ − ( t − t ′ ) − · Hence we get after integration(4.5) kT ( v ) k X ≤ k ˜ φ k H k ′ + C ( k ˜ φ k H k ′ ) ǫ − (cid:16) T ′ ǫ (cid:17) (1 + k ˜ φ k H k ′ ) ≤ k ˜ φ k H k ′ + C ( k ˜ φ k H k ′ ) ǫ − (cid:16) T ′ ǫ (cid:17) (1 + k ˜ φ k H k ′ ) ≤ k ˜ φ k H k ′ ) · One gets from (3.13) (4.6) (cid:13)(cid:13)(cid:13) h n i k ′ − (cid:16) \ A v ( t ′ )( n ) − \ A v ( t ′ )( n ) (cid:17)(cid:13)(cid:13)(cid:13) l ≤ C (cid:16) k ˜ φ k H k ′ (cid:17) k v ( t ′ ) − v ( t ′ ) k H k ′ Hence integrating in time we see that kT ( v ) − T ( v ) k X . C (cid:16) k ˜ φ k H k ′ (cid:17) ǫ − (cid:16) T ′ ǫ (cid:17) k v − v k L ∞ t H k ′ ([0 ,T ′ ǫ ]) ≤ k v − v k X Hence T is a contraction.We then claim that T v ∈ C (cid:16) [0 , T ′ ǫ ) , H k ′ (cid:17) . Indeed the dominated convergencetheorem allows to prove that t → e − ǫt∂ x ˜ φ in continuous in H k ′ . Let t ∈ [0 , T ′ ǫ )and let h > G ( t ) := R t e − ǫ ( t − t ′ ) ∂ x A v ( t ′ ) dt ′ . Writing G ( t + h ) − G ( t ) = H + H with H := R t + ht e − ǫ ( t + h − t ′ ) ∂ x A v ( t ′ ) dt ′ , and H := R t (cid:16) e − ǫ ( t + h − t ′ ) ∂ x − e − ǫ ( t − t ′ ) ∂ x (cid:17) A v ( t ′ ) dt ′ .(4.4), (4.5), and the dominated convergence theorem allow to prove that H → h →
0. In order to prove that H → (cid:12)(cid:12)(cid:12) e − ǫ ( t + h − t ′ ) n − e − ǫ ( t − t ′ ) n (cid:12)(cid:12)(cid:12) . e − ǫ ( t − t ′ ) n n hǫ . We also have (cid:12)(cid:12)(cid:12) e − ǫ ( t + h − t ′ ) n − e − ǫ ( t − t ′ ) n (cid:12)(cid:12)(cid:12) . e − ǫ ( t − t ′ ) n . Hence k H k H k ′ . X + Y with X := R t (cid:13)(cid:13)(cid:13) h n i e − ǫ ( t − t ′ ) n h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l ( n h ≥ dt ′ and Y := R t (cid:13)(cid:13)(cid:13) h n i n hǫe − ǫ ( t − t ′ ) n h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l ( n h ≤ dt ′ . There exists c > X . R t ( t − t ′ ) − sup n ′ ≥ ǫ ( t − t ′ ) h (cid:16) ( n ′ ) e − n ′ (cid:17) (cid:13)(cid:13)(cid:13) h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l dt ′ . (1 + k ˜ φ k H k ′ ) R t ( t − t ′ ) − e − cǫ ( t − t ′ ) h dt ′ . Hence, by making the change of variable u = ( t − t ′ ) cǫh , A → h →
0. We have Y . h R t (cid:13)(cid:13)(cid:13) sup n h ≤ (cid:16) n e − ǫ ( t − t ′ ) n (cid:17) h n i k ′ − \ A v ( t ′ )( n ) (cid:13)(cid:13)(cid:13) l ( n h ≤ dt ′ . h R t ( t − t ′ )( − − ) dt ′ (1 + k ˜ φ k H k ′ ) . h (1 + k ˜ φ k H k ′ ) · Hence B → h → T ′ ǫ as a continuous and de-creasing function of k ˜ φ k H k ′ .We consider now the union of all intervals [0 , T ′ ) such that there exists a (unique)solution u ∈ C (cid:16) [0 , T ′ ) , H k ′ (cid:17) satisfying the integral equation for (4.1) on [0 , T ′ ).Clearly there exists T ǫ := T ǫ ( ˜ φ ) ∈ (0 , ∞ ] such that this union is equal to [0 , T ǫ ).Assume now that T ǫ < ∞ and that lim inf t → T ǫ k u ( t ) k H k ′ < ∞ . Hence we can findan M < ∞ and 0 ≤ ˜ t < T ǫ close enough to T ǫ such that the following properties ONLINEAR THIRD-ORDER EQUATIONS 19 holds: k u (˜ t ) k H k ′ ≤ M , and (by proceeding similarly as in (4.5)) we get for all t ∈ [˜ t, T ǫ )(4.7) k u k L ∞ t H k ′ ([˜ t,t ]) ≤ M + C ( M ) ǫ − ( t − ˜ t ) (cid:16) k u k L ∞ t H k ′ ([˜ t,t ]) + 1 (cid:17) Hence a continuity argument shows that k u k L ∞ t H k ′ ([˜ t,t ]) . M + 1. Since we alsohave u ∈ C (cid:16) [0 , ˜ t ] , H k ′ (cid:17) this implies that there exists a constant (that we still denoteby M >
0) such that k u ( t ) k H k ′ ≤ M for all t ∈ [0 , T ǫ ). We then claim that thereexists M ′ > t ∈ [0 , T ǫ ) : k u ( t ) k H k ′ ≤ M ′ · Indeed let J := [ a, t ) ⊂ [0 , T ǫ ); then we get k u k L ∞ t H k ′ ([ a,t )) ≤ k u ( a ) k H k ′ + C ( M ) ǫ − ( t − a ) (cid:16) k u k L ∞ t H k ′ ([ a,t )) (cid:17) ;hence a continuity argument shows that there exists c := c ( M, ǫ ) > | J | ≤ c then k u k L ∞ t H k ′ ( J ) ≤ hk u ( a ) k H k ′ i ; hence, dividing [0 , T ǫ ) into subintervals J such that | J | = c except maybe the last one we see that (4.8) holds. Let ¯ T beclose enough to T ǫ from the left. Then k u ( ¯ T ) k H k ′ ≤ M ′ ; moreover by using thearguments above to construct a unique solution u ∈ C (cid:16) [0 , T ′ ǫ ] , H k ′ (cid:17) satisfying (4.2)we can construct a unique solution u ∈ C (cid:16) [ ¯ T ǫ , ¯ T ) , H k ′ (cid:17) for some ¯ T > T ǫ that satis-fies (4.2), replacing “ ˜ φ ”, “ R t ” with “ u ( ¯ T )”, “ R ¯ T ¯ T ” respectively. Hence there existsa unique solution u ∈ C (cid:16) [0 , ¯ T ) , H k ′ (cid:17) of (4.2), which contradicts the definition of T ǫ .Let { ˜ φ n } be a sequence such that k ˜ φ n − ˜ φ ∞ k H k ′ → n → ∞ . Let t ∈ (cid:20) , T ′∞ (cid:19) ,with T ′ ∞ := T ′ ǫ (cid:16) k ˜ φ ∞ k H k ′ (cid:17) . The above observation shows that T ′ ǫ (cid:16) k ˜ φ n k H k ′ (cid:17) > T ′∞ for n large. We have (cid:13)(cid:13)(cid:13) e − ǫt∂ x ( ˜ φ n − ˜ φ ∞ ) (cid:13)(cid:13)(cid:13) H k ′ . k ˜ φ n − ˜ φ ∞ k H k ′ We see from (4.6) ( with v := u n and v := u ∞ ) and the estimate k ˜ φ n k H k ′ . k ˜ φ ∞ k H k ′ we see that (cid:13)(cid:13)(cid:13) h n i k ′ − (cid:16) \ A u n ( t ′ )( n ) − \ A u ∞ ( t ′ )( n ) (cid:17)(cid:13)(cid:13)(cid:13) l ≤ C (cid:16) k ˜ φ k H k ′ (cid:17) k u n ( t ′ ) − u ∞ ( t ′ ) k H k ′ Integrating in time we get k u n − u k L ∞ t H k ′ (cid:18)(cid:20) , T ′∞ (cid:19)(cid:19) ≤ k ˜ φ n − ˜ φ k H k ′ + k u n − u k L ∞ t H k ′ (cid:18)(cid:20) , T ′∞ (cid:19)(cid:19) , which implies that k u n − u ∞ k L ∞ t H k ′ (cid:18)(cid:20) , T ′∞ (cid:19)(cid:19) → n → ∞ .Let 0 ≤ T < T ǫ ( ˜ φ ∞ ). Observe that the continuity of u ∞ on [0 , T ] implies thatthere exists M > t ∈ [0 , T ] : k u ∞ ( t ) k H k ′ ≤ M · Let T ∗ := sup ( t ∈ [0 , T ] : sup s ∈ [0 ,t ] k u n ( s ) − u ∞ ( s ) k H k ′ → n → ∞ ) . Since k u n − u ∞ k L ∞ t H k ′ (cid:18)(cid:20) , T ′∞ (cid:21)(cid:19) → n → ∞ , we have T ∗ >
0. Assume that T ∗ < T .Then choose 0 ≤ ˜ t < T ∗ that is close enough to T ∗ such that the statementsbelow hold. We have sup t ∈ [0 , ˜ t ] k u n ( t ) − u ∞ ( t ) k H k ′ → n → ∞ , and we alsohave k u n − u ∞ k L ∞ t H k ′ ([˜ t, ˜ t ′ ]) = sup ≤ h ≤ ˜ t ′ − ˜ t k u n (˜ t + h ) − u ∞ (˜ t + h ) k H k ′ → T ≥ ˜ t ′ > T ∗ as n → ∞ , by applying the above argument to u n (˜ t ) ( resp. u ∞ (˜ t ))for t ∈ [0 , ˜ t ′ − ˜ t ] ) instead of ˜ φ n (resp. ˜ φ ∞ ) for t ∈ [0 , T ′ ǫ ] ), taking into account(4.9). But then this would contradict the definition of T ∗ . Hence T ∗ = T . Wenow claim that sup t ∈ [0 ,T ∗ ] k u n ( t ) − u ∞ ( t ) k H k ′ → n → ∞ . Indeed we can choose0 ≤ ˜ t < T ∗ that is close enough to T ∗ such that sup t ∈ [0 , ˜ t ] k u n ( t ) − u ∞ ( t ) k H k ′ → n → ∞ and k u n − u ∞ k L ∞ t H k ′ ([˜ t, ˜ t ′ ]) → T ǫ ( ˜ φ ∞ ) > ˜ t ′ > T ∗ as n → ∞ .Hence sup t ∈ [0 ,T ] k u n ( t ) − u ∞ ( t ) k H k ′ → n → ∞ . Remark . We say that u is a smooth solution of (4.2) obtained by Proposition4.1 with data ˜ φ if ˜ φ is smooth, i.e ˜ φ ∈ T m ≥ k ′ H m = C ∞ . Observe that u ∈C ∞ ([0 , T ] , H m ) for all m ≥ k ′ and that u ( t ) ∈ C ∞ : this follows from a standardbootstrap argument applied to ∂ t u = − ǫ∂ x u + F ( ~u, t ), taking into account Lemma3.3. Let φ ∈ H k ′ . Let { ˜ φ n } be a sequence of smooth functions such that ˜ φ n → ˜ φ . Then Proposition 4.1 allows to approximate the solution u arbitrarily on anycompact interval [0 , T ] ⊂ [0 , T ǫ ) with the sequence of smooth solutions { u n } . In thesequel we will implicitly use a/ : smooth solutions to justify some computations thatallow to prove some estimates that involve these solutions b/ : the approximationof u with smooth solutions to show that these estimates also hold for u . (cid:3) Gauged energy
Let k ′ ≥
6. Let f and g be two functions. We define the gauged energy E k ′ ( f, t )of f (and more generally E k ′ ( f, g, t )) at a time t in the following fashion: E k ′ ( f, g, t ) := (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f − Φ k ′ ( g, t ) ∂ x g (cid:13)(cid:13) H k ′− + k f − g k H max ( k ′− ,
92 + ) , and E k ′ ( f, t ) := E k ′ ( f, , t ) · The following proposition shows that the gauged energy E k ′ ( f, t ) of f (resp. E k ′ ( f, g, t ))can be compared with the H k ′ − norm of f (resp. k f − g k H k ′ ): Proposition 5.1.
Let I ⊂ R be an interval such that | I | ≤ . Let ¯ δ > . Let K ∈ R + . Let k ′ > . The following holds:(1) Assume that inf x ∈ T (cid:12)(cid:12)(cid:12) ∂ ω F (cid:16) ~f , t (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≥ ¯ δ for all t ∈ I . This equality follows for the well-known Sobolev embedding H m ֒ → C p if m > p + ONLINEAR THIRD-ORDER EQUATIONS 21 • Assume also that k f k H
132 + ≤ K . Then there exists C := C ( K, ¯ δ ) > such that for all t ∈ I (5.1) E k ′ ( f, t ) ≤ C (cid:16) k f k H k ′ (cid:17) • Assume also that E + ( f, t ) ≤ K . Then there exists C := C ( K, ¯ δ ) > such that for all t ∈ I (5.2) k f k H k ′ ≤ C (1 + E k ′ ( f, t )) · (2) Assume that k f k H k ′ ≤ K and k g k H k ′ ≤ K . Assume also that for all t ∈ I and for all θ ∈ [0 ,
1] inf x ∈ T (cid:12)(cid:12)(cid:12) ∂ ω F (cid:16) −→ h θ , t (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≥ ¯ δ with h θ := θf + (1 − θ ) g .Then there exists C := C ( K, ¯ δ ) > such that for all t ∈ IC − k f − g k H k ′ ≤ E k ′ ( f, g, t ) ≤ C k f − g k H k ′ Remark . The proof shows that one can choose the constants depending on K to be continuous functions that increase as K increases. Proof.
Lemma 2.1 and Lemma 3.4 imply that (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f (cid:13)(cid:13) H k ′− . k Φ k ′ ( f, t ) k H
12 + k ∂ x f k H k ′− + k ∂ x f k H
12 + k Φ k ′ ( f, t ) k H k ′− . (cid:16) k f k H
72 + (cid:17) k f k H k ′ + k f k H
132 + (1 + k f k H k ′− ) . k f k H k ′ · Hence (5.1) holds. We have k D k ′ f k L . (cid:13)(cid:13) Φ − k ′ ( f, t )Φ k ′ ( f, t ) ∂ x f (cid:13)(cid:13) H k ′− . (cid:13)(cid:13) Φ − k ′ ( f, t ) (cid:13)(cid:13) H
12 + (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f (cid:13)(cid:13) H k ′− + (cid:13)(cid:13) Φ − k ′ ( f, t ) (cid:13)(cid:13) H k ′− (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f (cid:13)(cid:13) H
12 + . (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f (cid:13)(cid:13) H k ′− (cid:16) k f k H
72 + (cid:17) + (cid:16) k f k H k ′− (cid:17) E + ( f, t ) . E k ′ ( f, t )Hence (5.2) holds. We have (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f − Φ k ′ ( g, t ) ∂ x g (cid:13)(cid:13) H k ′− . k Φ k ′ ( f, t ) − Φ k ′ ( g, t ) k H k ′− k ∂ x f k H
12 + + k Φ k ′ ( f, t ) − Φ k ′ ( g, t ) k H
12 + k ∂ x f k H k ′− + k Φ k ′ ( g, t ) k H k ′− (cid:13)(cid:13) ∂ x ( f − g ) (cid:13)(cid:13) H
12 + + k Φ k ′ ( g, t ) k H
12 + (cid:13)(cid:13) ∂ x ( f − g ) (cid:13)(cid:13) H k ′− . k f − g k H k ′ It remains to show that k D k ′ ( f − g ) k L . E k ′ ( f, g, t ). We have (taking into accountthat k ′ − > ) k D k ′ ( f − g ) k L . (cid:13)(cid:13) Φ − k ′ ( f, t )Φ k ′ ( f, t ) ∂ x ( f − g ) (cid:13)(cid:13) H k ′− . (cid:13)(cid:13) Φ − k ′ ( f, t ) (cid:13)(cid:13) H k ′− (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x ( f − g ) (cid:13)(cid:13) H k ′− . (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f − Φ k ′ ( g, t ) ∂ x g (cid:13)(cid:13) H k ′− + k Φ k ′ ( g, t ) − Φ k ′ ( f, t ) k H k ′− k ∂ x g k H k ′− . (cid:13)(cid:13) Φ k ′ ( f, t ) ∂ x f − Φ k ′ ( f, t ) ∂ x g (cid:13)(cid:13) H k ′− + k f − g k H max( k ′− ,
92 +) . E k ′ ( f, g, t ) (cid:3) Gauged energy estimates
In this section we prove some gauged energy estimates. To this end we firstprove estimates for a function v that satisfies the equation (6.1): see Proposition6.1 below. Then we apply these estimates to v ( t ) := Φ k ′ ( u ( t ) , t ) u ( t ) (with u solutionof (4.1)) to get the gauged energy estimates.6.1. General estimates.
We prove the following proposition:
Proposition 6.1.
Let < ǫ ≪ . Let k ′ ∈ [6 , ∞ ) . Assume that v is a function onan interval [0 , T ′ ] that satisfies the form (6.1) ∂ t v + ǫ∂ x v = a ∂ x v + a ∂ x v + a ∂ x v + a + ǫb ∂ x v + ǫb ∂ x v + ǫb ∂ x v + ǫb + ǫ∂ x cv · There exists
C > such that for all t ∈ [0 , T ′ ](6.2) ∂ t k v ( t ) k H k ′− + 2 ǫ k ∂ k ′ − x v ( t ) k L + 2 R T R T (cid:0)(cid:0) − k ′ (cid:1) ∂ x a ( t ) − a ( t ) (cid:1) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx ≤ C k a ( t ) k H k ′− + k v ( t ) k H k ′− (cid:18) P m =1 k a m ( t ) k H ( m + 12 ) + (cid:19) + P m =2 k a m ( t ) k H k ′− m k v ( t ) k H
12 + + k a ( t ) k H k ′− k v ( t ) k H
32 + + ǫ (cid:16) k b ( t ) k H
32 + + k b ( t ) k H
12 + (cid:17) k ∂ x ˜ D k ′ − v ( t ) k L + ǫ k b ( t ) k H k ′− + ǫ (cid:18) P m =1 k b m ( t ) k H k ′− m + k c ( t ) k H k ′− (cid:19) k v ( t ) k H
12 + + ǫ (cid:18) P m =1 k b m ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k v ( t ) k H k ′− Proof.
Elementary considerations show that(6.3) h ∂ t v ( t ) + ǫ∂ x v ( t ) , ˜ D k ′ − v ( t ) i = ( − k ′− ∂ t k v ( t ) k H k ′− + ( − k ′ − ǫ (cid:13)(cid:13)(cid:13) ∂ k ′ − x v ( t ) (cid:13)(cid:13)(cid:13) L · On the other hand(6.4) h ∂ t v ( t ) + ǫ∂ x v ( t ) , ˜ D k ′ − v ( t ) i = ( − k ′ − ( A ( t ) + ... + A ( t ) + B ( t ) + ... + B ( t ) + C ( t )) , with A ( t ) := R ˜ D k ′ − ( a ( t )) ˜ D k ′ − v ( t ) dx,B ( t ) := ǫ R ˜ D k ′ − ( b ( t )) ˜ D k ′ − v ( t ) dx,A m ( t ) := R ˜ D k ′ − ( a m ( t ) ∂ mx v ( t )) ˜ D k ′ − v ( t ) dx,B m ( t ) := ǫ R ˜ D k ′ − ( b m ( t ) ∂ mx v ( t )) ˜ D k ′ − v ( t ) dx,C ( t ) := ǫ R ˜ D k ′ − (cid:0) ∂ x c ( t ) v ( t ) (cid:1) ˜ D k ′ − v ( t ) dx · (Here m ∈ { , , } ). We first estimate A m ( t ).We see from Lemma 3.1 that ONLINEAR THIRD-ORDER EQUATIONS 23 A ( t ) = A a ( t ) + A b ( t ) + A c ( t )+ O (cid:16) k v ( t ) k H k ′− (cid:16) k a ( t ) k H k ′− k v ( t ) k H
12 + + k a ( t ) k H
72 + k v ( t ) k H k ′− (cid:17)(cid:17) , with A a ( t ) := R a ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx,A b ( t ) := ( k ′ − R ˜ Da ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx, and A c ( t ) := ( k ′ − k ′ − R ˜ D a ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx · Integrations by parts show that A a ( t ) = − R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx − R a ( t ) ∂ x ˜ D k ′ − v ( t ) ∂ x ˜ D k ′ − v ( t ) dx = R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx + R ∂ x a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx = − R ∂ x a ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx + R ∂ x a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx · We have A b ( t ) = ( k ′ − R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx = − ( k ′ − R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx − ( k ′ − R ∂ x a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx = k ′ − R ∂ x a ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx − ( k ′ − R ∂ x a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx We have A c ( t ) = ( k ′ − k ′ − R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx = − ( k ′ − k ′ − R ∂ x a ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx · We can also write A ( t ) = A a ( t ) + A b ( t )+ O (cid:16) k a ( t ) k H k ′− k v ( t ) k H k ′− k v ( t ) k H
12 + (cid:17) + O (cid:16) k a ( t ) k H
52 + k v ( t ) k H k ′− (cid:17) with A a ( t ) := R a ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx, and A b ( t ) := ( k ′ − R ˜ Da ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx · We have A a ( t ) = − R ∂ x a ( t ) ∂ x ˜ D k ′ − v ( t ) ˜ D k ′ − v ( t ) dx − R a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx = − R a ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx + O (cid:16) k ∂ x a ( t ) k L ∞ k v ( t ) k H k ′− (cid:17) , and A b ( t ) = − k ′ − R ∂ x a ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx = O (cid:16) k ∂ x a ( t ) k L ∞ k v ( t ) k H k ′− (cid:17) · We have A ( t ) = R a ( t ) ˜ D k ′ − ∂ x v ( t ) ˜ D k ′ − v ( t ) dx + O (cid:16) k a ( t ) k H k ′− k v ( t ) k H
32 + k v ( t ) k H k ′− (cid:17) + O (cid:16) k a ( t ) k H
32 + k v ( t ) k H k ′− (cid:17) = − R ∂ x a ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx + O ( ... ) + O ( ... )= O (cid:16) k ∂ x a ( t ) k L ∞ k v ( t ) k H k ′− (cid:17) + O ( ... ) + O ( ... ) · We have | A ( t ) | . (cid:13)(cid:13)(cid:13) ˜ D k ′ − ( a ( t )) (cid:13)(cid:13)(cid:13) L k ˜ D k ′ − v ( t ) k L . k a ( t ) k H k ′− k v ( t ) k H k ′− · We then estimate B m ( t ) by a similar process to estimate A m ( t ). We get ǫ − B ( t ) = ( k ′ − − k ′ )4 R ∂ x b ( t ) (cid:16) ˜ D k ′ − v ( t ) (cid:17) dx + (cid:0) − k ′ (cid:1) R ∂ x b ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx + O (cid:16) k v ( t ) k H k ′− (cid:16) k b ( t ) k H k ′− k v ( t ) k H
12 + + k b ( t ) k H
72 + k v ( t ) k H k ′− (cid:17)(cid:17) ,ǫ − B ( t ) = − R b ( t ) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx + O (cid:16) k b ( t ) k H k ′− k v ( t ) k H k ′− k v ( t ) k H
12 + (cid:17) + O (cid:16) k b ( t ) k H
52 + k v ( t ) k H k ′− (cid:17) + O (cid:16) k ∂ x b ( t ) k L ∞ k v ( t ) k H k ′− (cid:17) , and ǫ − B ( t ) = O (cid:16) k ∂ x b ( t ) k L ∞ k v ( t ) k H k ′− (cid:17) + O (cid:16) k b ( t ) k H k ′− k v ( t ) k H
12 + k v ( t ) k H k ′− (cid:17) + O (cid:16) k b ( t ) k + k v ( t ) k H k ′− (cid:17) , and ǫ − | B ( t ) | . k b ( t ) k H k ′− k v ( t ) k H k ′− · We have ǫ − | C ( t ) | . (cid:16) k c ( t ) k H k ′− k v ( t ) k H
12 + + k c ( t ) k H
92 + k v ( t ) k H k ′− (cid:17) k v ( t ) k H k ′− · Next we divide (6.3) and (6.4) by ( − k ′ − . We see that we can bound from above d k v ( t ) k Hk − dt + 2 ǫ k ∂ k ′ − x v ( t ) k L + 2 R (cid:0)(cid:0) − k ′ (cid:1) ∂ x a ( t ) − a ( t ) (cid:1) (cid:16) ∂ x ˜ D k ′ − v ( t ) (cid:17) dx by the RHS of (6.2) by using the above estimates, the Sobolev embedding H + ֒ → L ∞ , and the Young inequality ab ≤ a + b . (cid:3) We also prove the proposition below:
Proposition 6.2.
Let < ǫ ≪ . Let k ′ ≥ , K ≥ , and j ∈ (cid:8) + , k ′ − (cid:9) . Let u be a solution of (4.2) obtained by Proposition 4.1 on an interval [0 , T ′ ] . Assumethat sup t ∈ [0 ,T ′ ] k u ( t ) k H
92 + ≤ K . Then there exists C := C ( K ) > such that (6.5) ∂ t k u ( t ) k H j ≤ C (cid:0) k u ( t ) k H j +3 (cid:1) · Assume that v is a function on an interval [0 , T ] that satisfies ONLINEAR THIRD-ORDER EQUATIONS 25 (6.6) ∂ t v + ǫ∂ x v = a ∂ x v + a ∂ x v + a ∂ x v + a + ǫb ∂ x v + ǫb ∂ x v + ǫb ∂ x v + ǫb + ǫ∂ x cv · There exists
C > such that for all t ∈ [0 , T ′ ](6.7) ∂ t k v ( t ) k L + 2 ǫ k ∂ x v ( t ) k L ≤ C k a ( t ) k L + (cid:18) P m =1 k a m ( t ) k H ( m + 12 ) + (cid:19) k v ( t ) k L + P m =2 k a m ( t ) k H m k v ( t ) k H
12 + + k a ( t ) k L k v ( t ) k H
32 + + (cid:16) k a ( t ) k H
32 + + k a ( t ) k H
12 + (cid:17) k ∂ x v ( t ) k L + ǫ (cid:16) k b ( t ) k H
32 + + k b ( t ) k H
12 + (cid:17) k ∂ x v ( t ) k L + ǫ k b ( t ) k L + ǫ (cid:18) P m =1 k b m ( t ) k H m + k c ( t ) k H (cid:19) k v ( t ) k H
12 + + ǫ (cid:18) P m =1 k b m ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k v ( t ) k L Remark . Regarding the constants in Proposition 6.2, we refer to Remark 3.1.
Proof.
Let m ∈ (cid:8) , + , k ′ − (cid:9) . Elementary considerations show that h ∂ t u ( t ) + ǫ∂ x u ( t ) , ˜ D m u ( t ) i = ( − m ∂ t (cid:18) k ˜ D m u ( t ) k L (cid:19) + ( − m ǫ k ∂ m +2 x u ( t ) k L On the other hand h F ( −−→ u ( t )) , ˜ D m u ( t ) i = ( − m h ˜ D m (cid:16) F ( −−→ u ( t )) (cid:17) , ˜ D m u ( t ) i . Hencewe see from Lemma 3.3, the Cauchy-Schwartz inequality, and elementary estimatesthat ∂ t k u ( t ) k H m . k u ( t ) k H m +3 · Repeating the proof of Proposition 6.1 (with k ′ replaced with 6) we see that d k v ( t ) k L dt + 2 ǫ k ∂ x v ( t ) k L + 2 R (cid:0) ∂ x a ( t ) − a ( t ) (cid:1) ( ∂ x v ( t )) dx is bounded by theRHS of (6.7) (with again k ′ replaced with 6). The estimate k f g k L . k f k L ∞ k g k L and the Sobolev embedding H + ֒ → L ∞ allow to show that the third term of thesum just above is bounded by the RHS of (6.7). (cid:3) Gauged energy estimates: statement.
In this subsection we prove thefollowing gauged energy estimates:
Proposition 6.3.
Let k ′ ≥ k ′ > . Let ˜ φ ∈ P + ,k ′ . Then there exist ≥ T ′ := T ′ (cid:18) k ˜ φ k H k ′ , ˜ δ ( −→ ˜ φ ) , ˜ δ ′ ( ˜ φ ) (cid:19) > and C := (cid:18) k ˜ φ k H k ′ , ˜ δ ( −→ ˜ φ ) (cid:19) such that if u is thesolution of (4.2) obtained by Proposition 4.1 on [0 , T ǫ ) then (6.8) T ǫ > T ′ ; sup t ∈ [0 ,T ′ ] (cid:18) E k ′ ( u ( t ) , t ) + R t R T Q (cid:16) u ( t ′ ) , t ′ ) (cid:17) (cid:16) ∂ x D k ′ − u ( t ′ ) (cid:17) dx dt ′ (cid:19) ≤ (cid:16) E k ′ ( ˜ φ, (cid:17) ; t ∈ [0 , T ′ ] : dE k ′ ( u ( t )) ,t ) dt ( t ) + R T Q ( u ( t ) , t ) (cid:16) ∂ x D k ′ − u ( t ) (cid:17) dx ≤ C (1 + E k ′ ( u ( t ) , t )) ,u ( t ) ∈ P + ,k ′ , δ (cid:16) −−→ u ( t ) , t (cid:17) & ˜ δ (cid:18) −→ ˜ φ (cid:19) , and δ ′ ( u ( t ) , t ) & ˜ δ ′ ( ˜ φ ) · Remark . The proof of Proposition 6.3 shows that T ′ (resp. C ) can be chosen as acontinuous function that decreases (resp. increases) as ˜ δ (cid:18) −→ ˜ φ (cid:19) decreases, decreases(resp. increases) as k ˜ φ k H k ′ increases. It also shows that T ′ can be chosen as acontinuous function that decreases as ˜ δ ′ ( ˜ φ ) decreases. Proof.
The proof relies upon the lemma below
Lemma 6.4.
Let K ≥ . Let u be a solution of (4.2) on an interval [0 , T ′′ ] with < T ′′ ≤ . Let ¯ δ > . Assume that that δ (cid:16) −−→ u ( t ) , t (cid:17) ≥ ¯ δ for all t ∈ [0 , T ′′ ] . Assumealso that sup t ∈ [0 ,T ′′ ] E + ( u ( t ) , t ) ≤ K . Then there exists C := C ( K, ¯ δ ) > suchthat (6.9) t ∈ [0 , T ′′ ] : dE k ′ ( u ( t ) ,t ) dt ( t ) + R T Q ( u ( t ) , t ) (cid:16) ∂ x D k ′ − u ( t ) (cid:17) dx ≤ C (1 + E k ′ ( u ( t ) , t )) · Remark . The proof of Lemma 6.4 shows that C can be chosen as a continuousfunction that increases as K increases, and that increases as ¯ δ decreases.We postpone the proof of Lemma 6.4 to Section 10.Let T ∗ ǫ := inf n t ≥ E k ′ ( u ( t ) , t ) > (cid:16) E k ′ ( ˜ φ, (cid:17)o We see from Proposition 4.1 and Proposition 5.1 that 0 < T ∗ ǫ < T ǫ . Moreover E k ′ ( u ( t ) , t ) ≤ (cid:16) E k ′ ( ˜ φ, (cid:17) for all t ∈ [0 , T ∗ ǫ ]. We have sup t ∈ [0 ,T ∗ ǫ ] E k ′ ( u ( t ) , t ) . k ˜ φ k Hk ′ t ∈ [0 ,T ∗ ǫ ] k u ( t ) k H k ′ . k ˜ φ k H k ′ . Let T ′ := T ′ (cid:18) ˜ δ (cid:18) −→ ˜ φ (cid:19) , ˜ δ ′ ( ˜ φ ) , k ˜ φ k H k ′ (cid:19) > t ∈ [0 , T ∗ ǫ ] if we assume that 0 < T ∗ ǫ ≤ T ′ .Claim: δ (cid:16) −−→ u ( t ) , t (cid:17) & ˜ δ (cid:18) −→ ˜ φ (cid:19) We see from Section 11 that (cid:13)(cid:13)(cid:13) X ( u ( t ) , ˜ φ, t, (cid:13)(cid:13)(cid:13) L ∞ . k u ( t ) − ˜ φ k H
72 + + t . We have t ≪ ˜ δ (cid:18) −→ ˜ φ (cid:19) . So it suffices to estimate k u ( t ) − ˜ φ k H
72 + . In fact we will estimate k u ( t ) − ˜ φ k H
92 + , since we will also use this estimate at the end of the proof. Let
ONLINEAR THIRD-ORDER EQUATIONS 27 u l ( t ) := e − ǫt∂ x ˜ φ and u nl ( t ) := R t e − ǫ ( t − t ′ ) ∂ x F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17) dt ′ . Elementary estimatesshow that | e − x − | . min( x,
1) if x ≥
0. Hence k u l ( t ) − ˜ φ k H
92 + . P n ∈ Z h n i ( + ) (cid:12)(cid:12)(cid:12) e − ǫt ( in ) − (cid:12)(cid:12)(cid:12) | b ˜ φ ( n ) | . ǫ t k ˜ φ k H k ′ ≪ ˜ δ (cid:18) −→ ˜ φ (cid:19) · The Minkowski inequality, the Parseval equality, and Lemma 3.3 show that k u nl ( t ) k H
92 + . R t k e − ǫ ( t − t ′ ) ∂ x F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17) k H
72 + dt ′ . k ˜ φ k Hk ′ t (cid:16) t ′ ∈ [0 ,t ] k u ( t ′ ) k H
152 + (cid:17) . k ˜ φ k Hk ′ t ≪ ˜ δ (cid:18) −→ ˜ φ (cid:19) · Hence we see from the above estimates and the triangle inequality that the claimholds.Claim: u ( t ) ∈ P + ,k ′ and δ ′ ( u ( t ) , t ) & ˜ δ ′ ( ˜ φ )From the claim above and similar arguments as those in Remark 1.2 we see that ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) has a constant sign. Hence Q ( u ( t )) = − (cid:12)(cid:12)(cid:12) ∂ ω F ( −−→ u ( t ) , t ) (cid:12)(cid:12)(cid:12) (cid:28) P ( u ( t ) ,t ) (cid:12)(cid:12)(cid:12) ∂ ω F ( −−→ u ( t ) ,t ) (cid:12)(cid:12)(cid:12) (cid:29) .We see from Section 11 and the above estimates that (cid:13)(cid:13)(cid:13) Y ( u ( t ) , ˜ φ, t, (cid:13)(cid:13)(cid:13) L ∞ . ˜ δ (cid:18) −→ ˜ φ , k ˜ φ k Hk ′ (cid:19) (cid:13)(cid:13)(cid:13) u ( t ) − ˜ φ (cid:13)(cid:13)(cid:13) H
92 + + t . ˜ δ (cid:18) −→ ˜ φ (cid:19) , k ˜ φ k Hk ′ (cid:0) t + ǫ t (cid:1) ≪ ˜ δ ′ ( ˜ φ ).Hence we can apply Lemma 6.4 and Gronwall inequality to get for some C ′ := C (cid:18) k ˜ φ k H k ′ , δ ( −→ ˜ φ ) (cid:19) > E k ′ ( u ( t ) , t ) ≤ e C ′ t (cid:16) E k ′ ( ˜ φ, (cid:17) ≤ (cid:16) E k ′ ( ˜ φ, (cid:17) · Hence by letting t = T ∗ ǫ in the above inequality, we see that we cannot have T ∗ ǫ ≤ T ′ . Hence T ∗ ǫ > T ′ . We then repeat the proof from ‘Claim: δ ( −−→ u ( t ) , t ) & ˜ δ ( −→ ˜ φ )’ to‘yields the claim.’, replacing ‘ δ ( −−→ u ( t ) , t ) & ˜ δ ( −→ ˜ φ ) and ‘ u ( t ) ∈ P + ,k ′ and δ ′ ( u ( t ) , t ) & ˜ δ ′ ( ˜ φ ) ’ with ‘ δ ( −−→ u ( t ) , t ) & ˜ δ ( −→ ˜ φ ), t ∈ [0 , T ′ ] ’ and ‘ u ( t ) ∈ P + ,k ′ and δ ′ ( u ( t ) , t ) & ˜ δ ′ ( ˜ φ ), t ∈ [0 , T ′ ] ’ respectively. Hence an application of the triangle inequality yields theclaim.Hence we can apply again Lemma 6.4 and Gronwall inequality to get (6.8). (cid:3) Gauged energy estimates for the difference of two solutions
In this section we prove gauged energy estimates for the difference of two so-lutions of an equation of the form (4.2). To this end we first prove estimates fora function ¯ v := v − v with v , v that satisfy (7.1): see Proposition 7.1 below.Then we apply these estimates to ¯ v ( t ) := Φ k ′ ( u ( t ) , t ) ∂ x u ( t ) − Φ k ′ ( u ( t ) , t ) ∂ x u ( t )( with u , u solutions of (4.1)) in order to get the gauged energy estimates for thedifference of two solutions.7.1. General estimates.
We prove the following proposition:
Proposition 7.1.
Let j ∈ { , } and < ǫ ≤ ǫ ≪ . Let k ′ ≥ . Assume that v j is a function that satisfies on an interval [0 , T ′ ](7.1) ∂ t v j + ǫ j ∂ x v j = a ,j ∂ x v j + a ,j ∂ x v j + a ,j ∂ x v j + a ,j + ǫ j b ,j ∂ x v j + ǫ j b ,j ∂ x v j + ǫ j b ,j ∂ x v j + ǫ j b ,j + ǫ j ∂ x c j v j · Let ¯ v := v − v . Then there exists C > such that for all time t ∈ [0 , T ′ ](7.2) ∂ t k ¯ v ( t ) k H k ′− + 2 ǫ k ∂ k ′ − x ¯ v ( t ) k L + 2 R T (cid:0)(cid:0) − k ′ (cid:1) ∂ x a , ( t ) − a , ( t ) (cid:1) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) dx ≤ C (cid:18) P m =1 k a m, ( t ) k H ( m + 12 ) + (cid:19) k ¯ v ( t ) k H k ′− + P m =2 k a m, ( t ) k H k ′− m k ¯ v ( t ) k H
12 + + k a , ( t ) k H k ′− k ¯ v ( t ) k H
32 + + ǫ (cid:16) k b , ( t ) k H
32 + + k b , ( t ) k H
12 + (cid:17) k ∂ x ˜ D k ′ − ¯ v ( t ) k L + ǫ k b , ( t ) k H k ′− + ǫ k b , ( t ) k H k ′− + ǫ (cid:18) P m =1 k b m, ( t ) k H k ′− m + k c ( t ) k H k ′− (cid:19) k ¯ v ( t ) k H
12 + + ǫ (cid:16) k c ( t ) − c ( t ) k H k ′− k v ( t ) k H
12 + + k c ( t ) − c ( t ) k H
92 + k v ( t ) k H k ′− (cid:17) + ǫ (cid:18) P m =1 k b m, ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k ¯ v ( t ) k H k ′− + ǫ P j =1 3 P m =1 (cid:18) k b m,j ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + + k b m,j ( t ) k H
12 + k v ( t ) k H k ′− m (cid:19) + k c ( t ) k H k ′− k v ( t ) k H
12 + + k c ( t ) k H
92 + k v ( t ) k H k ′− + k v ( t ) k H k ′− + k a , ( t ) − a , ( t ) k H k ′− + P m =1 k a m, ( t ) − a m, ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + + k a m, ( t ) − a m, ( t ) k H
12 + k v ( t ) k H m + k ′− ! Proof.
We have ∂ t ¯ v + ǫ ∂ x ¯ v = ( ǫ − ǫ ) ∂ x v + ( ǫ − ǫ ) ∂ x c v − ǫ ∂ x ( c − c ) v + ǫ ∂ x c ¯ v + P m =1 a m, ∂ mx ¯ v + P m =1 ( a m, − a m, ) ∂ mx v + ( a , − a , )+ ǫ P m =1 b m, ∂ mx ¯ v + ǫ P m =1 b m, ∂ mx v − ǫ P m =1 b m, ∂ mx v + ǫ b , − ǫ b , Elementary considerations show that
ONLINEAR THIRD-ORDER EQUATIONS 29 h ∂ t ¯ v ( t ) + ǫ ∂ x ¯ v ( t ) , ˜ D k ′ − ¯ v ( t ) i = ( − k ′ − (cid:18) ∂ t (cid:18) k ¯ v ( t ) k Hk ′− (cid:19) + ǫ k ∂ k ′ − x ¯ v ( t ) k L (cid:19) · The proof of Proposition 6.1, Lemma 3.1, and the Sobolev embedding H + ֒ → L ∞ show that P m =1 h a m, ( t ) ∂ mx ¯ v ( t ) , ˜ D k ′ − ¯ v ( t ) i = ( − k ′ − (cid:28)(cid:0) − k ′ (cid:1) ∂ x a , ( t ) − a , ( t ) , (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) (cid:29) + O k ¯ v ( t ) k H k ′− P m =1 k a m, ( t ) k H ( m + 12 ) + k ¯ v ( t ) k H k ′− + P m =2 k a m, ( t ) k H k ′− m k ¯ v ( t ) k H
12 + + k a , ( t ) k H k ′− k ¯ v ( t ) k H
32 + , (cid:12)(cid:12)(cid:12) h a , ( t ) − a , ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . k a , ( t ) − a , ( t ) k H k ′− k ¯ v ( t ) k H k ′− ǫ P m =1 h b m, ( t ) ∂ mx ¯ v ( t ) , ˜ D k ′ − ¯ v ( t ) i = ( − k ′ − ( X ( t ) + Y ( t ) + Z ( t )) , with ǫ − X ( t ) = ( k ′ − − k ′ )4 R ∂ x b , ( t ) (cid:16) ˜ D k ′ − ¯ v ( t ) (cid:17) dx + (cid:0) − k ′ (cid:1) R ∂ x b , ( t ) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) dx + O (cid:16) k ¯ v ( t ) k H k ′− (cid:16) k b , ( t ) k H k ′− k ¯ v ( t ) k H
12 + + k b , ( t ) k H
72 + k ¯ v ( t ) k H k ′− (cid:17)(cid:17) ,ǫ − Y ( t ) = − R b , ( t ) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) dx + O (cid:16) k ¯ v ( t ) k H k ′− (cid:16) k b , ( t ) k H k ′− k ¯ v ( t ) k H
12 + + k b , ( t ) k H
52 + k ¯ v ( t ) k H k ′− (cid:17)(cid:17) and ǫ − Z ( t ) = O (cid:16) k b , ( t ) k H
32 + k ¯ v ( t ) k H k ′− + k b , ( t ) k H k ′− k ¯ v ( t ) k H
12 + k ¯ v ( t ) k H k ′− (cid:17) · We also have (cid:12)(cid:12)(cid:12) ǫ h ∂ x c ( t )¯ v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( − k ′ − ǫ h ˜ D k ′ − (cid:0) ∂ x c ( t )¯ v ( t ) (cid:1) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ k ¯ v ( t ) k H k ′− (cid:16) k c ( t ) k H
92 + k ¯ v ( t ) k H k ′− + k c ( t ) k H k ′− k ¯ v ( t ) k H
12 + (cid:17) · We also have (cid:12)(cid:12)(cid:12) h ǫ b , ( t ) − ǫ b , ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ (cid:12)(cid:12)(cid:12) h b , ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) + ǫ (cid:12)(cid:12)(cid:12) h b , ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ( ǫ k b , ( t ) k H k ′− + ǫ k b , ( t ) k H k ′− ) k ¯ v ( t ) k H k ′− · Lemma 2.1 shows that (cid:12)(cid:12)(cid:12) h ( a m, ( t ) − a m, ( t )) ∂ mx v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( − k ′ − h ˜ D k ′ − (( a m, ( t ) − a m, ( t )) ∂ mx v ( t )) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . k a m, ( t ) − a m, ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + + k a m, ( t ) − a m, ( t ) k H
12 + k v ( t ) k H m + k ′− ! k ¯ v ( t ) k H k ′− · If j ∈ { , } then ǫ j (cid:12)(cid:12)(cid:12) h b m,j ( t ) ∂ mx v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ j k b m,j ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + + k b m,j ( t ) k H
12 + k v ( t ) k H k ′− m ! k ¯ v ( t ) k H k ′− · We have(7.3) ( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) h ∂ x c ( t ) v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) = ( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) ( − k ′ − h ˜ D k ′ − (cid:0) ∂ x c ( t ) v ( t ) (cid:1) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ (cid:16) k c ( t ) k H k ′− k v ( t ) k H
12 + + k c ( t ) k H
92 + k v ( t ) k H k ′− (cid:17) k ¯ v ( t ) k H k ′− , ( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) h ∂ x v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) = ( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) ( − k ′ − h ˜ D k ′ − ∂ x v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ k v ( t ) k H k ′− k ¯ v ( t ) k H k ′− , and(7.4) ǫ (cid:12)(cid:12)(cid:12) h ∂ x ( c ( t ) − c ( t )) v ( t ) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) = ǫ (cid:12)(cid:12)(cid:12) ( − k ′ − h ˜ D k ′ − (cid:0) ∂ x ( c ( t ) − c ( t )) v ( t ) (cid:1) , ˜ D k ′ − ¯ v ( t ) i (cid:12)(cid:12)(cid:12) . ǫ (cid:16) k c ( t ) − c ( t ) k H k ′− k v ( t ) k H
12 + + k c ( t ) − c ( t ) k H
92 + k v ( t ) k H k ′− (cid:17) k ¯ v ( t ) k H k ′− · The Young inequality ab ≤ a + b and the above estimates show that the LHS of(7.2) is bounded by the RHS of (7.2). (cid:3) We then prove the proposition below:
Proposition 7.2.
Let k ′ ≥ . The following hold:(1) Let j ∈ { , } . Let u j be a solution of (4.2) with ǫ := ǫ j on an interval [0 , T ′ ] . Let K ≥ . Let m ∈ (cid:8) + , k ′ − (cid:9) . Assume that sup t ∈ [0 ,T ′ ] k u j ( t ) k H m +3 ≤ K . Let ¯ u := u − u . Then there exists C := C ( K ) > d k ¯ u ( t ) k Hm dt ( t ) ≤ C (cid:0) k ¯ u ( t ) k H m +3 + ǫ k u ( t ) k H m +4 (cid:1) · (2) Let j ∈ { , } . Let < ǫ ≤ ǫ ≪ . Assume that v j is a function thatsatisfies on an interval [0 , T ′ ](7.6) ∂ t v j + ǫ j ∂ x v j = a ,j ∂ x v j + a ,j ∂ x v j + a ,j ∂ x v j + a ,j + ǫ j b ,j ∂ x v j + ǫ j b ,j ∂ x v j + ǫ j b ,j ∂ x v j + ǫ j b ,j + ǫ j ∂ x c j v j · Let ¯ v := v − v . Then there exists C > such that for all t ∈ [0 , T ′ ] ONLINEAR THIRD-ORDER EQUATIONS 31 (7.7) d k ¯ v ( t ) k L dt ( t ) + 2 ǫ k ∂ x ¯ v ( t ) k L ≤ C P m =2 k a m, ( t ) k H ( m − ) + k ∂ x ¯ v ( t ) k L + (cid:18) P m =1 k a m, ( t ) k H ( m + 12 ) + (cid:19) k ¯ v ( t ) k L + P m =2 k a m, ( t ) k H m k ¯ v ( t ) k H
12 + + k a , ( t ) k L k ¯ v ( t ) k H
32 + + ǫ (cid:16) k b , ( t ) k H
32 + + k b , ( t ) k H
12 + (cid:17) k ∂ x ¯ v ( t ) k L + ǫ k b , ( t ) k L + ǫ k b , ( t ) k L + ǫ (cid:18) P m =1 k b m, ( t ) k H m + k c ( t ) k H (cid:19) k ¯ v ( t ) k H
12 + + ǫ (cid:16) k c ( t ) − c ( t ) k H k v ( t ) k H
12 + + k c ( t ) − c ( t ) k H
92 + k v ( t ) k H (cid:17) + ǫ (cid:18) P m =1 k b m, ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k ¯ v ( t ) k L + ǫ P j =1 3 P m =1 (cid:18) k b m,j ( t ) k L k v ( t ) k H ( m + 12 ) + + k b m,j ( t ) k H
12 + k v ( t ) k H m (cid:19) + k c ( t ) k H k v ( t ) k H
12 + + k c ( t ) k H
92 + k v ( t ) k L + k v ( t ) k H + k a , ( t ) − a , ( t ) k L + P m =1 k a m, ( t ) − a m, ( t ) k L k v ( t ) k H ( m + 12 ) + + k a m, ( t ) − a m, ( t ) k H
12 + k v ( t ) k H m ! Remark . The proof shows that the constant C depending on K in Proposition7.2 can be chosen as a function of K that increases as K increases. Proof.
Let ¯ u := u − u . Let m ∈ (cid:8) , + , k ′ − (cid:9) . We have ∂ t ¯ u + ǫ ∂ x ¯ u = ( ǫ − ǫ ) ∂ x u + F ( −−−→ u ( t ) , t ) − F ( −−−→ u ( t ) , t )Elementary considerations show that h ∂ t ¯ u ( t ) + ǫ ∂ x ¯ u ( t ) , ˜ D m ¯ u ( t ) i = ( − m ∂ t (cid:18) k ˜ D m ¯ u ( t ) k L (cid:19) + ( − m ǫ ∂ t (cid:16) k ∂ m +2 x ¯ u ( t ) k L (cid:17) On the other hand the Young inequality ab ≤ a + b yields( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) h ∂ x u ( t ) , ˜ D m ¯ u ( t ) i (cid:12)(cid:12)(cid:12) = ( ǫ − ǫ ) (cid:12)(cid:12)(cid:12) ( − m h ∂ mx u ( t ) , ˜ D m ¯ u ( t ) i (cid:12)(cid:12)(cid:12) . ( ǫ − ǫ ) k u ( t ) k H m +4 k ¯ u ( t ) k H m . ǫ k u ( t ) k H m +4 + k ¯ u ( t ) k H m and h F ( −−−→ u ( t ) , t ) − F ( −−−→ u ( t ) , t ) , ˜ D m ¯ u ( t ) i = ( − m h ˜ D m (cid:16) F ( −−−→ u ( t ) , t ) − F ( −−−→ u ( t ) , t ) (cid:17) , ˜ D m ¯ u ( t ) i .Hence we see from Lemma 3.3 that ∂ t k ¯ u ( t ) k H m . k ¯ u ( t ) k H m +3 + ǫ k u ( t ) k H m +4 if m ∈ (cid:8) + , k ′ − (cid:9) and ∂ t k ¯ u ( t ) k L . k ¯ u ( t ) k H
72 + . Hence (7.5) holds.The Sobolev embedding H + ֒ → L ∞ shows that (cid:12)(cid:12) R (cid:0) ∂ x a , ( t ) − a , ( t ) (cid:1) ( ∂ x ¯ v ( t )) dx (cid:12)(cid:12) . ( k ∂ x a , ( t ) k L ∞ + k a , ( t ) k L ∞ ) k ∂ x ¯ v ( t ) k L . (cid:16) k a , ( t ) k H
32 + + k a , ( t ) k H
12 + (cid:17) k ∂ x ¯ v ( t ) k L Hence in view of the proof of Proposition 7.1 (with k ′ replaced with 6) we see that(7.7) holds. (cid:3) Gauged energy estimates for difference of solutions: statement.
Weprove the following proposition:
Proposition 7.3.
Let k ′ > , k ′ ≥ k ′ > , and ˜ φ ∈ P + ,k ′ . Let q ∈ { , } .There exists < c := c (cid:18) ˜ δ ( −→ ˜ φ ) , ˜ δ ′ ( ˜ φ ) , k ˜ φ k H k ′ (cid:19) ≪ such that if < ǫ ≤ ǫ ≤ c ,then the following holds: if u q is the solution of (4.2) obtained by Proposition4.1 with ǫ := ǫ q , replacing ˜ φ with ˜ φ q,k ′ := J ǫ q ,k ′ ˜ φ on [0 , T ǫ q ] , then there exist & T ′ := T ′ (cid:18) k ˜ φ k H k ′ , ˜ δ ( −→ ˜ φ ) , ˜ δ ′ ( ˜ φ ) (cid:19) > and C := C (cid:18) k ˜ φ k H k ′ , ˜ δ ( −→ ˜ φ ) (cid:19) > suchthat (7.8) T ǫ , T ǫ > T ′ ; (6 . holds, replacing u with u q ; and sup t ∈ [0 ,T ′ ] (cid:18) E k ′ ( u ( t ) , u ( t ) , t ) + R t R T Q ( u ( t ′ ) , t ′ ) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ′ ) (cid:17) dx dt ′ (cid:19) ≤ C (cid:16) k u (0) − u (0) k H k ′ + ǫ (cid:17) , with ¯ v ( t ) := Φ k ′ ( u ( t ) , t ) ∂ x u ( t ) − Φ k ′ ( u ( t ) , t ) ∂ x u ( t ) .Remark . The proof of Proposition 7.3 shows that one can choose T ′ (resp. C )to be a continuous function that decreases (resp. increases) as ˜ δ ( −→ ˜ φ ) decreases, andthat decreases (resp. increases) as k ˜ φ k H k ′ increases. It also shows that one canchoose T ′ as a continuous function that decreases as ˜ δ ′ ( ˜ φ ) decreases. Proof.
The proof relies upon the following lemma:
Lemma 7.4.
Let < ǫ ≤ ǫ ≪ . Let K ≥ and ¯ δ > . Consider the solutions u q on an interval [0 , T ′′ ] with < T ′′ ≤ . We define h θ ( t ) := θu ( t ) + (1 − θ ) u ( t ) for θ ∈ [0 , and for t ∈ [0 , T ′′ ] . Assume that t ∈ [0 , T ′′ ] : max (cid:16) sup t ∈ [0 ,T ′′ ] E k ′ ( u ( t ) , t ) , sup t ∈ [0 ,T ′′ ] E k ′ ( u ( t ) , t ) (cid:17) ≤ K . Assume also that δ (cid:16) −−−→ h θ ( t ) , t (cid:17) ≥ ¯ δ for all ( t, θ ) ∈ [0 , T ′′ ] × [0 , . Then thereexists C := C (cid:0) ¯ δ, K (cid:1) > such that for all t ∈ [0 , T ′′ ](7.9) dE k ′ ( u ( t ) ,u ( t ) ,t ) dt ( t ) + R T Q ( u ( t ) , t ) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) dx ≤ C E k ′ ( u ( t ) , u ( t ) , t ) + k u ( t ) − u ( t ) k H
72 + E k ′ +3 ( u ( t ) , t ) + k u ( t ) − u ( t ) k H
92 + E k ′ +2 ( u ( t ) , t )+ k u ( t ) − u ( t ) k H
132 + E k ′ +1 ( u ( t ) , t ) + k u ( t ) − u ( t ) k H k ′− E + ( u ( t ) , t )+ k u ( t ) − u ( t ) k H k ′− E + ( u ( t ) , t ) + ǫ (1 + E k ′ +4 ( u ( t ) , t ))+ ǫ (1 + E k ′ +1 ( u ( t ) , t ) + E k ′ +4 ( u ( t ) , t ))+ ǫ k u ( t ) − u ( t ) k H
132 + (1 + E k ′ +2 ( u ( t ) , t )) · ONLINEAR THIRD-ORDER EQUATIONS 33
Remark . The proof of Lemma 7.4 shows that one can choose C to be a contin-uous function that increases as ¯ δ decreases, and increases as K increases.We postpone the proof of Lemma 7.4 to Section 10.Let 0 < c := c (cid:18) ˜ δ ( −→ ˜ φ ) , ˜ δ ′ ( ˜ φ ) , k ˜ φ k H k ′ (cid:19) ≪ < ǫ ≤ ǫ ≤ c , thenthe estimates below hold.We see from Section 11 and Lemma 2.2 that (cid:12)(cid:12)(cid:12) X ( ˜ φ q,k ′ , ˜ φ, , (cid:12)(cid:12)(cid:12) . k ˜ φ q,k ′ − ˜ φ k H
72 + . ǫ k ′− (
72 + ) k ′ q k ˜ φ k H k ′ . From Remark 1.2 we seethat ∂ ω F (cid:18) −→ ˜ φ , (cid:19) has a constant sign. Hence Q ( ˜ φ,
0) = − (cid:12)(cid:12)(cid:12)(cid:12) ∂ ω F ( −→ ˜ φ , (cid:12)(cid:12)(cid:12)(cid:12) * P ( −→ ˜ φ , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ω F ( −→ ˜ φ , (cid:12)(cid:12)(cid:12)(cid:12) + .We see from Section 11 and the above estimates that (cid:12)(cid:12)(cid:12) Y ( ˜ φ q,k ′ , ˜ φ, , (cid:12)(cid:12)(cid:12) . ˜ δ ( ˜ φ ) , k ˜ φ k Hk ′ k ˜ φ q,k ′ − ˜ φ k H
92 + . ˜ δ ( ˜ φ ) , k ˜ φ k Hk ′ ǫ q k ˜ φ k H k ′ . Hence the triangle inequality shows that(7.10) ˜ δ ′ (cid:16) ˜ φ q,k ′ (cid:17) ≥ δ ′ ( ˜ φ )4 , ˜ δ (cid:18) −−→ ˜ φ q,k ′ (cid:19) ≥ δ ( −→ ˜ φ )4 and ˜ φ q,k ′ ∈ P + ,k ′ · Let m ≥ k ′ > . Proposition 6.3, Proposition 5.1, and the Parseval equality showthat there exists 1 & T ′ := T ′ (cid:18) ˜ δ ( −→ ˜ φ ) , ˜ δ ′ ( ˜ φ ) , k ˜ φ k H k ′ (cid:19) > T ǫ q > T ′ and C := C (cid:18) ˜ δ ( −→ ˜ φ ) , k ˜ φ k H k ′ (cid:19) > u with u q , and suchthat(7.11)sup t ∈ [0 ,T ′ ] k u q ( t ) k H m . t ∈ [0 ,T ′ ] E m ( u q ( t ) , t ) . E m ( u q (0) , · If it is necessary then we may WLOG reduce the size of T ′ (with T ′ still a functionof δ ( −→ ˜ φ ), δ ′ ( ˜ φ ), and k ˜ φ k H k ′ ) so that the estimates below hold. We see from Section11 that for t ∈ [0 , T ′ ] k X ( h θ ( t ) , u ( t ) , t, t ) k L ∞ . k ˜ φ k Hk ′ k h θ ( t ) − u ( t ) k H
72 + . k ˜ φ k Hk ′ k u ( t ) − u ( t ) k H
72 + ≪ ˜ δ ( −→ ˜ φ ) , where at the last line we use u ( t ) − u ( t ) = u ( t ) − ˜ φ ,k ′ + ˜ φ ,k ′ − ˜ φ ,k ′ + ˜ φ ,k ′ − u ( t ), the estimates below that can be derived from the arguments in the proofof Proposition 6.3 (see the text below ‘ Claim: δ ( −−→ u ( t ) , t ) & ˜ δ ( −→ ˜ φ ) ’), taking intoaccount (7.11):(7.12) k u q ( t ) − ˜ φ q,k ′ k H
72 + . k ˜ φ q k Hk ′ ǫ q t + t . ǫ q t + t, and the following estimate that follows from Lemma 2.2: k ˜ φ ,k ′ − ˜ φ ,k ′ k H
72 + . k ˜ φ ,k ′ − ˜ φ k H
72 + + k ˜ φ ,k ′ − ˜ φ k H
72 + . ǫ k ′− (
72 + ) k ′ k ˜ φ k H k ′ · Hence the assumptions of Lemma 7.4 hold.We then prove that u q ( t ) ∈ P + ,k ′ , t ∈ [0 , T ′ ]. Indeed by following the arguments toprove the claim ‘ δ ( −−→ u ( t ) , t ) & ˜ δ ( −→ ˜ φ ), t ∈ [0 , T ′ ] ’ and those below ‘Claim: u ( t ) ∈ P + ,k ′ and δ ′ ( u ( t ) , t ) & ˜ δ ′ ( ˜ φ ) , t ∈ [0 , T ′ ] ’ in the proof of Proposition 6.3, and by (7.12),we see that the claim holds.Let m ∈ N . We have E k ′ + m ( u q (0) , . k u q (0) k H k ′ + m . ǫ − mk ′ q (cid:16) k u q (0) k H k ′ (cid:17) Hence(7.13) E k ′ + m ( u q (0) , . ǫ − mk ′ q (cid:16) k ˜ φ k H k ′ (cid:17) · If 0 ≤ m ≤ k ′ then(7.14) E k ′ − m ( u (0) , u (0) , . k u (0) − u (0) k H k ′− m . k u (0) − ˜ φ k H k ′− m + k u (0) − ˜ φ k H k ′− m . ǫ mk ′ k ˜ φ k H k ′ · Let t ∈ [0 , T ′ ]. In the sequel we use Lemma 7.4 combined with Proposition 5.1,Proposition 6.3, (7.13), and (7.14). We have(7.15) dE ( k ′− − ( u ( t ) ,u ( t ) ,t ) dt ( t ) . E ( k ′ − − ( u ( t ) , u ( t ) , t ) (1 + E k ′ ( u (0) , ǫ (1 + E k ′ +1 ( u (0) , ǫ (1 + E k ′ ( u (0) ,
0) + E k ′ +1 ( u (0) , ǫ (1 + E k ′ ( u (0) , E ( k ′ − − ( u ( t ) , u ( t ) , t ) . E ( k ′ − − ( u ( t ) , u ( t ) , t ) + ǫ k ′− k ′ · The Gronwall inequality yields(7.16) E ( k ′ − − ( u ( t ) , u ( t ) , t ) . E ( k ′ − − ( u (0) , u (0) ,
0) + ǫ k ′− k ′ . ǫ k ′ +2 Hence dE k ′ ( u ( t ) ,u ( t ) ,t ) dt ( t ) + R T Q ( u ( t ) , t ) (cid:16) ∂ x ˜ D k ′ − ¯ v ( t ) (cid:17) dx . E k ′ ( u ( t ) , u ( t ) , t ) + ǫ k ′ +2 (1 + E k ′ +3 ( u (0) , ǫ (1 + E k ′ +4 ( u (0) , ǫ (1 + E k ′ +1 ( u (0) ,
0) + E k ′ +4 ( u (0) , ǫ (1 + E k ′ +2 ( u (0) , E k ′ ( u ( t ) , u ( t ) , t ) . E k ′ ( u ( t ) , u ( t ) , t ) + ǫ · ONLINEAR THIRD-ORDER EQUATIONS 35
Hence (7.8) holds. (cid:3) Proof of Theorem
In this section α denotes a function of which the values is allowed to change fromone line to the other one and such that lim x → α ( x ) = 0. Let m ∈ N ∗ .First we prove the local existence and the parabolic smoothing effect.Let T := T (cid:16) ˜ δ ( ~φ ) , ˜ δ ′ ( φ ) , k φ k H k (cid:17) > u m be the solution of (4.2) with ǫ := m , replac-ing φ with φ m := J m ,k φ on h , T m i . The dominated convergence shows thatlim m →∞ k φ m − φ k H k = 0. Hence we see from Proposition 7.3, Proposition 5.1, andthe triangle inequality that for m large and for p ∈ N , T m , T m + p > T and(8.1) t ∈ [0 , T ] : u m + p ( t ) ∈ P + ,k , δ ( −−−−−→ u m + p ( t ) , t ) & ˜ δ ( ~φ ) , δ ′ ( u m + p ( t ) , t ) & ˜ δ ′ ( φ ) , and k u m + p − u m k L ∞ t H k ([0 ,T ]) . (cid:0) m (cid:1) + α (cid:0) m (cid:1) · Hence { u m } is a Cauchy sequence so there exists u ∈ C (cid:0) [0 , T ] , H k (cid:1) such that u m → u . as m → ∞ . Letting p → ∞ and choosing m large enough in (8.1)we see from Section 11 with f := u m ( t ) and g := u that δ ( −−→ u ( t ) , t ) & ˜ δ ( ~φ ), δ ′ ( u ( t ) , t ) & ˜ δ ′ ( φ ), and u ( t ) ∈ P + ,k for t ∈ [0 , T ].We then claim that u ( t ) = u + R t F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17) dt ′ . To this end we write u m = u l,m + u nl,m with u l,m ( t ) := e − m t∂ x φ m and u nl,m ( t ) := − R t e − m ( t − t ′ ) ∂ x F (cid:16) −−−−→ u m ( t ′ ) , t ′ (cid:17) dt ′ for t ∈ [0 , T ]. Indeed k u l,m ( t ) − φ k H k . X + X , with X := (cid:13)(cid:13)(cid:13) ( e − m t∂ x − φ (cid:13)(cid:13)(cid:13) H k , and X := k φ m − φ k H k . Clearly X → m → ∞ . The Parseval equality and the dominated convergence theorem showsthat X → m → ∞ . Hence k u l,m ( t ) − φ k H k → m → ∞ . The Minkowskiinequality shows that (cid:13)(cid:13)(cid:13) u nl,m ( t ) − R t F ( −−→ u ( t ′ ) , t ′ ) dt ′ (cid:13)(cid:13)(cid:13) H k − . Y + Y , with Y := R t (cid:13)(cid:13)(cid:13) ( e − m ( t − t ′ ) ∂ x − F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17)(cid:13)(cid:13)(cid:13) H k − dt ′ and Y := R t (cid:13)(cid:13)(cid:13) F ( −−−−→ u m ( t ′ ) , t ′ ) − F (cid:16) −−→ u ( t ′ ) , t ′ (cid:17)(cid:13)(cid:13)(cid:13) H k − dt ′ .Lemma 3.3 and the dominated convergence theorem show that Y → m → ∞ .We also have Y . sup t ′ ∈ [0 ,T ] k u ( t ′ ) − u m ( t ′ ) k H k . Hence Y → m → ∞ .We then prove the parabolic smoothing effect. Let T > ν >
0. We see fromProposition 7.3 that R ν R T Q ( u m ( t ) , t ) (cid:16) ∂ x ˜ D k − (cid:0) Φ k ( u m ( t ) , t ) ∂ x u m ( t ) − Φ k ( u m + p ( t ) , t ) ∂ x u m + p ( t ) (cid:1)(cid:17) dx dt . (cid:0) m (cid:1) + α (cid:0) m (cid:1) · Observe from Lemma 3.3 that R T k F ( −−→ v ( t ′ ) , t ′ ) k H k − dt ′ < ∞ , so the integral makes sense asa Bochner integral We see from Section 11 with f := u m ( t ) and g := u ( t ) and from (8.1) that thereexists a positive constant c := c (cid:16) ˜ δ ( ~φ ) , ˜ δ ′ ( φ ) (cid:17) such that Q ( u m ( t ) , t ) ≥ c . Hence,using also Proposition 5.1 we see that { u m } is a Cauchy sequence in L t H k +1 ([0 , ν ])so there exists w ∈ L t H k +1 ([0 , ν ]) ⊂ L t H k ([0 , ν ]) such that u m → w as m → ∞ .Since it also converges in L ∞ t H k ([0 , ν ]) ⊂ L t H k ([0 , ν ]), w = u . Hence we can find¯ t ∈ [0 , ν ] such that k u (¯ t ) k H k +1 < ∞ . Recall that there exists c > δ (cid:16) −−−→ u (¯ t ) , ¯ t (cid:17) ≥ c ˜ δ ( ~φ ) and δ ′ ( u (¯ t ) , ¯ t ) ≥ c ˜ δ ′ ( φ ). Hence by adapting slightly thearguments in the proof of the local existence part with the new data u (¯ t ), wesee that u ∈ C (cid:0) [¯ t , T ] , H k +1 (cid:1) . We then apply the procedure again to find ¯ t suchthat ¯ t ≤ ¯ t ≤ ν and such that u ∈ C (cid:0) [¯ t , T ] , H k +2 (cid:1) , taking into account that δ (cid:16) −−−→ u (¯ t ) , ¯ t (cid:17) ≥ c ˜ δ ( ~φ ) and δ ′ ( u (¯ t ) , ¯ t ) ≥ c ˜ δ ′ ( φ ). Iterating m − times there exists0 ≤ ¯ t m ≤ ν such that u ∈ C (cid:0) [¯ t m , T ] , H k + m (cid:1) . So u ∈ C ([ ν, T ] , H p ) for all p ∈ N . Abootstrap argument (that also uses Lemma 3.3) applied to ∂ t u = F ( ~u, t ) show that u ∈ C ∞ ([ ν, T ] × T ). In other words u ∈ C ∞ ((0 , T ] × T ) .We then prove the uniqueness.In the proof we may assume WLOG that u q ( t ) ∈ P + ,k for t ∈ [0 , ˘ T ]. Let T ∗ :=max n t ∈ [0 , ˘ T ] : u ( t ) = u ( t ) o . Assume that T ∗ < ˘ T . We have u ( T ∗ ) = u ( T ∗ ).Let K ∈ R be such that k u ( t ) k H k + k u ( t ) k H k ≤ K for all t ∈ [0 , ˘ T ]. Recall that u q ( t ) = u q ( T ∗ ) + R tT ∗ F (cid:16) −−−→ u q ( t ′ ) , t ′ (cid:17) dt ′ · Then we claim that we can find ˘ T − T ∗ > T ′ := T ′ (cid:16) δ (cid:16) −−−−→ u q ( T ∗ ) , T ∗ (cid:17) , δ ′ ( u q ( T ∗ ) , T ∗ ) , k u ( T ∗ ) k H k ′ (cid:17) > δ (cid:16) −−−→ h θ ( t ) , t (cid:17) & δ (cid:16) −−−−→ u q ( T ∗ ) , T ∗ (cid:17) & δ ( u q ( T ∗ )) for t ∈ [ T ∗ , T ∗ + T ′ ] with h θ ( t ) := θu ( t ) + (1 − θ ) u ( t ). Indeed we see from Section11, the integral equation above, and Lemma 3.3 that if t ∈ [ T ∗ , T ∗ + T ′ ] then k X ( h θ ( t ) , u ( t ) , t, t ) k L ∞ . k h θ ( t ) − u ( t ) k H
72 + . k u ( t ) − u ( t ) k H
72 + . k u ( t ) − u q ( T ∗ ) k H
72 + + k u ( t ) − u q ( T ∗ ) k H
72 + . K ( t − T ∗ ) ≪ δ (cid:16) −−−−→ u q ( T ∗ ) , T ∗ (cid:17) · Hence by looking at the proof of Lemma 7.4 we see that it also works if ǫ = ǫ = 0and if k ′ is replaced with k ′ −
3. Hence letting ǫ = ǫ = 0 in (7.9) and replac-ing ‘ k ′ ’ with ‘ k −
3’ we see that dE k ′− ( u ( t ) ,u ( t ) ,t ) dt ( t ) . E k ′ − ( u ( t ) , u ( t ) , t ) for t ∈ [ T ∗ , T ∗ + T ′ ]. The Gronwall inequality yields E k ′ − ( u ( t ) , u ( t ) , t ) = 0 andconsequently u ( t ) = u ( t ) for t ∈ [ T ∗ , T ∗ + T ′ ].We then prove the continuous dependence.Let T ∗ := sup n t ∈ [0 , ˘ T ] : sup t ′ ∈ [0 ,t ] k u n ( t ′ ) − u ∞ ( t ′ ) k H k → n → ∞ o .Claim: T ∗ = ˘ T . Indeed assume that T ∗ < ˘ T . Then, by slightly adapting thearguments in the section Appendix we infer that δ : [0 , ˘ T ] → R : t → δ ( −−−→ u ∞ ( t ) , t ) ONLINEAR THIRD-ORDER EQUATIONS 37 and that δ ′ : [0 , ˘ T ] → R : t → δ ′ ( u ∞ ( t ) , t ) are continuous: hence there exists a¯ δ > (cid:16) δ (cid:16) −−−→ u ∞ ( t ) , t (cid:17) , δ ′ ( u ∞ ( t ) , t ) (cid:17) ≥ ¯ δ for all t ∈ [0 , ˘ T ]. We also inferthat for all t ∈ [0 , T ∗ ) (cid:16) δ (cid:16) −−−→ u n ( t ) , t (cid:17) , δ ′ ( u n ( t ) , t ) (cid:17) → (cid:16) δ (cid:16) −−−→ u ∞ ( t ) , t (cid:17) , δ ′ ( u ∞ ( t ) , t ) (cid:17) as n → ∞ , since k u n ( t ) − u ∞ ( t ) k H k → n → ∞ . Let q ∈ { n, ∞} . Let 0 ≤ ˜ t ≤ T ∗ be close enough to T ∗ such that all the statements below are true. Unless otherwisestated, let n , m be large enough such that all the statements below are true for all p ∈ N . Let u n,m (resp. u ∞ ,m ) be the solution of (4.2) with ǫ := m , replacing“ ˜ φ ” and “ R t ” with “ u n,m (˜ t ) := J m ,k u n (˜ t )” (resp. “ u ∞ ,m (˜ t ) := J m ,k u ∞ (˜ t )” ) and“ R t ˜ t ” respectively. Let q ∈ { n, ∞} . Let K := sup t ∈ [0 , ˘ T ] k u ∞ ( t ) k H k . We see fromProposition 5.1 and Proposition 7.3 that there exists T ′ := T ′ (cid:0) ¯ δ, K (cid:1) > T > ˜ t + T ′ > T ∗ and(8.2) t ∈ [˜ t, ˜ t + T ′ ] : k u q,m + p k L ∞ t H k ([˜ t, ˜ t + T ′ ]) . ,t ∈ [˜ t, ˜ t + T ′ ] : u q,m ( t ) ∈ P + ,k holds , δ (cid:16) −−−−→ u q,m ( t ) , t (cid:17) & ¯ δ, δ ′ ( u q,m ( t ) , t ) & ¯ δ, and k u q,m + p − u q,m k L ∞ t H k ([˜ t, ˜ t + T ′ ]) . (cid:0) m (cid:1) + α (cid:0) m (cid:1) · Since { u q,m } is a Cauchy sequence there exists v q ∈ C (cid:0) [˜ t, ˜ t + T ′ ] , H k (cid:1) such that u q,m → v q . We now claim that v q = u q . Indeed by using similar arguments asthose that appear in the proof of the local existence part of the theorem, we seethat v q ( t ) = u q (˜ t ) + R t ˜ t F (cid:16) −−−→ v q ( t ′ ) , t ′ (cid:17) dt ′ for t ∈ [˜ t, ˜ t + T ′ ]. Moreover letting p → ∞ in (8.2) we get(8.3) sup t ∈ [˜ t, ˜ t + T ′ ] k v q ( t ) − u q,m ( t ) k H k . (cid:0) m (cid:1) + α (cid:0) m (cid:1) Hence, we infer from Section 11 that v q ( t ) ∈ P + ,k for t ∈ [˜ t, ˜ t + T ′ ]. Hence v q = u q by using the uniqueness part of the theorem.We also see from Proposition 4.1 that lim m →∞ k u n,m − u ∞ ,m k L ∞ t H k ′ ([˜ t, ˜ t + T ′ ]) = 0 as m → ∞ . Let ǫ ′ >
0. We havesup t ∈ [˜ t, ˜ t + T ′ ] k u n ( t ) − u ∞ ( t ) k H k . sup t ∈ [˜ t, ˜ t + T ′ ] k u n ( t ) − u n,m ( t ) k H k + sup t ∈ [˜ t, ˜ t + T ′ ] k u n,m ( t ) − u ∞ ,m ( t ) k H k + sup t ∈ [˜ t, ˜ t + T ′ ] k u ∞ ,m ( t ) − u ∞ ( t ) k H k ≤ ǫ ′ · Hence lim n →∞ k u n ( t ) − u ∞ ( t ) k H k = 0 for ˜ t ≤ t ≤ ˜ t + T ′ . This is a contradiction.Hence T ∗ = ˘ T .Moreover by adapting slightly the argument above we see that lim n →∞ (cid:13)(cid:13)(cid:13) u n ( ˘ T ) − u ∞ ( ˘ T ) (cid:13)(cid:13)(cid:13) H k =0. 9. Proof of Corollary 1.2
We only prove the corollary for φ ∈ P + ,k , since a straightforward modificationof the arguments below would show that the conclusion of the corollary also holdsif φ ∈ P − ,k . Let − T < ˜ t < u ∈ C (cid:0) [ − T, , H k (cid:1) , then we see from Section 11 with ( f, g ) := (cid:0) u (˜ t ) , φ (cid:1) that δ (cid:16) −−→ u (˜ t ) , ˜ t (cid:17) & ˜ δ ( φ ) and inf x ∈ T Q (cid:0) u (˜ t ) , ˜ t (cid:1) ( x ) >
0. Now consider the problem w ( t ) = u (˜ t ) + R t F (cid:16) −−−→ w ( t ′ ) , t ′ (cid:17) dt ′ . We see from Theorem 1.1 and Remark 1.1 that thereexists T ′ > | ˜ t | such that for t ∈ [0 , T ′ ], w ( t ) = u ( t + ˜ t ) and w ∈ C ∞ ((0 , T ′ ] × T ). Inparticular φ ∈ C ∞ ( T ), which is a contradiction.10. Proof of Lemmas
In his section we prove Lemma 6.4 and Lemma 7.4.10.1.
Proof of Lemma 6.4.
An application of the Leibniz rule shows that ∂ x (cid:16) F ( −−→ u ( t ) , t ) (cid:17) = ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) ∂ x u ( t ) + P ( u ( t ) , t ) ∂ x u ( t ) + ˇ Q ( u ( t ) , t ) ∂ x u ( t ) + R ( u ( t ) , t ) , with ˇ Q ( u ( t ) , t ) polynomial in ∂ x u ( t ),..., u ( t ), and derivatives of F evaluated at (cid:16) −−→ u ( t ) , t (cid:17) (resp. R ( u ( t ) , t ) polynomial in ∂ x u ( t ),..., u ( t ), and derivatives of F eval-uated at (cid:16) −−→ u ( t ) , t (cid:17) ).Let v ( t ) := Φ k ′ ( u ( t ) , t ) ∂ x u ( t ). Combining the above equality with (4.1) we seethat ∂ t v + ǫ∂ x v = a ∂ x v + a ∂ x v + ¯ a ∂ x v + ¯ a + ǫ (cid:0) b ∂ x v + ¯ b ∂ x v + ¯ b ∂ x v + ∂ x cv (cid:1) , with c ( t ) := − Φ k ′ ( u ( t ) , t ),¯ a ( t ) :=Φ k ′ ( u ( t ) , t ) R ( u ( t ) , t ) + Φ k ′ ( u ( t ) , t ) ∂ x Φ − k ′ ( u ( t ) , t ) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) + ∂ x Φ − k ′ ( u ( t ) , t ) P ( u ( t ) , t )+ ∂ x Φ − k ′ ( u ( t ) , t ) ˇ Q ( u ( t ) , t ) ! + ∂ t Φ k ′ ( u ( t ) , t ) Φ − k ′ ( u ( t ) , t ) v ( t ) , ( a ( t ) , b ( t )) := (cid:16) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) , − k ′ ( u ( t ) , t ) ∂ x Φ − k ′ ( u ( t ) , t ) (cid:17)(cid:0) a ( t ) , ¯ b ( t ) (cid:1) := (cid:16) k ′ ( u ( t ) , t ) ∂ x Φ − k ′ ( u ( t ) , t ) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) + P ( u ( t ) , t ) , − k ′ ( u ( t ) , t ) ∂ x Φ − ( u ( t ) , t ) (cid:17) , and (cid:0) ¯ a ( t ) , ¯ b ( t ) (cid:1) := Φ k ′ ( u ( t ) , t ) (cid:16) ∂ x Φ − k ′ ( u ( t ) , t ) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) + 2 ∂ x Φ − k ′ ( u ( t ) , t ) P ( u ( t ) , t ) (cid:17) + ˇ Q ( u ( t ) , t ) , − k ′ ( u ( t ) , t ) ∂ x Φ − k ′ ( u ( t ) , t ) ! · Let ¯ P ( u ( t )) := P ( u ( t ) ,t ) ∂ω F ( −−→ u ( t ) ,t ) − (cid:28) P ( u ( t ) ,t ) ∂ω F ( −−→ u ( t ) ,t ) (cid:29) . We now rewrite ∂ t Φ k ′ ( u ( t ) , t ) Φ − k ′ ( u ( t ) , t ) v ( t )in such a way that we can apply Proposition 6.1.Claim: Let S ( u ( t ) , t ) be a polynomial in u ( t ), ∂ x u ( t ),..., ∂ x u ( t ), ∂ x u ( t ), ∂ t u ( t ), ∂ t ∂ x u ( t ), ∂ t ∂ x u ( t ), derivatives of F evaluated at (cid:16) −−→ u ( t ) , t (cid:17) , and (cid:16) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17)(cid:17) − .Let m ≥
0. Then(10.1) k S ( u ( t ) , t ) k H m . hk u ( t ) k H m +6 i· ONLINEAR THIRD-ORDER EQUATIONS 39
Let S (cid:16) −−→ u ( t ) , t (cid:17) be a polynomial in u ( t ),..., ∂ x u ( t ), derivatives of F evaluated at (cid:16) −−→ u ( t ) , t (cid:17) , and (cid:16) ∂ ω F ( −−→ u ( t ) , t ) (cid:17) − . Let m ≥
0. Then(10.2) k S (cid:16) −−→ u ( t ) , t (cid:17) k H m . hk u ( t ) k H m +3 i· Proof.
We only prove (10.1). The proof of (10.2) follows from that of (10.1): there-fore, it is left to the reader.One can find I ,..., I , I , I , J J , J finite subsets of { , , ... } × ... × { , , ... } such that for all ~i := ( i , ..., i ) ∈ I × ... × I and for all ~j := ( j , j , j ) ∈ J × J × J , one can find a ~i,~j ∈ R such that a ~i,~j is a constant multiplied by apower of (cid:16) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17)(cid:17) − that is multiplied by derivatives of F evaluated at (cid:16) −−→ u ( t ) , t (cid:17) such that S ( u ( t ) , t ) = P ~i ∈ I × ... × I ~j ∈ J × J × J a ~i,~j ( ∂ x u ( t )) i ... ( u ( t )) i ( ∂ t ∂ x u ( t )) j ( ∂ t ∂ x u ( t )) j ( ∂ t u ( t )) j If p ∈ { , , } then ∂ t ∂ px u ( t ) = − ǫ∂ px u ( t ) + ∂ px (cid:16) F (cid:16) −−→ u ( t ) , t (cid:17)(cid:17) . An applicationof the Leibnitz rule shows that ∂ px (cid:16) F ( ~f , t ) (cid:17) and ∂ x (cid:16) F ( ~f , t ) (cid:17) are polynomials in ∂ x f ,..., f , and derivatives of F evaluated at ( ~f , t ). Applying (several times) Lemma2.1 and 3.3, and using Proposition 5.1, we see that there exists α ∈ N such that k S ( u ( t ) , t ) k H m . hk u ( t ) k H m +6 ihk u ( t ) k H
132 + i α . hk u ( t ) k H m +6 i· (cid:3) Throughout the sequel of the proof, let S ( u ( t ) , t ) denotes a function that is apolynomial in u ( t ), ∂ x u ( t ),..., ∂ x u ( t ), ∂ t ∂ x u ( t ), ∂ t ∂ x u ( t ), derivatives of F evaluatedat (cid:16) −−→ u ( t ) , t (cid:17) , and (cid:16) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17)(cid:17) − for a function u depending on the variables x and t . For sake of simplicity we allow the value of S ( u ( t ) , t ) to change from oneline to the other one. If several functions share the same properties as those statedjust above appear within the same equation then S ( u ( t ) , t ), S ( u ( t ) , t ),... denotethe first function, the second function,..., respectively. We also allow the value of S ( u ( t ) , t ), S ( u ( t ) , t ),... to vary from one line to the other one.An application of the Leibnitz rule shows that ∂ t P ( u ( t ) , t ) = ¯ X P ( −−→ u ( t ) , t ) ∂ t ∂ x u ( t ) + ¯ Y P ( −−→ u ( t ) , t ) ∂ t ∂ x u ( t ) + S ( u ( t ) , t ) , and ∂ t ∂ ω F ( −−→ u ( t ) , t ) = ¯ Y F ( −−→ u ( t ) , t ) ∂ t ∂ x u ( t ) + S ( u ( t ) , t ) , with ¯ X P ( ~f , t ) := 2 ∂ ω F ( ~f , t ), ¯ Y P ( ~f , t ) := 2 P m =0 ∂ ω ω ω m F ( ~f , t ) ∂ m +1 x f + ∂ ω ω ω − F ( ~f , t )+3 ∂ ω ω F ( ~f , t ), and ¯ Y F ( ~f , t ) := ∂ ω F ( ~f , t ).We have ∂ t ∂ x u ( t ) = − ǫ∂ x u ( t ) + ∂ x (cid:16) F ( −−→ u ( t )) (cid:17) . Observe that ∂ x u ( t ) = ∂ x (cid:0) Φ − k ′ ( u ( t ) , t ) v ( t ) (cid:1) = ∂ x Φ − k ′ ( u ( t ) , t ) v ( t ) + 2 ∂ x Φ − k ′ ( u ( t ) , t ) ∂ x v ( t ) + Φ − k ′ ( u ( t ) , t ) ∂ x v ( t ) · An application of the Leibnitz rule shows that ∂ x (cid:16) F ( −−→ u ( t ) , t ) (cid:17) = ∂ ω F (cid:16) −−→ u ( t ) (cid:17) ∂ x u + S ( u ( t ) , t ) · We also have ∂ t ∂ x u ( t ) = − ǫ∂ x u ( t ) + S ( u ( t ) , t ) with ∂ x u ( t ) = ∂ x (cid:0) Φ − k ′ ( u ( t ) , t ) v ( t ) (cid:1) = ∂ x Φ − k ′ ( u ( t ) , t ) v ( t ) + Φ − k ′ ( u ( t ) , t ) ∂ x v ( t ) · Hence ∂ t ∂ x u ( t ) = A ( t ) ∂ x v ( t ) + B ( t ) ∂ x v ( t ) + C ( t ) with A ( t ) := − ǫ Φ − k ′ ( u ( t ) , t ), B ( t ) := ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) Φ − k ′ ( u ( t ) , t ) − ǫ∂ x Φ − k ′ ( u ( t ) , t ), and C ( t ) := (cid:16) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) ∂ x Φ − k ′ ( u ( t ) , t ) − ǫ∂ x Φ − k ′ ( u ( t ) , t ) (cid:17) v ( t ) + S ( u ( t ) , t ). Wealso have ∂ t ∂ x u ( t ) = B ( t ) ∂ x v ( t )+ C ( t ) with B ( t ) := − ǫ Φ − k ′ ( u ( t ) , t ) and C ( t ) := − ǫ∂ x Φ − k ′ ( u ( t ) , t ) v ( t ) + S ( u ( t ) , t ).Hence ∂ t P ( u ( t ) , t ) = X P ( u ( t ) , t ) ∂ x v ( t ) + Y P ( u ( t ) , t ) ∂ x v ( t ) + Z P ( u ( t ) , t ) and ∂ t ∂ ω F ( −−→ u ( t ) , t ) = Y F ( u ( t ) , t ) ∂ x v ( t ) + Z F ( u ( t ) , t ) , with X P ( u ( t ) , t ) := ¯ X P ( −−→ u ( t )) A ( t ), Y P ( u ( t ) , t ) := ¯ X P ( −−→ u ( t ) , t ) B ( t )+ ¯ Y P ( −−→ u ( t )) B ( t ), Z P ( u ( t ) , t ) := ¯ X P ( u ( t ) , t ) C ( t )+ C ( t ) ¯ Y P ( −−→ u ( t ) , t )+ S ( u ( t ) , t ), Y F ( u ( t ) , t ) := ¯ Y F ( −−→ u ( t )) B ( t ),and Z F ( u ( t ) , t ) := ¯ Y F ( −−→ u ( t ) , t ) C ( t ) + S ( u ( t ) , t ).HenceΦ − k ′ ( u ( t ) , t ) ∂ t Φ k ′ ( u ( t ) , t ) v ( t ) = ˜ a ( t ) ∂ x v ( t ) + ˜ a ( t ) + ǫ (cid:16) ˜ b ( t ) ∂ x v ( t ) + ˜ b ( t ) ∂ x v ( t ) + b ( t ) (cid:17) , with˜ a ( t ) := S (cid:16) −−→ u ( t ) , t (cid:17) Φ − k ′ ( u ( t ) , t ) v ( t ) , ˜ a ( t ) := S ( u ( t ) , t ) + S ( u ( t ) , t ) v ( t ) , ˜ b ( t ) := S (cid:16) −−→ u ( t ) , t (cid:17) Φ − k ′ ( u ( t ) , t ) v ( t ) , ˜ b ( t ) := Φ − k ′ ( u ( t ) , t ) v ( t ) S ( u ( t ) , t ) , and b ( t ) := S ( u ( t ) , t ) + S ( u ( t ) , t ) v ( t ) · Hence ∂ t v + ǫ∂ x v = a ∂ x v + a ∂ x v + a ∂ x v + a + ǫ (cid:0) b ∂ x v + b ∂ x v + b ∂ x v + b + ∂ x cv (cid:1) , with ONLINEAR THIRD-ORDER EQUATIONS 41 a ( t ) := ¯ a ( t ) + ˜ a ( t ) ,a ( t ) := Φ k ′ ( u ( t ) , t ) R ( u ( t ) , t )+Φ k ′ ( u ( t ) , t ) ∂ x Φ − k ′ ( u ( t ) , t ) ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) + ∂ x Φ − k ′ ( u ( t ) , t ) P ( u ( t ) , t )+ ∂ x Φ − k ′ ( u ( t ) , t ) ˇ Q ( u ( t ) , t ) ! v ( t ) + ˜ a ( t ) ,b ( t ) := ¯ b ( t ) + ˜ b ( t ) , and b ( t ) := ¯ b ( t ) + ˜ b ( t ) · We get from (2.2)(10.3) (cid:0) − k ′ (cid:1) ∂ x a ( t ) − a ( t ) = − ∂ ω F (cid:16) −−→ u ( t ) , t (cid:17) h P ( u ( t ) ,t ) ∂ ω F (cid:16) −−→ u ( t ) ,t (cid:17) i· In the sequel we use the above claim (in particular (10.1)), Lemma 2.1, Lemma 3.4,Proposition 5.1, and (3.20). We have(10.4)max ( k a ( t ) k H k ′− , k a ( t ) k H k ′− , k a ( t ) k H k ′− , k a ( t ) k H k ′− ) . E k ′ ( u ( t )) , and P m =1 k a m ( t ) k H ( m + 12 ) + . · We have(10.5) k b ( t ) k H k ′− . E k ′ ( u ( t )) , and P m =1 k b m ( t ) k H ( m + 12 ) + . · We have k b ( t ) k H k ′− . k u ( t ) k H k ′ +1 . We also have k v ( t ) k L . k Φ k ′ ( u ( t )) k L ∞ k ∂ x u ( t ) k L . k Φ k ′ ( u ( t ) , t ) k H
12 + k u ( t ) k H . · Hence k b ( t ) k H k ′− . k v ( t ) k H k ′− + 1 + k u ( t ) k H k ′ +1 . k v ( t ) k ˙ H k ′− + 1 + k u ( t ) k H k ′ +1 · Similarly k b ( t ) k H k ′− . k v ( t ) k ˙ H k ′− + 1 + k u ( t ) k H k ′ +1 · We have k u ( t ) k H k ′ +1 . k u ( t ) k L + k u ( t ) k ˙ H k ′ +1 . k u ( t ) k L + k Φ − k ′ ( u ( t ) , t ) v ( t ) k ˙ H k ′− . k u ( t ) k L + k Φ − k ′ ( u ( t ) , t ) k H k ′− k v ( t ) k H
12 + + k Φ − k ′ ( u ( t ) , t ) k H
12 + k v ( t ) k H k ′− . E k ′ ( u ( t ) , t ) + k v ( t ) k ˙ H k ′− · Hence(10.6)max ( k b ( t ) k H k ′− , k b ( t ) k H k ′− , k b ( t ) k H k ′− ) . E k ′ ( u ( t ) , t ) + k v ( t ) k ˙ H k ′− We have(10.7) k c ( t ) k H k ′− . k u ( t ) k H k ′ +1 . E k ′ ( u ( t ) , t ) + k v ( t ) k ˙ H k ′− , and k c ( t ) k H
92 + . · (10.5), the interpolation inequality k v ( t ) k ˙ H k ′− . k v ( t ) k ˙ H k ′− k v ( t ) k ˙ H k ′− , and theYoung inequality ab ≤ a + b yield(10.8) ǫ (cid:16) k b ( t ) k H
32 + + k b ( t ) k H
12 + (cid:17) k ∂ x ˜ D k ′ − v ( t ) k L . ǫ k ∂ k ′ − x v ( t ) k L + E k ′ ( u ( t ) , t )Hence by application of Proposition 6.1 and Proposition 6.2 we get (6.9), takinginto account that all the terms ǫ k ∂ k ′ − x v ( t ) k L that appear can be absorbed by theterm 2 ǫ k ∂ k ′ − x v ( t ) k L of (6.2).10.2. Proof of Lemma 7.4. If G is a function that depends on f and t thenwe let △ G ( t ) := G ( u ( t ) , t ) − G ( u ( t ) , t ). In particular if G is a function thatdepends on ~f and t then we let △ G ( t ) = G ( −−−→ u ( t ) , t ) − G ( −−−→ u ( t ) , t ). If s ∈ R is aparameter and G s is a function of f that depends on a parameter s then we let △ G s ( t ) := G s ( u ( t ) , t ) − G s ( u ( t ) , t ). If y q,s is a mathematical expression depend-ing on q and s then △ y q,s := y ,s − y ,s . Let α be a constant that is allowed tochange from one line to the other one and such that all the statements and esti-mates below are true. Let ¯ u := u − u and ¯ v := v − v .The proof relies upon the result below.10.2.1. One result.
Result 10.1.
Let ≤ r ≤ k ′ − . Let S ( u q ( t )) be a polynomial in u q ( t ) , ∂ x u q ( t ) ..., ∂ x u q ( t ) , ∂ t ∂ x u q ( t ) , ∂ t ∂ x u q ( t ) , ∂ t u q ( t ) , derivatives of F evaluated at −−−→ u q ( t ) , and (cid:16) ∂ ω F ( −−−→ u q ( t )) (cid:17) − .Then (10.9) k△ S ( t ) k H r . k ¯ u ( t ) k H r +6 · Let S ( −−−→ u q ( t ) , t ) be a polynomial in u q ( t ) ,..., ∂ x u q ( t ) , derivatives of F evaluated at −−−→ u q ( t ) , and (cid:16) ∂ ω F ( −−−→ u q ( t )) (cid:17) − . Then (10.10) k△ S ( t ) k H r . k ¯ u ( t ) k H r +3 · We postpone the proof of this result to the last subsection.10.2.2.
The proof.
Recall (see the proof of Lemma 6.4) that ∂ t v q + ǫ∂ x v q = a ,q ∂ x v q + a ,q ∂ x v q + a ,q ∂ x v q + a ,q + ǫ q (cid:0) b ,q ∂ x v q + b ,q ∂ x v q + b ,q ∂ x v q + b ,q + ∂ x c q v q (cid:1) , with ONLINEAR THIRD-ORDER EQUATIONS 43 ( a ,q ( t ) , b ,q ( t )) := (cid:16) ∂ ω F (cid:16) −−−→ u q ( t ) , t (cid:17) , − k ′ ( u q ( t ) , t ) ∂ x Φ − k ′ ( u q ( t ) , t ) (cid:17) , ( a ,q ( t ) , b ,q ( t )) := (cid:16) k ′ ( u q ( t ) , t ) ∂ x Φ − k ′ ( u q ( t ) , t ) ∂ ω F (cid:16) −−−→ u q ( t ) , t (cid:17) + P ( u q ( t ) , t ) , ¯ b ,q ( t ) + ˜ b ,q ( t ) (cid:17) , ( a ,q ( t ) , b ,q ( t )) := (cid:16) ¯ a ,q ( t ) + ˜ a ,q ( t ) , ¯ b ,q ( t ) + ˜ b ,q ( t ) (cid:17) ,a ,q ( t ) := Φ k ′ ( u q ( t )) R ( u q ( t ) , t )+Φ k ′ ( u q ( t ) , t ) ∂ x Φ − k ′ ( u q ( t ) , t ) ∂ ω F (cid:16) −−−→ u q ( t ) , t (cid:17) + ∂ x Φ − k ′ ( u q ( t ) , t ) P ( u q ( t ) , t )+ ∂ x Φ − k ′ ( u q ( t ) , t ) ˇ Q ( u q ( t ) , t ) ! v q ( t )+˜ a ,q ( t ) ,b ,q ( t ) := S ( u q ( t ) , t ) + S ( u q ( t ) , t ) v q ( t ) , and c q ( t ) := − Φ k ′ ( u q ( t ) , t ). Here ˜ a ,q ( t ) := S ( −−−→ u q ( t ) , t )Φ − k ′ ( u q ( t ) , t ) v q ( t ), ˜ a ,q ( t ) := S ( u q ( t ) , t ) + S ( u q ( t ) , t ) v q ( t ),¯ a ,q ( t ) := Φ k ′ ( u q ( t ) , t ) (cid:16) ∂ x Φ − k ′ ( u q ( t ) , t ) ∂ ω F (cid:16) −−−→ u q ( t ) , t (cid:17) + 2 ∂ x Φ − k ′ ( u q ( t ) , t ) P ( u q ( t ) , t ) (cid:17) +ˇ Q ( u q ( t ) , t ), ˜ b ,q ( t ) := S (cid:16) −−−→ u q ( t ) , t (cid:17) Φ − k ′ ( u q ( t ) , t ) v q ( t ), ¯ b ,q ( t ) := − k ′ ( u q ( t ) , t ) ∂ x Φ − k ′ ( u q ( t ) , t ),˜ b ,q ( t ) := Φ − k ′ ( u q ( t ) , t ) v q ( t ) S ( u q ( t )), and ¯ b ,q ( t ) := − k ′ ( u q ( t ) , t ) ∂ x Φ − k ′ ( u q ( t ) , t ).We would like to apply Proposition 7.1 and Proposition 7.2. To this end we esti-mate some norms.Let r ∈ { k ′ − , , + } . The triangle inequality, Lemma 3.4, and Proposition 5.1yield(10.11) k△ c ( t ) k H r . P q =1 k u q ( t ) k H r +3 . P q =1 E k ′ +1 ( u q ( t ) , t )Let r ∈ { k ′ − , + , } . Lemma 3.3 yields(10.12) k△ a ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + ) · Lemma 2.1 yields k△ a ( t ) k H r . P i =1 X i + P i =1 Y i + P i =1 Z i + k△ P ( t ) k H r with X := k△ Φ k ′ ( t ) k H r k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + ,X := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + ,X := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17) k H r ,Y := k Φ k ′ ( u ( t ) , t ) k H r k△ ∂ x Φ − k ′ ( t ) k H
12 + k ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17) k H
12 + ,Y := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t ) k H r k ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17) k H
12 + ,Y := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t ) k H
12 + k ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17) k H r ,Z := k Φ k ′ ( u ( t ) , t ) k H r k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k△ ∂ ω F ( t ) k H
12 + ,Z := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k△ ∂ ω F ( t ) k H
12 + , and Z := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k△ ∂ ω F ( t ) k H r · We have X . k u ( t ) − u ( t ) k H ¯ r (cid:16) k u ( t ) k H
92 + (cid:17) (cid:16) k u ( t ) k H
72 + (cid:17) . k u ( t ) − u ( t ) k H ¯ r ,X . k u ( t ) − u ( t ) k H
92 + (1 + k u ( t ) k H r +4 ) (cid:16) k u ( t ) k H
72 + (cid:17) . k u ( t ) − u ( t ) k H
92 + , and X . k u ( t ) − u ( t ) k H
92 + (cid:16) k u ( t ) k + (cid:17) (1 + k u ( t ) k H r +3 ) . k u ( t ) − u ( t ) k H
92 + · We also have Y + Y . k u ( t ) − u ( t ) k H
92 + , Y . k u ( t ) − u ( t ) k H r +1 , Z + Z . k u ( t ) − u ( t ) k H
72 + , and Z . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + ). Hence, taking also intoaccount (3.22), we get(10.13) k△ a ( t ) k H r . k u ( t ) − u ( t ) k H max (
92 + ,r +4 ,r +1 ) · We have k△ ¯ a ( t ) k H r . P q ∈{ , , } X q + X ′ q + Y q + Y ′ q + Z q + Z ′ q with X := k△ Φ k ′ ( t ) k H r k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + , X := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + , X := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k ∂ ω F ( −−−→ u ( t ) , t ) k H r , Y := k Φ k ′ ( u ( t ) , t ) k H r k△ ∂ x Φ − k ′ ( t )) k H
12 + k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + , Y := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t )) k H r k ∂ ω F ( −−−→ u ( t ) , t ) k H
12 + , Y := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t )) k H
12 + k ∂ ω F ( −−−→ u ( t ) , t ) k H r , Z := k Φ k ′ ( u ( t ) , t ) k H r k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k△ ∂ ω F ( t ) k H
12 + , Z := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k△ ∂ ω F ( t ) k H
12 + , Z := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k△ ∂ ω F ( t ) k H r , X ′ := k△ Φ k ′ ( t ) k H r k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k P ( u ( t )) k H
12 + , X ′ := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k P ( u ( t )) k H
12 + , X ′ := k△ Φ k ′ ( t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k P ( u ( t )) k H r , Y ′ := k Φ k ′ ( u ( t ) , t ) k H r k△ ∂ x Φ − k ′ ( t ) k H
12 + k P ( u ( t ) , t ) k H
12 + , Y ′ := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t ) k H r k P ( u ( t ) , t ) k H
12 + , Y ′ := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ ∂ x Φ − k ′ ( t ) k H
12 + k P ( u ( t ) , t ) k H r , Z ′ := k Φ k ′ ( u ( t ) , t ) k H r k ∂ x Φ − k ′ ( u ( t )) k H
12 + k△ P ( t ) k H
12 + , Z ′ := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H r k△ P ( t ) k H
12 + , Z ′ := k Φ k ′ ( u ( t ) , t ) k H
12 + k ∂ x Φ − k ′ ( u ( t ) , t ) k H
12 + k△ P ( t ) k H r , and X ′′ := k△ Q ( t ) k H r .Then taking into account (10.9) and proceeding similarly as previously we canestimate X q , X ′ q , Y q , Y ′ q , Z q , Z ′ q , and X ′′ . We get(10.14) k△ ¯ a ( t ) k H r . k u ( t ) − u ( t ) k H r +6 ONLINEAR THIRD-ORDER EQUATIONS 45
We have k△ ˜ a ( t ) k H r . P q ∈{ , , } X q + Y q + Z q with X := k△ S ( t ) k H r k Φ − k ′ ( u ( t ) , t ) k H
12 + k v ( t ) k H
12 + , X := k△ S ( t ) k H
12 + k Φ − k ′ ( u ( t ) , t ) k H r k v ( t ) k H
12 + , X := k△ S ( t ) k H
12 + k Φ − k ′ ( u ( t ) , t ) k H
12 + k v ( t ) k H r , Y := k S ( u ( t )) k H r k△ Φ − k ′ ( t ) k H
12 + k v ( t ) k H
12 + , Y := k S ( u ( t )) k H
12 + k△ Φ − k ′ ( t ) k H r k v ( t ) k H
12 + , Y := k S ( u ( t )) k H
12 + k△ Φ − k ′ ( t ) k H
12 + k v ( t ) k H r , Z := k S ( u ( t )) k H r k Φ − k ′ ( u ( t ) , t ) k H
12 + k△ v ( t ) k H
12 + , Z := k S ( u ( t )) k H
12 + k Φ − k ′ ( u ( t ) , t ) k H r k△ v ( t ) k H
12 + ,and Z := k S ( u ( t )) k H
12 + k Φ − k ′ ( u ( t ) , t ) k H
12 + k△ v ( t ) k H r .Again we proceed similarly as previously to estimate X j , Y j , and Z j . We also useProposition 5.1. We get(10.15) k△ ˜ a ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + )Hence we see from the above estimates that(10.16) k△ a k H r . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + )We have k△ a ( t ) k H r . X ,a + X ,a + Y ,a + Y ,b + P l ∈{ a,b,c, } P j =1 W ~i j ,l + X ~i j ,l + Y ~i j ,l + Z ~i j ,l + k△ ˜ a k H r , Here X ,a , Y ,a , X ,a , and Y ,a are defined as follows: X ,a := k△ Φ k ′ ( t ) k H r k R ( u ( t )) k H
12 + , X ,a := k△ Φ k ′ ( t ) k H
12 + k R ( u ( t )) k H r ,Y ,a := k Φ k ′ ( u ( t ) , t ) k H r k△ R ( t ) k H
12 + , and Y ,a := k Φ k ′ ( u ( t ) , t ) k H
12 + k△ R ( t ) k H r , Here ~i j := ( i , i , i , i ) ∈ R with i k := + if k = j and i j := r . Here W ~i j ,a := k△ Φ k ′ ( t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k ∂ ω F ( −−−→ u ( t ) , t ) k H i k v ( t ) k H i ,X ~i j ,a := k Φ k ′ ( u ( t )) k H i k△ ∂ x Φ − k ′ ( t ) k H i k ∂ ω F ( −−−→ u ( t ) , t ) k H i k v ( t ) k H i ,Y ~i j ,a := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k△ ∂ ω F ( t ) k H i k v ( t ) k i ,Z ~i j ,a := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k ∂ ω F ( u ( t )) k H i k△ v ( t ) k H i ,W ~i j ,b := k△ Φ k ′ ( t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k P ( u ( t ) , t ) k H i k v ( t ) k H i X ~i j ,b := k Φ k ′ ( u ( t ) , t ) k H i k△ ∂ x Φ − k ′ ( t ) k H i k P ( u ( t ) , t ) k H i k v ( t ) k H i ,Y ~i j ,b := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k△ P ( t ) k H i k v ( t ) k H i ,Z ~i j ,b := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k P ( u ( t ) , t ) k H i k△ v ( t ) k H i ,W ~i j ,c := k△ Φ k ′ ( t ) k H i k ∂ x Φ − k ′ ( u ( t ) , t ) k H i k ˇ Q ( u ( t ) , t ) k H i k v ( t ) k H i ,X ~i j ,c := k Φ k ′ ( u ( t ) , t ) k H i k△ ∂ x Φ − k ′ ( t ) k H i k ˇ Q ( u ( t ) , t ) k H i k v ( t ) k H i Y ~i j ,c := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t )) k H i k△ ˇ Q ( t ) k H i k v ( t ) k H i , and Z ~i j ,c := k Φ k ′ ( u ( t ) , t ) k H i k ∂ x Φ − k ′ ( u ( t )) k H i k ˇ Q ( u ( t ) , t ) k H i k△ v ( t ) k H i · By using also (10.1), we get X ,a + Y ,a + X ,b + Y ,b . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + ), P l ∈{ a,b,c } P j =1 W ~i j ,l . k u ( t ) − u ( t ) k H max ( ¯ r,
92 + ) , P l ∈{ a,b,c } P j =1 X ~i j ,l . k u ( t ) − u ( t ) k H max ( r +3 ,
132 + ,r +6 ) , P l ∈{ a,b,c } P j =1 Y ~i j ,l . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + ) , and P l ∈{ a,b,c } P j =1 Z ~i j ,l . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + ) · We also have k△ ˜ a ( t ) k H r . k△ S ( t ) k H r + k△ S ( t ) k H r k v ( t ) k H
12 + + k△ S ( t ) k H
12 + k v ( t ) k H r + k S ( u ( t )) k H r k ¯ v ( t ) k H
12 + + k S ( u ( t )) k H
12 + k ¯ v ( t ) k H r · Hence k△ ˜ a ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + ) , and k△ a ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +6 ,r +3 ,
132 + ) · Hence we see from the above estimates that P m =1 k△ a m ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + . k u ( t ) − u ( t ) k H k ′− E + ( u ( t ) , t ) + k u ( t ) − u ( t ) k H k ′− E + ( u ( t ) , t ) + E k ′ ( u ( t ) , u ( t ) , t ) , P m =1 k△ a m ( t ) k H
12 + k v ( t ) k H m + k ′− . k u ( t ) − u ( t ) k H
72 + E k ′ +3 ( u ( t ) , t ) + k u ( t ) − u ( t ) k H
92 + E k ′ +2 ( u ( t ) , t )+ k u ( t ) − u ( t ) k H
132 + E k ′ +1 ( u ( t ) , t ) , and k△ a ( t ) k H k ′− . E k ′ ( u ( t ) , u ( t ) , t ) · ONLINEAR THIRD-ORDER EQUATIONS 47
In the sequel we use implicitly Lemma 2.1, Proposition 5.1, Lemma 3.4, and theclaim in the proof of Lemma 6.4 in order to prove the estimates below. We have ǫ k c ( t ) k H k ′− k v ( t ) k H
12 + . ǫ (1 + E k ′ +1 ( u ( t ) , t )) , andWe have ǫ (cid:16) k c ( t ) k H
92 + k v ( t ) k H k ′− + k v ( t ) k H k ′− (cid:17) . ǫ E k ′ +4 ( u ( t ) , t ) · We have ǫ
22 2 P j =1 3 P m =1 k b m,j ( t ) k H k ′− k v ( t ) k H ( m + 12 ) + . ǫ E + ( u ( t ) , t ) · We have ǫ
22 2 P j =1 3 P m =1 k b m,j ( t ) k H
12 + k v ( t ) k H k ′− m . ǫ E k ′ +3 ( u ( t ) , t )We have ǫ (cid:18) P m =1 k b m, ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k ¯ v ( t ) k H k ′− . ǫ E k ′ ( u ( t ) , u ( t ) , t ) · We get from the triangle inequality ǫ (cid:16) k△ c ( t ) k H k ′− k v ( t ) k H
12 + + k△ c ( t ) k H
92 + k v ( t ) k H k ′− (cid:17) . ǫ (cid:16) k u ( t ) − u ( t ) k H k ′ +1 + k u ( t ) − u ( t ) k H
152 + E k ′ +4 ( u ( t ) , t ) (cid:17) . ǫ ( E k ′ +1 ( u ( t ) , t ) + E k ′ +4 ( u ( t ) , t ))We have ǫ k b , ( t ) k H k ′− + ǫ k b , ( t ) k H k ′− . ǫ (cid:16) E k ′ ( u ( t ) , t ) (cid:17) + ǫ (cid:16) E k ′ ( u ( t ) , t ) (cid:17) . ǫ · We have P m =2 k a m, ( t ) k H ( m − ) + k ∂ x ¯ v ( t ) k L . E k ′ ( u ( t ) , u ( t ) , t ) · Then we use similar estimates that appear between below (10.3) and ‘ by the term2 ǫ k ∂ k ′ − x v ( t ) k L of (6.2)’ below (10.8). We have (cid:0) − k ′ (cid:1) ∂ x a , ( t ) − a , ( t ) = − ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17) h P ( u ( t ) ,t ) ∂ω F ( −−−→ u t ) ,t ) i , (cid:18) P m =1 k a m, ( t ) k H ( m + 12 ) + (cid:19) k ¯ v ( t ) k H k ′− . E k ′ ( u ( t ) , u ( t ) , t ) , P m =2 k a m, ( t ) k H k ′− m k ¯ v ( t ) k H
12 + + k a , ( t ) k H k ′− k ¯ v ( t ) k H
12 + . E k ′ ( u ( t ) , u ( t ) , t ) ,ǫ (cid:16) k b , ( t ) k H
32 + + k b , ( t ) k H
12 + (cid:17) k ∂ x ˜ D k ′ − ¯ v ( t ) k L . ǫ k ∂ k ′ − x ¯ v ( t ) k L + E k ′ ( u ( t ) , u ( t ) , t ) , ǫ (cid:18) P m =1 k b m, ( t ) k H k ′− m + k c ( t ) k H k ′− (cid:19) k ¯ v ( t ) k H
12 + . ǫ (cid:16) E k ′ ( u ( t ) , t ) + k v ( t ) k H k ′− (cid:17) k u ( t ) − u ( t ) k H
132 + | . ǫ (1 + E k ′ +2 ( u ( t ) , t )) , P m =2 k a m, ( t ) k H k ′− m k ¯ v ( t ) k H
12 + . E k ′ ( u ( t ) , u ( t ) , t ) , k a , k H k ′− k ¯ v ( t ) k H
32 + . E k ′ ( u ( t ) , u ( t ) , t ) , and ǫ (cid:18) P m =1 k b m, ( t ) k H ( m + 12 ) + + k c ( t ) k H
92 + (cid:19) k ¯ v ( t ) k H k ′− . ǫ E k ′ ( u ( t ) , u ( t ) , t ) · Observe again that the term ǫ (cid:13)(cid:13)(cid:13) ∂ k ′ − x ¯ v ( t ) (cid:13)(cid:13)(cid:13) L can be absorbed by the term 2 ǫ (cid:13)(cid:13)(cid:13) ∂ k ′ − x ¯ v ( t ) (cid:13)(cid:13)(cid:13) L of (7.2). Combining all the estimates above, we see from Proposition 7.1 and Propo-sition 7.2 that Lemma 7.4 holds.10.2.3. Proof of Result 10.1.
We only prove (10.9). The proof of (10.10) is similarand is left as an exercise to the reader.One can find I ,..., I , J , J , and J finite subsets of { , , ... } × ... × { , , ... } suchthat for all ~i := ( i , ..., i ) ∈ I × ... × I and for all ~j := ( j , j , j ) ∈ J × J × J ,one can find a ~i,~j (cid:16) −−−→ u q ( t ) , t (cid:17) ∈ R such that a ~i,~j (cid:16) −−−→ u q ( t ) , t (cid:17) is a constant multiplied bya finite product of derivatives of F and the inverse of ∂ ω F evaluated at (cid:16) −−−→ u q ( t ) , t (cid:17) and S ( u q ( t ) , t ) = P ~i ∈ I × ... × I ~j ∈ J × J × J a ~i,~j (cid:16) −−−→ u q ( t ) , t (cid:17) ( ∂ x u q ( t )) i ... ( u q ( t )) i ( ∂ t ∂ x u q ( t )) j ( ∂ t ∂ x u q ( t )) j ( ∂ t u q ( t )) j · We prove two claims.Claim: Let p ∈ { , .., } . Let β ∈ { , , ... } . Then(10.17) k△ ( ∂ px u ) β ( t ) k H r . k u ( t ) − u ( t ) k H max ( r + p, ( p + 12 ) + ) Proof. If β ∈ { , , ... } elementary algebraic identities show that there exists apolynomial P in ∂ px u ( t ),..., u ( t ), ∂ px u ( t ),..., u ( t ) such that △ ( ∂ px u ) β ( t ) := P ( ∂ px u ( t ) , ..., u ( t ) , ∂ px u ( t ) , ..., u ( t )) △ ∂ px u ( t ) · Taking into account Proposition 5.1, applying several times Lemma 2.1 and pro-ceeding similarly as the end of the proof of (10.1) we see that there exists α > k P ( ∂ px u ( t ) , .., u ( t ) , ∂ px u ( t ) , ..., u ( t )) k H r . ( k u ( t ) k H r + p + k u ( t ) k H r + p ) hk u ( t ) k H ( p + 12 ) + i α hk u ( t ) k H ( p + 12 ) + i α . k u ( t ) k H r + p + k u ( t ) k H r + p Hence
ONLINEAR THIRD-ORDER EQUATIONS 49 k△ ( ∂ px u ( t )) β k H r . k P ( ∂ px u ( t ) , ..., u ( t ) , ∂ px u ( t ) , ..., u ( t )) k H r k△ ∂ px u ( t ) k H
12 + + k P ( ∂ px u ( t ) , ..., u ( t ) , ∂ px u ( t ) , ..., u ( t )) k H
12 + k△ ∂ px u ( t ) k H r . ( k u ( t ) k H r + p + k u ( t ) k H r + p ) k△ ∂ px u ( t ) k H
12 + + (cid:16) k u ( t ) k H ( p + 12 ) + + k u ( t ) k H ( p + 12 ) + (cid:17) k△ ∂ px u ( t ) k H r . k u ( t ) − u ( t ) k H max ( r + p, ( p + 12 ) + ) (cid:3) Claim: Let m ∈ { , , } . Let β ∈ { , , ... } . Then we have(10.18) k△ ( ∂ t ∂ mx u ( t )) β k H r . ǫ k u ( t ) − u ( t ) k H max ((
92 + m ) + ,r + m +4 )+( ǫ − ǫ ) k u ( t ) k H max ((
92 + m ) + ,r + m +4 ) + k u ( t ) − u ( t ) k H max ( r + m +3 , (
72 + m ) + ) Proof. (4.1) yields △ ∂ t ∂ mx u ( t ) = − ǫ △ ∂ mx u ( t ) − ( ǫ − ǫ ) ∂ mx u ( t )+ △ ∂ mx F (cid:16) −−→ u ( t ) , t (cid:17) .Lemma 3.3 and Proposition 5.1 yield the claim with β = 1.Let β >
1. Elementary algebraic properties show that △ ( ∂ t ∂ mx u ( t )) β = P ( ∂ t ∂ mx u ( t ) , ..., u ( t ) , ∂ t ∂ mx u ( t ) , .., u ( t )) △ ∂ t ∂ mx u ( t ) , with P a polynomial in ∂ t ∂ mx u ( t ),..., u ( t ), ∂ t ∂ mx u ( t ),..., and u ( t ). Proceedingsimilarly as in the proof of (10.1) we get (cid:13)(cid:13) P (cid:0) ∂ t ∂ x u ( t ) , ..., u ( t ) , ∂ t ∂ x u ( t ) , .., u ( t ) (cid:1)(cid:13)(cid:13) H r . hk u ( t ) k H r +6 i + hk u ( t ) k H r +6 i· Hence by application of Lemma 2.1 k△ ( ∂ t ∂ mx u ( t )) β k H r . (cid:13)(cid:13) P (cid:0) ∂ t ∂ x u ( t ) , ..., u ( t ) , ∂ t ∂ x u ( t ) , .., u ( t ) (cid:1)(cid:13)(cid:13) H r k△ ∂ t ∂ mx u ( t ) k H
12 + + (cid:13)(cid:13) P (cid:0) ∂ t ∂ x u ( t ) , ..., u ( t ) , ∂ t ∂ x u ( t ) , .., u ( t ) (cid:1)(cid:13)(cid:13) H
12 + k△ ∂ t ∂ mx u ( t ) k H r . R.H.S of (10 . · (cid:3) We continue the proof of (10.9). We have △ S ( t ) := △ S ( t ) + P m =0 △ S m ( t ) + P m =0 △ ¯ S m ( t )(10.19) △ S ( t ) := P ~i ∈ I × ... × I ~j ∈ J × J × J △ a ~i,~j ( t )( ∂ x u ( t )) i ... ( u ( t )) i ( ∂ t ∂ x u ( t )) j ( ∂ t ∂ x u ( t )) j ( ∂ t u ( t )) j , In (10.20) we use the following convention: we ignore the terms that do not make sense forsome values of m ( such as (cid:16) ∂ − mx u ( t ) (cid:17) i − m if m = 0 ) (10.20) △ S m ( t ) := P ~i ∈ I × ... × I ~j ∈ J × J × J " a ~i,~j (cid:16) −−−→ u ( t ) , t (cid:17) ( ∂ x u ( t )) i ... ( ∂ − mx u ( t )) i − m △ (cid:0) ∂ − mx u ( t ) (cid:1) i − m ( ∂ − mx u ( t )) i − m ... ( u ( t )) i ( ∂ t ∂ x u ( t )) j ( ∂ t ∂ x u ( t )) j ( ∂ t u ( t )) j , and △ ¯ S m ( t ) := P ~i ∈ I × ... × I ~j ∈ J × J × J Q ~i,~j ( t ) c m,~j ( t ) , with Q ~i,~j ( t ) := a ~i,~j (cid:16) −−−→ u ( t ) , t (cid:17) ( ∂ x u ( t )) i ... ( u ( t )) i ,c ,~j ( t ) := △ ( ∂ t ∂ x u ( t )) j ( ∂ t ∂ x u ( t )) j ( ∂ t u ( t )) j ,c ,~j ( t ) := ( ∂ t ∂ x u ( t )) j △ ( ∂ t ∂ x u ( t )) j ( ∂ t u ( t )) j , and c ,~j ( t ) := ( ∂ t ∂ x u ( t )) j ( ∂ t ∂ x u ( t )) j △ ( ∂ t u ( t )) j · Hence △ S ( t ), △ S m ( t ), and △ ¯ S m ( t ) are finite sums made of terms that are uniquelycharacterized by ~i and ~j .In the sequel let α be a constant of which the value is allowed to change fromone line to the other one and even within the same line and such all the statementsbelow are true.Claim: Let r such that 0 ≤ r ≤ k ′ −
3. Then(10.21) k a ~i,~j (cid:16) −−−→ u q ( t ) , t (cid:17) k H r . , and(10.22) k△ a ~i,~j ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + ) Proof.
We can write a ~i,~j (cid:16) −−−→ u q ( t ) (cid:17) = R ( −−−→ u q ( t ) ,t ) (cid:16) ∂ ω F ( −−−→ u q ( t )) ,t (cid:17) α with R ( −−−→ u q ( t ) , t ) a product ofderivatives of F evaluated at ( −→ u q ( t ) , t ). Taking into account Proposition 5.1 we get(10.21) by applying Lemma 3.2, Lemma 3.3, and Lemma 2.1 more than once ifnecessary. We have △ a ~i,~j ( t ) = X + Y with X := R ( −−−→ u ( t ) ,t ) − R ( −−−→ u ( t ) ,t ) (cid:16) ∂ ω F ( −−−→ u ( t ) ,t ) (cid:17) α , and Y := R ( −−−→ u ( t ) ,t ) (cid:16)(cid:16) ∂ ω F ( −−−→ u ( t ) ,t ) (cid:17) α − (cid:16) ∂ ω F ( −−−→ u ( t ) ,t ) (cid:17) α (cid:17)(cid:16) ∂ ω F ( −−−→ u ( t ) ,t ) (cid:17) α (cid:16) ∂ ω F ( −−−→ u ( t ) ,t ) (cid:17) α We have k X k H r . k R ( −−−→ u ( t ) , t ) − R ( −−−→ u ( t ) , t ) k H max ( r,
12 + ) . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + ) · We also have, taking into account Lemma 2.1
ONLINEAR THIRD-ORDER EQUATIONS 51 k Y k H r . k R ( −−−→ u ( t ) , t ) (cid:16)(cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α − (cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α (cid:17) k H max ( r,
12 + ) . k R ( −−−→ u ( t ) , t ) k H max ( r,
12 + ) (cid:13)(cid:13)(cid:13)(cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α − (cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α (cid:13)(cid:13)(cid:13) H
12 + + k R ( −−−→ u ( t ) , t ) k H
12 + (cid:13)(cid:13)(cid:13)(cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α − (cid:16) ∂ ω F ( −−−→ u ( t ) , t ) (cid:17) α (cid:13)(cid:13)(cid:13) H max ( r,
12 + ) . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + )Hence (10.22) holds. (cid:3)
Let Z ~i,~j ( t ) be the term of the sum △ S ( t ) determined by ~i and ~j . We see fromLemma 2.1 that(10.23) k Z ~i,~j ( t ) k H r . (cid:13)(cid:13)(cid:13) △ a ~i,~j ( t ) (cid:13)(cid:13)(cid:13) H r hk u ( t ) k H
132 + i α sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α + (cid:13)(cid:13)(cid:13) △ a ~i,~j ( t ) (cid:13)(cid:13)(cid:13) H
12 + k u ( t ) k H r hk u ( t ) k H
132 + i α sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α + (cid:13)(cid:13)(cid:13) △ a ~i,~j ( t ) (cid:13)(cid:13)(cid:13) H
12 + hk u ( t ) k H
132 + i α sup m ∈{ , , } k ∂ t ∂ mx u ( t ) k H r sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α Hence we see from (10.1) and (10.22) that k△ S ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +3 ,
72 + )Let Z ~i,~j ( t ) be the term of the sum △ S m ( t ) determined by ~i and ~j . We have k Z ~i,~j ( t ) k H r . k a ~i,~j ( −−−→ u ( t ) , t ) k H r hk u ( t ) k H
132 + i α hk u ( t ) k H
132 + i α sup m ∈ [0 ,.., k△ ( ∂ − mx u ( t )) i − m k H
12 + sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α + k a ~i,~j ( −−−→ u ( t ) , t ) k H
12 + hk u ( t ) k H
132 + i α sup m ∈ [0 ,.., k△ ( ∂ − mx u ( t )) i − m k H r sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α + k a ~i,~j ( −−−→ u ( t ) , t ) k H
12 + P q ∈{ , } k u q ( t ) k H r hk u ( t ) k H
132 + i α hk u ( t ) k H
132 + i α sup m ∈ [0 ,.., k△ ( ∂ − mx u ( t )) i − m k H
12 + sup m ∈{ , , } hk ∂ t ∂ mx u ( t ) k H
12 + i α + k a ~i,~j ( −−−→ u ( t ) , t ) k H
12 + hk u ( t ) k H
132 + i α hk u ( t ) k H
132 + i α sup m ∈{ , , } k ∂ t ∂ mx u ( t ) k H r h sup m ∈{ , , } k ∂ t ∂ mx u ( t ) k H
12 + i α Hence we see from (10.17) and (10.21) that k△ S m ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + )Let Z ~i,~j ( t ) be the term of △ ¯ S m ( t ) defined by ~i and ~j . We have k Z ~i,~j ( t ) k H r . k Q ~i,~j ( t ) k H r k c m,~j ( t ) k H
12 + + k Q ~i,~j ( t ) k H
12 + k c m,~j ( t ) k H r We have k Q ~i,~j ( t ) k H r . hk u ( t ) k H
132 + i α (cid:16) k a ~i,~j ( −−−→ u ( t ) , t ) k H r + k a ~i,~j ( −−−→ u ( t ) , t ) k H
12 + k u ( t ) k H r (cid:17) . · There exists a polynomial S ( u ( t ) , t ) in u ( t ), ∂ x u ( t ),..., ∂ x u ( t ), ∂ t ∂ x u ( t ), ∂ t ∂ x u ( t ),derivatives of F evaluated at (cid:16) −−−→ u ( t ) , t (cid:17) , and (cid:16) ∂ ω F (cid:16) −−−→ u ( t ) , t (cid:17)(cid:17) − such that c m,~j ( t ) := △ (cid:0) ∂ t ∂ x u ( t ) (cid:1) j m S ( u ( t ) , t ). (10.18) yields k c m,~j ( t ) k H r . k△ ( ∂ t ∂ mx u ( t )) j m k H r k S ( u ( t ) , t ) k H
12 + + k△ ( ∂ t ∂ mx u ( t )) j m k H
12 + k S ( u ( t ) , t ) k H r . ǫ k u ( t ) − u ( t ) k H max ((
92 + m ) + ,r + m +4 )+( ǫ − ǫ ) k u ( t ) k H max ((
92 + m ) + ,r + m +4 ) + k u ( t ) − u ( t ) k H max ( r + m +3 , (
72 + m ) + )Hence we also get k△ ¯ S m ( t ) k H r . k u ( t ) − u ( t ) k H max ( r +6 ,
132 + ) · This implies that (10.9) holds. 11.
Appendix
In this appendix we prove the following lemma:
Lemma 11.1.
Let I be an interval such that | I | ≤ . Let k ′ ≥ . Let K ∈ R + .Let f and g be two functions such that k ( f, g ) k H
92 + × H
92 + ≤ K . Then the followingholds: • There exists C := C ( K ) > such that for all t , t ∈ I k X ( f, g, t , t ) k L ∞ ≤ C (cid:16) | t − t | + k f − g k H
72 + (cid:17) · Here X ( f, g, t , t ) := (cid:12)(cid:12)(cid:12) ∂ ω F ( ~f , t ) (cid:12)(cid:12)(cid:12) − | ∂ ω F ( ~g, t ) | . • Let ¯ δ > . There exists C := C ( K, ¯ δ ) > such that for all t , t ∈ I satisfying δ ( ~f , t ) ≥ ¯ δ and δ ( ~g, t ) ≥ ¯ δ we have k Y ( f, g, t , t ) k L ∞ ≤ C (cid:16) | t − t | + k f − g k H
92 + (cid:17) · Here Y ( f, g, t , t ) := (cid:12)(cid:12)(cid:12) ∂ ω F ( ~f , t ) (cid:12)(cid:12)(cid:12) (cid:28) P ( f,t ) | ∂ ω F ( ~f,t ) | (cid:29) − | ∂ ω F ( ~g, t ) | (cid:28) P ( g,t ) | ∂ ω F ( ~g,t ) | (cid:29) · Remark . The proof shows that one can choose the constants C to be increasingfunctions as K increases. Moreover one can choose the constant C in the secondestimate to be an increasing function as ¯ δ decreases. Proof.
The bound for X ( f, g, t , t ) results from the Sobolev embedding, the trian-gle inequality | X ( f, g, t , t ) | . (cid:12)(cid:12)(cid:12) ∂ ω F ( ~f , t ) − ∂ ω F ( ~g, t ) (cid:12)(cid:12)(cid:12) , the Sobolev embedding H + ֒ → L ∞ , and Lemma 3.3.We write Y ( f, g, t , t ) = A + A + A with ONLINEAR THIRD-ORDER EQUATIONS 53 A := (2 π ) − (cid:16)(cid:12)(cid:12)(cid:12) ∂ ω F (cid:16) ~f , t (cid:17)(cid:12)(cid:12)(cid:12) − | ∂ ω F ( ~g, t ) | (cid:17) R T P ( f,t ) | ∂ ω F ( ~f,t ) | dxA := (2 π ) − | ∂ ω F ( ~g, t ) | R T P ( f,t ) − P ( g,t ) | ∂ ω F ( ~f,t ) | dx, and A := (2 π ) − | ∂ ω F ( ~g, t ) | R T P ( g, t ) (cid:18) | ∂ ω F ( ~f,t ) | − | ∂ ω F ( ~g,t ) | (cid:19) dx · We see from the embeddings L ∞ ֒ → H + and L ֒ → L , from Lemma 3.3, andfrom (3.22) that(11.1) | A | . ¯ δ − (cid:13)(cid:13)(cid:13) ∂ ω F (cid:16) ~f , t (cid:17) − ∂ ω F ( ~g, t ) (cid:13)(cid:13)(cid:13) L ∞ k P ( f, t ) k L . k f − g k H
92 + + | t − t |· Similarly | A | . ¯ δ − k ∂ ω F ( ~g, t ) k L ∞ k P ( f, t ) − P ( g, t ) k L . k f − g k H
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American University of Beirut, Department of Mathematics
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