# Global Well-Posedness for Coupled System of mKdV Equations in Analytic Spaces

Authors: Khaled Zennir; Aissa Boukarou; Rehab Nasser Alkhudhayr
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The main result in this paper is to prove, in Bourgain type spaces, the existence of unique local solution to system of initial value problem described by integrable equations of modified Korteweg-de Vries (mKdV) by using linear and trilinear estimates, together with contraction mapping principle. Moreover, owing to the approximate conservation law, we prove the existence of global solution.


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## Body

## 1. Introduction and Main Results

For an effective approach to solving problems arising in modern science and technology, one cannot do without researching nonlinear problems of mathematical physics. The rapid development of new technology and the emergence of its high speed allow researchers to build and consider increasingly complex multidimensional models describing various phenomena, which are modeled, as a rule, using nonlinear partial differential equations (systems). However, now it has become clear that without the development of analytical methods, it is impossible to get a complete idea of the essence of the phenomenon. Analytical methods provide not only a reliable tool for debugging and comparing various numerical methods but also sometimes anticipate some scientific discoveries, make it possible to study the properties of models, to detect the presence of certain effects as a result of the existence or nonexistence of objects (solutions) with the required properties. Therefore, at present, fundamental research is being intensively carried out aimed at proving theorems of existence, uniqueness, and regularity of solutions of nonlinear partial differential equations.In the present paper, a coupled system of modified Korteweg-de Vries equations is considered as follows:(1)∂tu+∂x3u+∂xuv2=0,∂tv+β∂x3v+∂xu2v=0,x,t∈ℝ2,0<β<1,ux,0=u0x,vx,0=v0x.The dynamics of solutions in the Korteweg-de Vries equations (KdV) and the modified Korteweg-de Vries equations (mKdV) are well studied due to the complete integrability of these equations (see [1–6]). For KdV equations, the studies date back to the 1970s, although some results have been obtained very recently (please see [7]). We extend the results in [7] and consider a coupled system of mKdV-type equations on the line in Equation (1).For mKdV equations, many problems have been studied. It is proved that the mKdV equation is locally [8] and globally [9] well-posed in HsT for s≥1/2. Global well-posedness in L2T is shown in [10].For0<β<1, the author in [7] proved that the IVP (Equation (1)) is locally well-posed for the given data u0,v0∈Hsℝ×Hsℝ, s>−1/2. Oh in [11] used the Fourier transform restriction norm method and proved that the next IVP
(2)∂tu+∂x3u+∂xv2=0,ux,0=u0x,∂tv+β∂x3v+∂xuv=0,vx,0=v0x,0<β<1,is locally well-posed for data with regularity s≥0.Forβ=1, the system (Equation (1)) reduces to a special case of a broad class of nonlinear evolution equations considered by Ablowitz et al. [12] in the inverse scattering context. In this case, the well-posedness issues along with existence and stability of solitary waves for this system are widely studied in the literature, using the technique developed by Kenig et al. in [13, 14].Well-posedness for the nonperiodic gKdV equation in spaces of analytic functions has been proved by Grujic and Kalisch [15].A class of suitable analytic functions for our analysis is the analytic Gevrey classGδ,sℝ=Gδ,s introduced by Foias and Temam [16], defined as follows:
(3)Gδ,s=f∈L2;fGδ,s=∫ℝe2δζζ2sf∧ζ2dζ<∞,for s∈ℝ and δ>0 with ·=1+·. For δ=0, the space Gδ,s coincides with the standard Sobolev space Hs. For all 0<δ′<δ and s,s′∈ℝ, we have
(4)Gδ,s⊂Gδ′,s′,i.e.,fGδ′,s′≤cs,s′,δ,δ′fGδ,s,which is the embedding property of the Gevrey spaces.New minimal conditions are used to show the local well-posedness of solution by using linear and trilinear estimates, together with contraction mapping principle. By imposing a more appropriate conditions with the help of the approximate conservation law, we obtain an unusual global existence result in Gevery spaces.Proposition 1 (Paley-Wiener Theorem) [17].
Letδ>0, s∈ℝ. Then, f∈Gδ,s if and only if it is the restriction to the real line of a function F which is holomorphic in the strip x+iy:x,y∈ℝ,y<δ and satisfies
(5)sup∣y∣<δFx+iyHxs<∞.Remark 2.
In the view of the Paley-Wiener Theorem, it is natural to take initial data inGδ,s, to obtain the best behavior of solution and may be extended to be globally in time. It means that given u0,v0∈Gδ,s×Gδ,s for some initial radius δ>0, we then estimate the behavior of the radius of analyticity δT over time.The first main result on local well-posedness of Equation (1) in analytic spaces reads as follows.Theorem 3.
Letδ>0 and s>−1/2. Then for any u0,v0∈Gδ,s×Gδ,s, there exists T=Tu0,v0Gδ,s×Gδ,s and unique solution u,v of Equation (1) on 0,T such that
(6)u,v∈C0,T,Gδ,s×C0,T,Gδ,s.
Moreover, the solution depends onu0,v0, where
(7)T=116C3+16C3u0,v0Gδ,s×Gδ,s21/ε.
Furthermore, the solution satisfies the following:(8)u,vXδ,s,b×Xδ,s,bβ≤2Cu0,v0Gδ,s×Gδ,s,b=12+ε,with constant C>0 depending only on s and b.An effective method for studying lower bounds on the radius of analyticity, including this type of problem, was introduced in [18] for 1D Dirac-Klein-Gordon equations. It was applied in [19] to the modified Kawahara equation and in [20] to the nonperiodic KdV equation (for more details, please see [20–23]).The second result for the problem (Equation(1)) is given in the next theorem.Theorem 4.
Lets>−1/2,0<β<1, and δ0>0. Assume that u0,v0∈Gδ,s×Gδ,s, then the solution in Theorem 3 can be extended to be global in time and for any T′>0, we have the following:
(9)u,v∈C0,T′,GδT′,s×C0,T′,GδT′,s,with
(10)δT′=minδ0,C1T′−2+σ0,where σ0>0 can be taken arbitrarily small and C1>0 is a constant depending on w0,δ0,s, and σ0.The third result is Gevrey’s temporal regularity of the unique solution obtained in the Theorem3. A nonperiodic function fx is the Gevrey class of order r, i.e., fx∈Gr, if there exists a constant C>0 such that
(11)∣∂xlfx∣≤Cl+1l!rl=0,1,2,⋯,if r=1fx is analytic.Here, we will show that forx∈ℝ, for every t∈0,T and j,l∈0,1,2,⋯, there exist C>0 such that
(12)∂tj∂xlux,t≤Cj+l+1j!3l!,∂tj∂xlvx,t≤Cj+l+1j!3l!,i.e., u·,t,v·,t∈G1ℝ×G1ℝ in spacial variable and ux,·,vx,·∈G30,T×G30,T in time variable. Also,
(13)∂tj∂xlux,t≤Cj+l+1j!dl!,(14)∂tj∂xlvx,t≤Cj+l+1j!dl!,where Equations (13) and (14) do not hold for1≤d<3.Theorem 5.
Lets>−1/2,0<β<1, and δ>0. If u0,v0∈Gδ,s×Gδ,s, then the solution u,v∈C0,T,GδT,s×C0,T,GδT,s given by Theorem 4 belongs to the Gevrey class G30,T×G30,T in time variable. Furthermore, it is not belong to Gd0,T×Gd0,T, 1≤d<3 in t.The proof of Theorem5 is similar to that in [1].The paper is organized as follows. In Section2, we define the function spaces and linear and trilinear estimates. In Section 3, we prove Theorem 3, using the linear and trilinear estimates, together with contraction mapping principle. In Section 4, we prove the existence of fundamental approximate conservation law. In the last section, Theorem 4 will be proved using the approximate conservation law.


## 2. Preliminary Tools and Analytic Function Spaces

### 2.1. Function Spaces

We define the analytic Bourgain spaces related to the modified Korteweg-de Vries type equations. The completion of the Schwartz classSℝ2 is given by Xδ,s,bβℝ2=Xδ,s,bβ, for s,b∈ℝ, δ>0, subjected to the norm:
(15)wXδ,s,bβ=∫ℝ2e2δζζ2sη−βζ32bw∧ζ,η2dζdη1/2.We often use without mention, the definitionXδ,s,b1=Xδ,s,b, where
(16)wXδ,s,b=∫ℝ2e2δζζ2sη−ζ32bw∧ζ,η2dζdη1/2.For any intervalI, we define the localized spaces Xδ,s,bβℝ×I=Xδ,s,bβ,I with norm:
(17)wXδ,s,bβ,I=infWXδ,s,bβ;Wℝ×I=w.


### 2.2. Linear Estimates

We have the trilinear estimate (Equations (15) and (16)) defined in the analytic Bourgain spaces. Since the spaces Xδ,s,bβ is continuously embedded in C0,T,Gδ,s, provided b>1/2.Lemma 6.
Letb>1/2, s∈ℝ, and δ>0. Then, for all T>0, we have the following:
(18)Xδ,s,b↪C0,T,Gδ,s.Proof.
First, we note that the operatorA defined by
(19)Aw∧xζ,t=eδζw∧xζ,t,satisfies
(20)wXδ,s,bβ=AwXs,bβ,wGδ,s=AwHs,where Xs,bβ is introduced in [7]. We observe that Aw belongs to Cℝ,Hs and for some C>0, we have the following:
(21)AwCℝ,Hs≤CAwXs,bβ.
Thus, it follows thatw∈C0,T,Gδ,s and
(22)wC0,T,Gδ,s≤CwXδ,s,bβ.Taking the Fourier transform with respect tox of the Cauchy problems (Equation (1)), after an ordinary calculation, we localize in t by using a cut-off function, satisfying ψ∈C0∞, with ψ=1 in −1,1, suppψ⊂−2,2, and ψTt=ψt/T. We consider the operator Λ,Γ given by the following:
(23)Λu,vt=ψtStu0−ψTt∫0tSt−ν∂xF1νdν,Γu,vt=ψtSβtv0−ψTt∫0tSβt−ν∂xF2νdν,where St=e−t∂x3 and Sβt=e−tβ∂x3 are the unitary groups associated with the linear problems.The nonlinear terms defined byF1=uv2 and F2=u2v will be treated in the next lemmas.Lemma 7.
Lets,b∈ℝ and δ>0. For some constant C>0, we have the following:
(24)ψtStu0Xδ,s,b≤Cu0Gδ,s,ψtSβtv0Xδ,s,bβ≤Cv0Gδ,s,for all u0,v0∈Gδ,s.Proof.
By definition, we have the following:(25)ψtSβtu0=Cψt∫ℝeixζ+tβζ3u0^ζdζ=C∫ℝ2eixζ+tηψ^η−βζ3u0^ζdζdη.
It follows that(26)ψtStu0Xδ,s,bβ2=C∫ℝ2e2δ∣ζ∣1+∣ζ∣2s1+η−βζ32bψ∧η−βζ32u0∧ζ2dζdη=C∫ℝe2δ∣ζ∣1+ζ2su0∧ζ2∫ℝψ∧η−βζ321+η−βζ32bdηdζ.
Sinceb>1/2, we have the following:
(27)∫ℝψ∧η−βζ321+∣η−βζ3∣2bdη≤C∫ℝψ∧η−βζ32dη+C∫ℝψ∧η−βζ321+∣η−βζ3∣2bdη≤C.Lemma 8.
Lets∈ℝ, −1/2<b′≤0≤b<b′+1, 0≤T≤1, and δ>0, then for some constant C>0, we have the following:
(28)ψTt∫0tSt−ν∂xF1x,νdνXδ,s,b≤CT1−b+b′∂xF1Xδ,s,b′,ψTt∫0tSβt−ν∂xF2x,νdνXδ,s,bβ≤CT1−b+b′∂xF2Xδ,s,b′β.Proof.
Define(29)W=ψTt∫0tSβt−ν∂xF2x,νdν.
We have, by Equation (19), the following:
(30)AW∧xζ,t=ψTt∫0te−it−νβζ3eδ∣ζ∣∂xF2∧xζ,νdν=ψTt∫0tSβt−ν∂xAF2∧xζ,νdν.
Thus,(31)WXδ,s,bβ=AWXs,bβ=ψTt∫0tSβt−ν∂xAF2x,νdνXs,bβ.
Owing to Lemma6 in [7], we get the following:
(32)ψTt∫0tSβt−ν∂xAF2x,νdνXs,bβ≤CT1−b+b′∂xAF2Xs,b′β=CT1−b+b′∂xF2Xδ,s,b′β.This completes the proof.Lemma 9.
LetΘ∈Sℝ be a Schwartz function in time, s∈ℝ, and δ≥0. If −1/2<b≤b′<1/2, then for any T>0, we have the following:
(33)ΘTtwXδ,s,b≤CTb′−bwXδ,s,b′,ΘTtwXδ,s,bβ≤CTb′−bwXδ,s,b′β,where C depends only on b and b′.Proof.
The proof of Lemma9 for δ=0 can be found in Lemma 13 of [14], for δ>0 as one merely has to replace w by Aw, where the operator is defined in Equation (19).Lemma 10 [20].
Lets∈ℝ, δ≥0, −1/2<b<1/2, and T>0. Then, for any time interval I⊂0,T, we have the following:
(34)χItwXδ,s,b≤CwXδ,s,bT,χItwXδ,s,bβ≤CwXδ,s,bβ,T,where χIt is the characteristic function of I and C depends only on b.


### 2.3. Trilinear Estimates

We have the trilinear estimate in the following lemmas.Lemma 11.
Lets>−1/2, δ>0, b>1/2, and b′ be as in Lemma 8. Then,
(35)∂xuv2Xδ,s,b′≤CuXδ,s,bvXδ,s,bβ2,∂xu2vXδ,s,b′β≤CuXδ,s,b2vXδ,s,bβ.Proof.
We observe, by considering the operatorA in (19), that
(36)eδ∣ζ∣uuv^=2π−2eδ∣ζ∣u^∗u^∗v^≤2π−2∫ℝ4eδ∣ζ−ζ1∣u^ζ−ζ1,η−η1eδ∣ζ1−ζ2∣u^ζ1−ζ2,η1−η2eδζ2v^ζ2,η2dζ1dζ2dη1dη2=AuAuAv^,since δζ≤δζ−ζ1+δζ1−ζ2+δζ2.
Then,(37)∂xu2vXδ,s,b′β=eδ∣ζ∣ζsη−βζ3b∂xuuv^ζ,ηLζ,η2≤∂xAuAuAvXs,b′β.
Thanks to Proposition 2.3 of [7], for some C>0, we have the following:
(38)∂xAuAuAvXs,b′β≤CAuXs,b2AvXs,bβ=CuXδ,s,b2vXδ,s,bβ.This completes the proof.


## 2.1. Function Spaces

We define the analytic Bourgain spaces related to the modified Korteweg-de Vries type equations. The completion of the Schwartz classSℝ2 is given by Xδ,s,bβℝ2=Xδ,s,bβ, for s,b∈ℝ, δ>0, subjected to the norm:
(15)wXδ,s,bβ=∫ℝ2e2δζζ2sη−βζ32bw∧ζ,η2dζdη1/2.We often use without mention, the definitionXδ,s,b1=Xδ,s,b, where
(16)wXδ,s,b=∫ℝ2e2δζζ2sη−ζ32bw∧ζ,η2dζdη1/2.For any intervalI, we define the localized spaces Xδ,s,bβℝ×I=Xδ,s,bβ,I with norm:
(17)wXδ,s,bβ,I=infWXδ,s,bβ;Wℝ×I=w.


## 2.2. Linear Estimates

We have the trilinear estimate (Equations (15) and (16)) defined in the analytic Bourgain spaces. Since the spaces Xδ,s,bβ is continuously embedded in C0,T,Gδ,s, provided b>1/2.Lemma 6.
Letb>1/2, s∈ℝ, and δ>0. Then, for all T>0, we have the following:
(18)Xδ,s,b↪C0,T,Gδ,s.Proof.
First, we note that the operatorA defined by
(19)Aw∧xζ,t=eδζw∧xζ,t,satisfies
(20)wXδ,s,bβ=AwXs,bβ,wGδ,s=AwHs,where Xs,bβ is introduced in [7]. We observe that Aw belongs to Cℝ,Hs and for some C>0, we have the following:
(21)AwCℝ,Hs≤CAwXs,bβ.
Thus, it follows thatw∈C0,T,Gδ,s and
(22)wC0,T,Gδ,s≤CwXδ,s,bβ.Taking the Fourier transform with respect tox of the Cauchy problems (Equation (1)), after an ordinary calculation, we localize in t by using a cut-off function, satisfying ψ∈C0∞, with ψ=1 in −1,1, suppψ⊂−2,2, and ψTt=ψt/T. We consider the operator Λ,Γ given by the following:
(23)Λu,vt=ψtStu0−ψTt∫0tSt−ν∂xF1νdν,Γu,vt=ψtSβtv0−ψTt∫0tSβt−ν∂xF2νdν,where St=e−t∂x3 and Sβt=e−tβ∂x3 are the unitary groups associated with the linear problems.The nonlinear terms defined byF1=uv2 and F2=u2v will be treated in the next lemmas.Lemma 7.
Lets,b∈ℝ and δ>0. For some constant C>0, we have the following:
(24)ψtStu0Xδ,s,b≤Cu0Gδ,s,ψtSβtv0Xδ,s,bβ≤Cv0Gδ,s,for all u0,v0∈Gδ,s.Proof.
By definition, we have the following:(25)ψtSβtu0=Cψt∫ℝeixζ+tβζ3u0^ζdζ=C∫ℝ2eixζ+tηψ^η−βζ3u0^ζdζdη.
It follows that(26)ψtStu0Xδ,s,bβ2=C∫ℝ2e2δ∣ζ∣1+∣ζ∣2s1+η−βζ32bψ∧η−βζ32u0∧ζ2dζdη=C∫ℝe2δ∣ζ∣1+ζ2su0∧ζ2∫ℝψ∧η−βζ321+η−βζ32bdηdζ.
Sinceb>1/2, we have the following:
(27)∫ℝψ∧η−βζ321+∣η−βζ3∣2bdη≤C∫ℝψ∧η−βζ32dη+C∫ℝψ∧η−βζ321+∣η−βζ3∣2bdη≤C.Lemma 8.
Lets∈ℝ, −1/2<b′≤0≤b<b′+1, 0≤T≤1, and δ>0, then for some constant C>0, we have the following:
(28)ψTt∫0tSt−ν∂xF1x,νdνXδ,s,b≤CT1−b+b′∂xF1Xδ,s,b′,ψTt∫0tSβt−ν∂xF2x,νdνXδ,s,bβ≤CT1−b+b′∂xF2Xδ,s,b′β.Proof.
Define(29)W=ψTt∫0tSβt−ν∂xF2x,νdν.
We have, by Equation (19), the following:
(30)AW∧xζ,t=ψTt∫0te−it−νβζ3eδ∣ζ∣∂xF2∧xζ,νdν=ψTt∫0tSβt−ν∂xAF2∧xζ,νdν.
Thus,(31)WXδ,s,bβ=AWXs,bβ=ψTt∫0tSβt−ν∂xAF2x,νdνXs,bβ.
Owing to Lemma6 in [7], we get the following:
(32)ψTt∫0tSβt−ν∂xAF2x,νdνXs,bβ≤CT1−b+b′∂xAF2Xs,b′β=CT1−b+b′∂xF2Xδ,s,b′β.This completes the proof.Lemma 9.
LetΘ∈Sℝ be a Schwartz function in time, s∈ℝ, and δ≥0. If −1/2<b≤b′<1/2, then for any T>0, we have the following:
(33)ΘTtwXδ,s,b≤CTb′−bwXδ,s,b′,ΘTtwXδ,s,bβ≤CTb′−bwXδ,s,b′β,where C depends only on b and b′.Proof.
The proof of Lemma9 for δ=0 can be found in Lemma 13 of [14], for δ>0 as one merely has to replace w by Aw, where the operator is defined in Equation (19).Lemma 10 [20].
Lets∈ℝ, δ≥0, −1/2<b<1/2, and T>0. Then, for any time interval I⊂0,T, we have the following:
(34)χItwXδ,s,b≤CwXδ,s,bT,χItwXδ,s,bβ≤CwXδ,s,bβ,T,where χIt is the characteristic function of I and C depends only on b.


## 2.3. Trilinear Estimates

We have the trilinear estimate in the following lemmas.Lemma 11.
Lets>−1/2, δ>0, b>1/2, and b′ be as in Lemma 8. Then,
(35)∂xuv2Xδ,s,b′≤CuXδ,s,bvXδ,s,bβ2,∂xu2vXδ,s,b′β≤CuXδ,s,b2vXδ,s,bβ.Proof.
We observe, by considering the operatorA in (19), that
(36)eδ∣ζ∣uuv^=2π−2eδ∣ζ∣u^∗u^∗v^≤2π−2∫ℝ4eδ∣ζ−ζ1∣u^ζ−ζ1,η−η1eδ∣ζ1−ζ2∣u^ζ1−ζ2,η1−η2eδζ2v^ζ2,η2dζ1dζ2dη1dη2=AuAuAv^,since δζ≤δζ−ζ1+δζ1−ζ2+δζ2.
Then,(37)∂xu2vXδ,s,b′β=eδ∣ζ∣ζsη−βζ3b∂xuuv^ζ,ηLζ,η2≤∂xAuAuAvXs,b′β.
Thanks to Proposition 2.3 of [7], for some C>0, we have the following:
(38)∂xAuAuAvXs,b′β≤CAuXs,b2AvXs,bβ=CuXδ,s,b2vXδ,s,bβ.This completes the proof.


## 3. Proof of Theorem3

### 3.1. Existence of Solution

We estimate terms in Equation (23). For this end, we defineBδ,s,b=Xδ,s,b×Xδ,s,bβ and Nδ,s=Gδ,s×Gδ,s, with norms u,vBδ,s,b=maxuXδ,s,b,vXδ,s,bβ and similar for Nδ,s.Lemma 12.
Lets>−1/2, δ>0, and b>1/2. Then, for all u0,v0∈Nδ,s and 0<T<1, with some constant C>0, we have the following:
(39)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+Tεu,vBδ,s,b3,(40)Λu,v−Λu∗,v∗,Γu,v−Γu∗,v∗Bδ,s,b≤CTεu−u∗,v−v∗Bδ,s,bu,vBδ,s,b2+u,vBδ,s,bu∗,v∗Bδ,s,b+v∗,v∗Bδ,s,b2,for all u,v,u∗,v∗∈Bδ,s,b.Proof.
To prove estimate of Equation (39), we have the following:
(41)Λu,vXδ,s,b≤Cu0Gδ,s+CTεuXδ,s,bvXδ,s,bβ2≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3,(42)Γu,vXδ,s,bβ≤Cv0Gδ,s+CTεuXδ,s,b2vXδ,s,bβ≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3.
Therefore, from Equations (41) and (42), we obtain the following:
(43)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+Tεu,vBδ,s,b3.
For the estimate of Equation (40), we observe that
(44)Λu,v−Λu∗,v∗=ψTt∫0tSt−ν∂xuv2−u∗v∗2x,νdν,Γu,v−Γu∗,v∗=ψTt∫0tSβt−ν∂xu2v−u∗2v∗x,νdν,where
(45)ω=∂xu2v−u∗2v∗=∂xvu+u∗u−u∗+u∗2v−v∗,ω′=∂xuv2−u∗v∗2=∂xuv+v∗v−v∗+v∗2u−u∗.We will show thatΛ×Γ is a contraction on the ball B0,R to B0,R, where B0,R is given in Equation (46).Lemma 13.
Lets≥−1/4, δ>0, and b>1/2. Then, for all u0,v0∈Nδ,s, such that the map Λ×Γ:B0,R⟶B0,R is a contraction, where B0,R is given by the following:
(46)B0,R=u,v∈Bδ,s,b;u,vBδ,s,b≤R,with R=2Cu0,v0Nδ,s.Proof.
From Lemma12, for all u,v∈B0,R, we have the following:
(47)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3≤R2+CTεR3.
We chooseT sufficiently small such that Tε≤1/4CR2; hence,
(48)Λu,v,Γu,vBδ,s,b≤R,∀u,v∈B0,R.
Thus,Λ×Γ maps B0,R into B0,R, which is a contraction, since
(49)Λu,v−Λu∗,v∗,Γu,v−Γu∗,v∗Bδ,s,b≤CTεu−u∗,v−v∗Bδ,s,bu,vBδ,s,b2+u,vBδ,s,bu∗,v∗Bδ,s,b+v∗,v∗Bδ,s,b2,≤3CTεR2u−u∗,v−v∗Bσ,δ,s,b≤34u−u∗,v−v∗Bσ,δ,s,b,for all u,v∈B0,R. Hence, Λ,Γ:B0,R⟶B0,R is a contraction.


### 3.2. The Uniqueness

Uniqueness of the solution inC0,T,Gδ,s×C0,T,Gδ,s can be proved by the following standard argument.Suppose thatu,v,u∗,v∗∈C0,T,Gδ,s×C0,T,Gδ,s are solutions to Equation (1) with v·,0,u·,0=v∗·,0,u∗·,0 in Gδ,s×Gδ,s. Setting ϑ=u−u∗ and ω=v−v∗, we see that ϑ,ω solves the Cauchy problem:
(50)∂tϑ+∂x3ϑ+∂xuv2−u∗v2=0,ϑ0=0,(51)∂tω+∂x3ω+∂xu2v−u2v∗=0,ω0=0.Thus, by Equation (50), we have the following:
(52)12∂tϑt,·L22=12∂t∫ℝϑ2t,xdx=∫ℝϑt,x∂tϑt,xdx=−∫ℝϑt,x∂xuv2−u∗v2dx=0,since we have the following:
(53)∫ℝϑt,x∂x3ϑt,xdx=0.Thanks to Equation (53), we have the following:
(54)∂tϑt,·L22=−2∫ℝϑt,x∂xv2ϑt,xdx.Integrating by parts of the last integral, we obtain the following:(55)∂tϑt,·L22=−∫ℝ∂xv2t,xϑ2t,xdx,from which we deduce the inequality as follows:
(56)∂tϑt,·L22≤∂xv2L∞ϑtL22.Sinceu,u∗∈C0,T,Gδ,s, we have that u and u∗ are continuous in t on the compact set 0,T and are Gδ,s in x. Thus, we can conclude that
(57)∂xv2L∞≤c<∞.Therefore, from Equations (56) and (57), we obtain the differential inequality:
(58)∂tϑt,·L22≤cϑtL22,0≤t≤T.Solving it gives the following:(59)ϑtL22≤ecϑ0L22,0≤t≤T.Since∥ϑ0∥L22=0, from Equation (59), we obtain that ϑt=0, 0≤t≤T, or u=u∗.Now by Equation (51), we have the following:
(60)∂tωt,·L22≤cωtL22,0≤t≤T.Solving it gives the following:(61)ωtL22≤ecω0L22,0≤t≤T.Since∥ω0∥L22=0, from Equation (61), we obtain that ωt=0, 0≤t≤T, or v=v∗.


### 3.3. Continuous Dependence of the Initial Data

To prove continuous dependence of the initial data, we will prove the following.Lemma 14.
Lets>−1/2, δ>0, and b>1/2. Then, for all u0,v0,u0∗,v0∗∈Nδ,s, if u,v and u∗,v∗ are two solutions to Equation (1) corresponding to initial data u0,v0 and u0∗,v0∗, we have the following:
(62)u−u∗,v−v∗C0,T,Gδ,s2≤4C0Cu0−u0∗,v0−v0∗Nδ,s.Proof.
Ifu,v and u∗,v∗ are two solutions to Equation (1), corresponding to initial data u0,v0 and u0∗,v0∗, we have from Lemma 6 as follows:
(63)u−u∗C0,T,Gδ,s≤C0u−u∗Xδ,s,b,v−v∗C0,T,Gδ,s≤C0v−v∗Xδ,s,bβ.
By takingu,v,u∗,v∗∈B0,R and Tε≤1/4CR, we have the following:
(64)u−u∗Xδ,s,b≤Cu0−u0∗,v0−v0∗Nδ,s+34u−u∗,v−v∗Bδ,s,b,v−v∗Xδ,s,bβ≤Cu0−u0∗,v0−v0∗Nδ,s+34u−u∗,v−v∗Bδ,s,b.
Thus,(65)u−u∗,v−v∗Bδ,s,b≤4Cu0−u0∗,v0−v0∗Nδ,s.Then,
(66)u−u∗,v−v∗C0,T,Gδ,s2≤4C0Cu0−u0∗,v0−v0∗Nδ,s.This completes the proof of Theorem3.


## 3.1. Existence of Solution

We estimate terms in Equation (23). For this end, we defineBδ,s,b=Xδ,s,b×Xδ,s,bβ and Nδ,s=Gδ,s×Gδ,s, with norms u,vBδ,s,b=maxuXδ,s,b,vXδ,s,bβ and similar for Nδ,s.Lemma 12.
Lets>−1/2, δ>0, and b>1/2. Then, for all u0,v0∈Nδ,s and 0<T<1, with some constant C>0, we have the following:
(39)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+Tεu,vBδ,s,b3,(40)Λu,v−Λu∗,v∗,Γu,v−Γu∗,v∗Bδ,s,b≤CTεu−u∗,v−v∗Bδ,s,bu,vBδ,s,b2+u,vBδ,s,bu∗,v∗Bδ,s,b+v∗,v∗Bδ,s,b2,for all u,v,u∗,v∗∈Bδ,s,b.Proof.
To prove estimate of Equation (39), we have the following:
(41)Λu,vXδ,s,b≤Cu0Gδ,s+CTεuXδ,s,bvXδ,s,bβ2≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3,(42)Γu,vXδ,s,bβ≤Cv0Gδ,s+CTεuXδ,s,b2vXδ,s,bβ≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3.
Therefore, from Equations (41) and (42), we obtain the following:
(43)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+Tεu,vBδ,s,b3.
For the estimate of Equation (40), we observe that
(44)Λu,v−Λu∗,v∗=ψTt∫0tSt−ν∂xuv2−u∗v∗2x,νdν,Γu,v−Γu∗,v∗=ψTt∫0tSβt−ν∂xu2v−u∗2v∗x,νdν,where
(45)ω=∂xu2v−u∗2v∗=∂xvu+u∗u−u∗+u∗2v−v∗,ω′=∂xuv2−u∗v∗2=∂xuv+v∗v−v∗+v∗2u−u∗.We will show thatΛ×Γ is a contraction on the ball B0,R to B0,R, where B0,R is given in Equation (46).Lemma 13.
Lets≥−1/4, δ>0, and b>1/2. Then, for all u0,v0∈Nδ,s, such that the map Λ×Γ:B0,R⟶B0,R is a contraction, where B0,R is given by the following:
(46)B0,R=u,v∈Bδ,s,b;u,vBδ,s,b≤R,with R=2Cu0,v0Nδ,s.Proof.
From Lemma12, for all u,v∈B0,R, we have the following:
(47)Λu,v,Γu,vBδ,s,b≤Cu0,v0Nδ,s+CTεu,vBδ,s,b3≤R2+CTεR3.
We chooseT sufficiently small such that Tε≤1/4CR2; hence,
(48)Λu,v,Γu,vBδ,s,b≤R,∀u,v∈B0,R.
Thus,Λ×Γ maps B0,R into B0,R, which is a contraction, since
(49)Λu,v−Λu∗,v∗,Γu,v−Γu∗,v∗Bδ,s,b≤CTεu−u∗,v−v∗Bδ,s,bu,vBδ,s,b2+u,vBδ,s,bu∗,v∗Bδ,s,b+v∗,v∗Bδ,s,b2,≤3CTεR2u−u∗,v−v∗Bσ,δ,s,b≤34u−u∗,v−v∗Bσ,δ,s,b,for all u,v∈B0,R. Hence, Λ,Γ:B0,R⟶B0,R is a contraction.


## 3.2. The Uniqueness

Uniqueness of the solution inC0,T,Gδ,s×C0,T,Gδ,s can be proved by the following standard argument.Suppose thatu,v,u∗,v∗∈C0,T,Gδ,s×C0,T,Gδ,s are solutions to Equation (1) with v·,0,u·,0=v∗·,0,u∗·,0 in Gδ,s×Gδ,s. Setting ϑ=u−u∗ and ω=v−v∗, we see that ϑ,ω solves the Cauchy problem:
(50)∂tϑ+∂x3ϑ+∂xuv2−u∗v2=0,ϑ0=0,(51)∂tω+∂x3ω+∂xu2v−u2v∗=0,ω0=0.Thus, by Equation (50), we have the following:
(52)12∂tϑt,·L22=12∂t∫ℝϑ2t,xdx=∫ℝϑt,x∂tϑt,xdx=−∫ℝϑt,x∂xuv2−u∗v2dx=0,since we have the following:
(53)∫ℝϑt,x∂x3ϑt,xdx=0.Thanks to Equation (53), we have the following:
(54)∂tϑt,·L22=−2∫ℝϑt,x∂xv2ϑt,xdx.Integrating by parts of the last integral, we obtain the following:(55)∂tϑt,·L22=−∫ℝ∂xv2t,xϑ2t,xdx,from which we deduce the inequality as follows:
(56)∂tϑt,·L22≤∂xv2L∞ϑtL22.Sinceu,u∗∈C0,T,Gδ,s, we have that u and u∗ are continuous in t on the compact set 0,T and are Gδ,s in x. Thus, we can conclude that
(57)∂xv2L∞≤c<∞.Therefore, from Equations (56) and (57), we obtain the differential inequality:
(58)∂tϑt,·L22≤cϑtL22,0≤t≤T.Solving it gives the following:(59)ϑtL22≤ecϑ0L22,0≤t≤T.Since∥ϑ0∥L22=0, from Equation (59), we obtain that ϑt=0, 0≤t≤T, or u=u∗.Now by Equation (51), we have the following:
(60)∂tωt,·L22≤cωtL22,0≤t≤T.Solving it gives the following:(61)ωtL22≤ecω0L22,0≤t≤T.Since∥ω0∥L22=0, from Equation (61), we obtain that ωt=0, 0≤t≤T, or v=v∗.


## 3.3. Continuous Dependence of the Initial Data

To prove continuous dependence of the initial data, we will prove the following.Lemma 14.
Lets>−1/2, δ>0, and b>1/2. Then, for all u0,v0,u0∗,v0∗∈Nδ,s, if u,v and u∗,v∗ are two solutions to Equation (1) corresponding to initial data u0,v0 and u0∗,v0∗, we have the following:
(62)u−u∗,v−v∗C0,T,Gδ,s2≤4C0Cu0−u0∗,v0−v0∗Nδ,s.Proof.
Ifu,v and u∗,v∗ are two solutions to Equation (1), corresponding to initial data u0,v0 and u0∗,v0∗, we have from Lemma 6 as follows:
(63)u−u∗C0,T,Gδ,s≤C0u−u∗Xδ,s,b,v−v∗C0,T,Gδ,s≤C0v−v∗Xδ,s,bβ.
By takingu,v,u∗,v∗∈B0,R and Tε≤1/4CR, we have the following:
(64)u−u∗Xδ,s,b≤Cu0−u0∗,v0−v0∗Nδ,s+34u−u∗,v−v∗Bδ,s,b,v−v∗Xδ,s,bβ≤Cu0−u0∗,v0−v0∗Nδ,s+34u−u∗,v−v∗Bδ,s,b.
Thus,(65)u−u∗,v−v∗Bδ,s,b≤4Cu0−u0∗,v0−v0∗Nδ,s.Then,
(66)u−u∗,v−v∗C0,T,Gδ,s2≤4C0Cu0−u0∗,v0−v0∗Nδ,s.This completes the proof of Theorem3.


## 4. Approximate Conservation Law

We have the following:(67)u,vL2=∫ℝu2+v2dx,which is conserved for a solution u,v of Equation (1). We are going to show an approximate conservation law for a solution to Equation (1) based on the conservation of the L2ℝ norm of solution.Theorem 15.
Letκ∈0,1/2 and 0<T1<T0<1, T0 be as in Theorem 3 with s=0; there exist b=1/2+ε and C>0, such that for any δ>0 and any solution u,v∈Bδ,0,bT0 to the Cauchy problem (Equation (1)) on the time interval 0,T1, we have the estimate:
(68)supt∈0,T1ut,vtNδ,02≤u0,v0Nδ,02+Cδκu,vBδ,0,b4.
Moreover, we have the following:(69)supt∈0,T1ut,vtNδ,02≤u0,v0Nδ,02+Cδκu0,v0Nδ,04.We need the following estimate.Lemma 16.
Givenκ∈0,−1/2, there exist b=1/2+ε, C>0, and u,v∈Bδ,0,b; we have the following:
(70)G1,G2B0,b−1≤Cδκu,vBδ,0,b3,where G1=∂xAuAvAv−Auv2, G2=∂xAuAuAv−Au2v, and the operator A is given by Equation (19).Proof.
LetL1=AuAvAv−Auv2. Then,
(71)G1X0,b−1=ζη−ζ31−bL1^ζ,ηLζ,η2=∫ℝ2ζ2η−ζ321−bL1∧ζ,η2dζdη.
We shall calculate the Fourier transform ofL1 as follows:
(72)L^ζ,η=AuAvAv^−Auv2^=Ceδζu^∗eδζv^∗eδζv^∗ζ,η−eδζu^∗v^∗v^∗ζ,η=C∫ℝ4eδζ1u^ζ1,η1eδζ2v^ζ2,η2eδζ−ζ1−ζ2v^ζ−ζ1−ζ2,η−η1−η2−eδζu^ζ1,η1v^ζ2,η2v^ζ−ζ1−ζ2,η−η1−η2dζ1dζ2dη1dη2≤C∫ℝ4eδζ1eδζ2eδζ−ζ1−ζ2−eδζ×u^ζ1,η1v^ζ2,η2v^ζ−ζ1−ζ2,η−η1−η2dζ1dζ2dη1dη2.
Now using Corollary 7.3 in [21], let θ∈0,1; we have the following:
(73)eδζ1eδζ2eδζ−ζ1−ζ2−eδζ≤4δζ−ζ1−ζ2ζ1ζ2ζθeδζ1eδζ2eδζ−ζ1−ζ2.
Forκ∈0,1/2⊂0,1, one can see that
(74)G1X0,b−12=ζη−ζ31−bL1^ζ,ηLζ,η22≤C4δ2κ∫ℝ2ζ2η−ζ321−b∫ℝ4ζ−ζ1−ζ2ζ1ζ2ζκ×eδζ1eδζ2eδζ−ζ1−ζ2×u∧ζ1,η1v∧ζ2,η2v∧ζ−ζ1−ζ2,η−η1−η2dζ1dζ2dη1dη22dζdη=C4δ2κζζ−κη−ζ31−b∫ℝ4eδζ1ζ1κu^ζ1,η1eδζ2ζ2κv^ζ2,η2×eδζ−ζ1−ζ2ζ−ζ1−ζ2κv^ζ−ζ1−ζ2,η−η1−η2dζ1dζ2dη1dη2Lζ,η22.
Now by takings=−κ∈−1/2,0, we obtain the following:
(75)G1X0,b−1≤C4δκζζsη−ζ31−b∫ℝ4eδ∣ζ1∣u^ζ1,η1ζ1seδ∣ζ2∣v^ζ2,η2ζ2s·eδζ−ζ1−ζ2v^ζ−ζ1−ζ2,η−η1−η2ζ−ζ1−ζ2sdζ1dζ2dη1dη2Lζ,η2≤C4δκζζsη−ζ31−b∫ℝ4eδ∣ζ1∣η1−ζ13bu^ζ1,η1ζ1sη1−ζ13beδ∣ζ2∣η2−βζ23bv^ζ2,η2ζ2sη2−βζ23b×eδ∣ζ−ζ1−ζ2∣η−η1−η2−βζ−ζ1−ζ23bv^ζ−ζ1−ζ2,η−η1−η2ζ−ζ1−ζ2sη−η1−η2−βζ−ζ1−ζ23b×dζ1dζ2dη1dη2Lζ,η2.
Then,(76)G1X0,b−1≤CδκAuX0,bAvX0,bβ2=CδκuXδ,0,bvXδ,0,bβ2≤Cδκu,vBδ,0,b3.
Now letL2=AuAuAv−Au2v. Then,
(77)G2X0,b−1≤CδκAuX0,b2AvX0,bβ=CδκuXδ,0,b2vXδ,0,bβ≤Cδκu,vBδ,0,b3.
By Equations (76) and (77), we have the following:
(78)G1,G2B0,b−1≤Cδκu,vBδ,0,b3.Proof (Theorem15).
LetUt,x=Aut,x,Vt,x=Avt,x which are real-valued since the multiplier A is even and u,v are real-valued. Applying A to Equation (1), we obtain the following:
(79)∂tU+∂x3U+∂xUV2=G1,(80)∂tU+∂x3U+∂xU2V=G2,where G1=∂xAuAvAv−Auv2 and G2=∂xAuAuAv−Au2v.
By multiplying both sides of Equation (79) by U and Equation (80) by V and integrating with respect to space variable, we get the following:
(81)∫ℝU∂tUdx+∫ℝU∂x3Udx+∫ℝU∂xUV2dx=∫ℝUG1dx,∫ℝV∂tVdx+∫ℝV∂x3Vdx+∫ℝV∂xU2Vdx=∫ℝVG2dx.
Then,(82)∫ℝU∂tU+V∂tVdx+∫ℝU∂x3U+V∂x3Vdx+∫ℝU∂xUV2+V∂xU2Vdx=∫ℝUG1+VG2dx,∫ℝU∂tU+V∂tVdx+∫ℝ∂x∂xU∂xU+∂xV∂xVdx+∫ℝ∂xU2V2dx=∫ℝUG1+VG2dx.
Noting that∂xjUx,t⟶0 as ∣x∣⟶∞ (see [20]), we use integration by parts to obtain the following:
(83)12∂t∫ℝU2+V2dx=∫ℝUG1+VG2dx.
Integrating the last equality with respect tot∈0,T1, we obtain the following:
(84)∫ℝU2T1,x+V2T1,xdx=∫ℝU20,x+V20,xdx+2∫ℝ2χ0,T1tUG1+VG2dxdt.
Thus,(85)uT1Gδ,02+vT1Gδ,02=u0Gδ,02+v0Gδ,02+2∫ℝ2χ0,T1tUG1+VG2dxdt.
By using Holder’s inequality, Lemma10, Lemma 9, and the fact that
(86)12<1−b<12,12<b−1<12.
Sinceb>1/2+ε, we obtain the following:
(87)∫ℝ2χ0,T1tUG1+VG2dxdt≤χ0,T1tUX0,1−bχ0,T1tG1X0,b−1+χ0,T1tVX0,1−bβχ0,T1tG2X0,b−1β≤CUX0,1−bT1G1X0,b−1T1+CVX0,1−bβ,T1G2X0,b−1β,T1≤CΘT1UX0,1−bΘT1G1X0,b−1+CΘT1VX0,1−bβΘT1G2X0,b−1β≤CUX0,1−bG1X0,b−1+CVX0,1−bβG2X0,b−1β,where ΘT1=1 for t∈0,T1; we can conclude from Lemma 16:
(88)∫ℝ2χ0,T1tUG1+VG2dxdt≤CUX0,1−bG1X0,b−1+CVX0,1−bβG2X0,b−1β≤CδκuXδ,0,b2vXδ,0,bβ2+CδκuXδ,0,b2vXδ,0,bβ2=2CδκuXδ,0,b2vXδ,0,bβ2≤2Cδκu,vBδ,0,b4.
Therefore,(89)uT1Gδ,02+vT1Gδ,02≤u0Gδ,02+v0Gδ,02+2Cδκu,vBδ,0,b4,2uT1,vT1Nδ,02≤2u0,v0Nδ,02+2Cδκu,vBδ,0,b4,supt∈0,T1ut,vtNδ,02≤u0,v0Nδ,02+Cδκu,vBδ,0,b4.
Finally, by using Equation (8), we conclude that
(90)supt∈0,T1ut,vtNδ,02≤u0,v0Nδ,02+Cδκu0,v0Nδ,04.


## 5. Proof of Theorem4

Letδ0>0, s>−1/2, and κ∈0,1/2 be fixed, and u0,v0∈Nδ0,s. Then, we have to prove that the solution u,v of Equation (1) satisfies the following:
(91)u,v∈C0,T′,GδT′,s×C0,T′,GδT′,s,where
(92)δT′=minδ0,C1T′−1/κ,forallT′>0,and C1>0 is a constant depending on u0,v0,δ0,s, and κ. By Theorem 3, there is a maximal time T∗=T∗u0,v0,δ0,s∈0,∞, such that
(93)u,v∈C0,T∗,Gδ0,s×C0,T∗,Gδ0,s.IfT∗=∞, it is done.IfT∗<∞, as we assume henceforth, it remains to prove the following:
(94)u,v∈C0,T′,GC1T′−1/κ,s×C0,T′,GC1T′−1/κ,s,forallT′≥T∗.


### 5.1. The CaseS=0

FixedT′≥T∗; we will show that, for δ>0, sufficiently small
(95)supt∈0,T′u0,v0Nδ,02≤2u0,v0Nδ0,02.In this case, by Theorem3 and Theorem 15 with
(96)T0=116C3+32C3u0,v0Nδ0,021/ε,the smallness conditions on δ will be
(97)δ<δ0,2T′T0Cδκ22u0,v0Nδ0,02≤1,C>0.Here,C is the constant in Theorems 15.By induction, we check that(98)supt∈0,nT0ut,vtNδ,02≤u0,v0Nδ,02+nCδκ22u0,v0Nδ0,04,(99)supt∈0,nT0ut,vtNδ,02≤2u0,v0Nδ0,02,for n∈1,⋯,m+1, where m∈ℕ is chosen so that T′∈mT0,m+1T0. This m does exist; by Theorem 3 and the definition of T∗, we have the following:
(100)T0<116C3+16C3u0,v0Nδ0,021/ε<T∗,henceT0<T′.In the first step, we cover the interval0,T0, and by Theorem 15, we have the following:
(101)supt∈0,T0ut,vtNδ,02≤u0,v0Nδ,02+Cδκu0,v0Nδ,04≤u0,v0Nδ,02+Cδκu0,v0Nδ0,04,since δ≤δ0; we used the following:
(102)u0,v0Nδ,0≤u0,v0Nδ0,0.This satisfies Equation (98) for n=1, and Equation (99) is following and using again u0,v0Nδ,0≤u0,v0Nδ0,0 as well as the following:
(103)Cδκu0,v0Nδ0,02≤1.Suppose now that Equations (98) and (99) hold for some n∈1,⋯,m and we prove that it holds for n+1, we estimate the following:
(104)supt∈nT0,n+1T0ut,vtNδ,02≤wnT0Nδ,02+CδκunT0,vnT0Nδ,04≤unT0nT0,vnT0Nδ,02+Cδκ22u0,v0Nδ0,04≤u0,v0Nδ,02+nCδκ22u0,v0Nδ0,04+Cδκ22u0,v0Nδ0,04,satisfying Equation (98) with n replaced by n+1. To get Equation (99) with n replaced by n+1, it is then enough to have the following:
(105)n+1Cδκ22u0,v0Nδ0,02≤1,but this holds by Equation (97), since n+1≤m+1≤T′/T0+1<2T′/T0.Finally, Equation (97) is satisfied for δ∈0,δ0 such that
(106)2T′T0Cδκ22u0,v0Nδ0,02=1.Thus,δ=C1T′−1/κ, where
(107)C1=1C23u0,v0Nδ0,0216C3+32C3u0,v0Nδ0,021/ε1/κ.


### 5.2. The General Case

For alls, by Equation (4), we have u0,v0∈Gδ0,s⊂Gδ0/2,0.For cases=0, it is proved that there is a T2>0, such that
(108)u,v∈C0,T2,Gδ0/2,0×C0,T2,Gδ0/2,0,u,v∈C0,T′,G2σT′−1/κ,0×C0,T′,G2σT′−1/κ,0,forT′≥T2,where σ>0 depends on u0,v0,δ0, and κ.Applying again the embedding Equation (4), we now conclude that
(109)u,v∈C0,T2,Gδ0/4,s×C0,T2,Gδ0/4,s,u,v∈C0,T′,GσT′−1/κ,s×C0,T′,GσT′−1/κ,s,forT′≥T2,which imply Equation (94). The proof of Theorem 4 is now completed.


## 5.1. The CaseS=0

FixedT′≥T∗; we will show that, for δ>0, sufficiently small
(95)supt∈0,T′u0,v0Nδ,02≤2u0,v0Nδ0,02.In this case, by Theorem3 and Theorem 15 with
(96)T0=116C3+32C3u0,v0Nδ0,021/ε,the smallness conditions on δ will be
(97)δ<δ0,2T′T0Cδκ22u0,v0Nδ0,02≤1,C>0.Here,C is the constant in Theorems 15.By induction, we check that(98)supt∈0,nT0ut,vtNδ,02≤u0,v0Nδ,02+nCδκ22u0,v0Nδ0,04,(99)supt∈0,nT0ut,vtNδ,02≤2u0,v0Nδ0,02,for n∈1,⋯,m+1, where m∈ℕ is chosen so that T′∈mT0,m+1T0. This m does exist; by Theorem 3 and the definition of T∗, we have the following:
(100)T0<116C3+16C3u0,v0Nδ0,021/ε<T∗,henceT0<T′.In the first step, we cover the interval0,T0, and by Theorem 15, we have the following:
(101)supt∈0,T0ut,vtNδ,02≤u0,v0Nδ,02+Cδκu0,v0Nδ,04≤u0,v0Nδ,02+Cδκu0,v0Nδ0,04,since δ≤δ0; we used the following:
(102)u0,v0Nδ,0≤u0,v0Nδ0,0.This satisfies Equation (98) for n=1, and Equation (99) is following and using again u0,v0Nδ,0≤u0,v0Nδ0,0 as well as the following:
(103)Cδκu0,v0Nδ0,02≤1.Suppose now that Equations (98) and (99) hold for some n∈1,⋯,m and we prove that it holds for n+1, we estimate the following:
(104)supt∈nT0,n+1T0ut,vtNδ,02≤wnT0Nδ,02+CδκunT0,vnT0Nδ,04≤unT0nT0,vnT0Nδ,02+Cδκ22u0,v0Nδ0,04≤u0,v0Nδ,02+nCδκ22u0,v0Nδ0,04+Cδκ22u0,v0Nδ0,04,satisfying Equation (98) with n replaced by n+1. To get Equation (99) with n replaced by n+1, it is then enough to have the following:
(105)n+1Cδκ22u0,v0Nδ0,02≤1,but this holds by Equation (97), since n+1≤m+1≤T′/T0+1<2T′/T0.Finally, Equation (97) is satisfied for δ∈0,δ0 such that
(106)2T′T0Cδκ22u0,v0Nδ0,02=1.Thus,δ=C1T′−1/κ, where
(107)C1=1C23u0,v0Nδ0,0216C3+32C3u0,v0Nδ0,021/ε1/κ.


## 5.2. The General Case

For alls, by Equation (4), we have u0,v0∈Gδ0,s⊂Gδ0/2,0.For cases=0, it is proved that there is a T2>0, such that
(108)u,v∈C0,T2,Gδ0/2,0×C0,T2,Gδ0/2,0,u,v∈C0,T′,G2σT′−1/κ,0×C0,T′,G2σT′−1/κ,0,forT′≥T2,where σ>0 depends on u0,v0,δ0, and κ.Applying again the embedding Equation (4), we now conclude that
(109)u,v∈C0,T2,Gδ0/4,s×C0,T2,Gδ0/4,s,u,v∈C0,T′,GσT′−1/κ,s×C0,T′,GσT′−1/κ,s,forT′≥T2,which imply Equation (94). The proof of Theorem 4 is now completed.



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*Source: 6614375-2021-03-05.xml*