1.Introduction YongshengMi, ChunlaiMu, andWeianTao OntheCauchyProblemfortheTwo-ComponentNovikovEquation ResearchArticle

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Volume 2013, Article ID 810725,11pages http://dx.doi.org/10.1155/2013/810725

Research Article

On the Cauchy Problem for the Two-Component Novikov Equation

Yongsheng Mi,

^1,2

Chunlai Mu,

¹

and Weian Tao

²

1College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

2College of Mathematics and Computer Sciences, Yangtze Normal University, Chongqing, Fuling 408100, China

Correspondence should be addressed to Yongsheng Mi; [email protected] Received 9 January 2013; Accepted 27 March 2013

Academic Editor: M. Lakshmanan

Copyright © 2013 Yongsheng Mi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin’s cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.

1. Introduction

In this paper, we consider the following Cauchy problem of the two-component Novikov equation

𝑚_𝑡+ 𝑢V𝑚_𝑥+ 3V𝑢_𝑥𝑚 = 0, 𝑡 > 0, 𝑥 ∈R, 𝑛_𝑡+ 𝑢V𝑛_𝑥+ 3𝑢V_𝑥𝑛 = 0, 𝑡 > 0, 𝑥 ∈R, 𝑚 = 𝑢 − 𝑢_𝑥𝑥, 𝑛 =V−V_𝑥𝑥, 𝑡 > 0, 𝑥 ∈R, 𝑢 (0, 𝑥) = 𝑢₀(𝑥) , V(0, 𝑡) =V₀(𝑥) , 𝑥 ∈R.

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The two-component system in (1) was found by Geng and Xue [1]. It was shown in [1] that the system (1) is exactly a negative flow in the hierarchy and admits exact solutions with 𝑁-peakons and an infinite sequence of conserved quantities.

Moreover, a reduction of this hierarchy and its Hamiltonian structures are discussed.

ForV= 1, (1) becomes the Degasperis-Procesi equation 𝑚_𝑡+ 𝑢𝑚_𝑥+ 3𝑢_𝑥𝑚 = 0, 𝑡 > 0, 𝑥 ∈R,

𝑚 = 𝑢 − 𝑢_𝑥𝑥, 𝑡 > 0, 𝑥 ∈R, 𝑢 (0, 𝑥) = 𝑢0(𝑥) , 𝑥 ∈R.

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Degasperis et al. [2] proved the formal integrability of (2) by constructing a Lax pair. They also showed that it has bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peakon solutions. The direct and inverse scattering approach to pursue it can be seen in [3]. Moreover, in [4], they also presented that the Degasperis-Procesi equation has a bi-Hamiltonian structure and an infinite number of conservation laws and admits exact peakon solutions which are analogous to the Camassa- Holm peakons. It is worth pointing out that solutions of this type are not mere abstractizations: the peakons replicate a feature that is characteristic for the waves of great height- waves of the largest amplitude that are exact solutions of the governing equations for irrotational water waves (cf. the papers [5–7]). The Degasperis-Procesi equation is a model for nonlinear shallow water dynamics (cf. the discussion in [8]). The numerical stability of solitons and peakons, the multisoliton solutions, and their peakon limits, together with an inverse scattering method to compute 𝑁-peakon solutions to Degasperis-Procesi equation, have been investigated, respectively, in [9–11]. Furthermore, the traveling wave solutions and the classification of all weak traveling wave solutions to Degasperis-Procesi equation were presented in [12,13].

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For𝑢 =V, (1) becomes the Novikov equation 𝑚_𝑡+ 𝑢²𝑚_𝑥+ 3𝑢𝑢_𝑥𝑚 = 0, 𝑡 > 0, 𝑥 ∈R,

𝑚 = 𝑢 − 𝑢_𝑥𝑥, 𝑡 > 0, 𝑥 ∈R, 𝑢 (0, 𝑥) = 𝑢₀(𝑥) , 𝑥 ∈R,

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which has been recently discovered by Vladimir Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity [14]. The perturbative symmetry approach yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate (3) and find its first few symmetries, and he subsequently found a scalar Lax pair for it, then proved that the equation is integrable, which can be thought as a generalization of the Camassa-Holm equation. In [15], it is shown that the Novikov equation admits peakon solutions like the Camassa-Holm. Also, it has a Lax pair in matrix form and a bi-Hamiltonian structure. Furthermore, it has infinitely many conserved quantities, like Camassa-Holm. The most important quantity conserved by a solution 𝑢 to Novikov equation is its𝐻¹-norm‖𝑢‖²_𝐻1 = ∫_𝑅(𝑢²+ 𝑢²_𝑥), which plays an important role in the study of (1). In [16–19], the authors study well-posedness and dependence on initial data for the Cauchy problem for Novikov equation. Recently, in [20], a global existence result and conditions on the initial data were considered. Existence and uniqueness of global weak solution to Novikov equation with initial data under some conditions was proved in [21]. The Novikov equation with dissipative term was considered in [22]. Multipeakon solutions were studied in [15, 23]. The Cauchy problem of the Novikov equation on the circle was investigated in [24]. An alternative modified Camassa-Holm equation was introduced in [25].

Motivated by the references cited above, the goal of the present paper is to establish the local well-posedness for the strong solutions to the Cauchy problem (1). The proof of the local well-posedness is inspired by the argument of approximate solutions by Danchin [26] in the study of the local wellposedness to the Camassa-Holm equation.

However, one problematic issue is that we here deal with two-component system with a higher order nonlinearity in the Besov spaces, making the proof of several required nonlinear estimates somewhat delicate. These difficulties are nevertheless overcome by careful estimates for each iterative approximation of solutions to (1). Moreover, we also prove the analyticity of its solutions𝑢 = 𝑢(𝑡, 𝑥)in both variables, with 𝑥inR and𝑡in an interval around zero, provided that the initial profile𝑢₀is an analytic function on the real line. Hence, this analytic result can be viewed as a Cauchy-Kowalevski theorem for (1).

Now we are in the position to state the local existence result and analyticity result, where the definition of Besov spaces𝐵^𝑠_𝑝,𝑟,𝐸^𝑠_𝑝,𝑟(𝑇), and and𝐸_𝑠₀will be given in Sections2 and3.

Theorem 1. Let𝑝, 𝑟 ∈ [1, ∞]and𝑠 > max{5/2, 2 + (1/𝑝)}.

Assume that(𝑢₀,V₀) ∈ 𝐵^𝑠_𝑝,𝑟× 𝐵^𝑠_𝑝,𝑟. There exists a time𝑇 > 0 such that the initial-value problem(1)has a unique solution

(𝑢,V) ∈ 𝐸^𝑠_𝑝,𝑟(𝑇) × 𝐸^𝑠_𝑝,𝑟(𝑇)and the map(𝑢₀,V₀) 󳨃→ (𝑢,V)is continuous from a neighborhood of(𝑢₀,V₀)in𝐵^𝑠_𝑝,𝑟× 𝐵^𝑠_𝑝,𝑟into

𝐶 ([0, 𝑇] ; 𝐵^𝑠_𝑝,𝑟^󸀠 ) ∩ 𝐶¹([0, 𝑇] ; 𝐵^𝑠_𝑝,𝑟^󸀠⁻¹)

× 𝐶 ([0, 𝑇] ; 𝐵^𝑠_𝑝,𝑟^󸀠 ) ∩ 𝐶¹([0, 𝑇] ; 𝐵_𝑝,𝑟^𝑠^󸀠⁻¹)

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for every𝑠^󸀠< 𝑠when𝑟 = ∞and𝑠^󸀠= 𝑠whereas𝑟 < ∞.

Theorem 2. If the initial data(^𝑢V₀⁰)is real analytic on the line Rand belongs to a space𝐸_𝑠₀, for some0 < 𝑠₀ ≤ 1, then there exist an𝜀 > 0and a unique solution(^𝑢_V)to the Cauchy problem (1)that is analytic on(−𝜀, 𝜀) ×R.

The rest of this paper is organized as follows. InSection 2, we prove the local well-posedness of the initial value problem (1) in the Besov space.Section 3is devoted to the study of the analyticity of the Cauchy problem (1) based on a contraction type argument in a suitably chosen scale of the Banach spaces.

2. Local Well-Posedness in the Besov Spaces

In this section, we will establish local well-posedness of the initial value problem (1) in the Besov spaces.

First, for the convenience of the readers, we recall some facts on the Littlewood-Paley decomposition and some useful lemmas.

Notation.Sstands for the Schwartz space of smooth functions overR^𝑑whose derivatives of all order decay at infinity.

The setS^󸀠of temperate distributions is the dual set ofSfor the usual pairing. We denote the norm of the Lebesgue space 𝐿^𝑝(R)by‖ ⋅ ‖_𝐿^𝑝with1 ≤ 𝑝 ≤ ∞, and the norm in the Sobolev space𝐻^𝑠(R)with𝑠 ∈Rby‖ ⋅ ‖_𝐻^𝑠.

Proposition 3 (Littlewood-Paley decomposition [27]). Let B≐ {𝜉 ∈R^𝑑,|𝜉| ≤ 4/3}andC≐ {𝜉 ∈R^𝑑,4/3 ≤ |𝜉| ≤ 8/3}.

There exist two radial functions𝜒 ∈ 𝐶^∞_𝑐 (B)and𝜑 ∈ 𝐶^∞_𝑐 (C) such that

𝜒 (𝜉) + ∑

𝑞≥0

𝜑 (2^−𝑞𝜉) = 1, ∀𝜉 ∈R^𝑑,

󵄨󵄨󵄨󵄨󵄨𝑞 − 𝑞^󸀠󵄨󵄨󵄨󵄨󵄨 ≥ 2 󳨐⇒Supp𝜑 (2^−𝑞⋅) ∩Supp𝜑 (2^−𝑞^󸀠⋅) = 0, 𝑞 ≥ 1 󳨐⇒Supp𝜒 (⋅) ∩Supp𝜑 (2⁻^q^󸀠⋅) = 0,

1

3 ≤ 𝜒(𝜉)²+ ∑

𝑞≥0

𝜑(2^−𝑞𝜉)²≤ 1, ∀𝜉 ∈R^𝑑.

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Furthermore, letℎ ≐ F⁻¹𝜑and̃ℎ ≐ F⁻¹𝜒. Then for all 𝑓 ∈S^󸀠(R^𝑑), the dyadic operatorsΔ_𝑞and𝑆_𝑞can be defined as follows:

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Δ_𝑞𝑓 ≐ 𝜑 (2^−𝑞𝐷) 𝑓

= 2^𝑞𝑑∫

R^𝑑ℎ (2^𝑞𝑦) 𝑓 (𝑥 − 𝑦) 𝑑𝑦 for𝑞 ≥ 0, 𝑆_𝑞𝑓 ≐ 𝜒 (2^−𝑞𝐷) 𝑓

= ∑

−1≤𝑘≤𝑞−1

Δ_𝑘 = 2^𝑞𝑑∫

R^𝑑̃ℎ (2^𝑞𝑦) 𝑓 (𝑥 − 𝑦) 𝑑𝑦, Δ₋₁𝑓 ≐ 𝑆₀𝑓, Δ_𝑞𝑓 ≐ 0 for 𝑞 ≤ −2.

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Hence,

𝑓 = ∑

𝑞≥0Δ_𝑞𝑓 inS^󸀠(R^𝑑) , (7)

where the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of𝑓.

Lemma 4 (Bernstein’s inequality [28]). LetBbe a ball with center0inR^𝑑andCa ring with center0inR^𝑑. A constant𝐶 exists so that, for any positive real number𝜆, any nonnegative integer𝑘, any smooth homogeneous function𝜎of degree𝑚, and any couple of real numbers(𝑎, 𝑏)with𝑏 ≥ 𝑎 ≥ 1, there hold

Supp̂𝑢 ⊂ 𝜆B

󳨐⇒sup

|𝛼|=𝑘󵄩󵄩󵄩󵄩𝜕^𝛼𝑢󵄩󵄩󵄩󵄩𝐿^𝑎≤ 𝐶^𝑘+1𝜆𝑘+𝑑((1/𝑎)−(1/𝑏))‖𝑢‖_𝐿^𝑎, Supp̂𝑢 ⊂ 𝜆C

󳨐⇒ 𝐶^−𝑘−1𝜆^𝑘‖𝑢‖_𝐿^𝑎 ≤sup

|𝛼|=𝑘󵄩󵄩󵄩󵄩𝜕^𝛼𝑢󵄩󵄩󵄩󵄩𝐿^𝑎≤ 𝐶^𝑘+1𝜆^𝑘‖𝑢‖_𝐿^𝑎, Supp̂𝑢 ⊂ 𝜆C

󳨐⇒ ‖𝜎 (𝐷) 𝑢‖_𝐿^𝑏≤ 𝐶_𝜎,𝑚𝜆𝑚+𝑑((1/𝑎)−(1/𝑏))‖𝑢‖_𝐿^𝑎,

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for any function𝑢 ∈ 𝐿^𝑎.

Definition 5 (Besov space). Let𝑠 ∈ R, 1 ≤ 𝑝, 𝑟 ≤ ∞.

The inhomogeneous Besov space𝐵^𝑠_𝑝,𝑟(R^𝑑)(𝐵^𝑠_𝑝,𝑟for short) is defined by

𝐵^𝑠_𝑝,𝑟≐ {𝑓 ∈S^󸀠(R^𝑑) ; 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟< ∞} , (9) where

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟≐ {{ {{ {{ {{ {

(∑

𝑞∈Z

2^𝑞𝑠𝑟󵄩󵄩󵄩󵄩󵄩Δ^𝑞𝑓󵄩󵄩󵄩󵄩󵄩^𝑟𝐿_𝑝)

1/𝑟

, for𝑟 < ∞, sup𝑞∈Z2^𝑞𝑠󵄩󵄩󵄩󵄩󵄩Δ^𝑞𝑓󵄩󵄩󵄩󵄩󵄩𝐿𝑝, for𝑟 = ∞.

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If𝑠 = ∞,𝐵^∞_𝑝,𝑟≐ ∩_𝑠∈R𝐵_𝑝,𝑟^𝑠 .

Proposition 6 (see [28]). Suppose that𝑠 ∈R,1 ≤ 𝑝, 𝑟,𝑝_𝑖, 𝑟_𝑖≤

∞ (𝑖 = 1, 2). One has the following.

(1)Topological properties:𝐵^𝑠_𝑝,𝑟is a Banach space which is continuously embedded inS^󸀠.

(2)Density:𝐶_𝑐^∞is dense in𝐵^𝑠_𝑝,𝑟⇔ 1 ≤ 𝑝,𝑟 ≤ ∞.

(3)Embedding:𝐵^𝑠_𝑝₁_,𝑟₁ 󳨅→ 𝐵_𝑝^{𝑠−𝑛(1/𝑝}₂_,𝑟₂ ¹^)−(1/𝑝²⁾, if𝑝₁ ≤ 𝑝₂and 𝑟₁≤ 𝑟₂.

𝐵^𝑠_𝑝,𝑟²₂ 󳨅→ 𝐵^𝑠_𝑝,𝑟¹₁locally compact, if𝑠₁< 𝑠₂.

(4)Algebraic properties: for all𝑠 > 0,𝐵^𝑠_𝑝,𝑟 ∩ 𝐿^∞ is an algebra. Moreover, 𝐵^𝑠_𝑝,𝑟 is an algebra, provided that 𝑠 > 𝑛/𝑝or𝑠 ≥ 𝑛/𝑝and𝑟 = 1.

(5)Complex interpolation:

‖𝑢‖_𝐵𝜃𝑠1+(1−𝜃)𝑠2 𝑝,𝑟

≤ 𝐶‖𝑢‖^𝜃_𝐵^𝑠1_𝑝,𝑟‖𝑢‖^1−𝜃_𝐵𝑠2

𝑝,𝑟, ∀𝑢 ∈ 𝐵^𝑠_𝑝,𝑟¹ ∩ 𝐵_𝑝,𝑟^𝑠² , ∀𝜃 ∈ [0, 1] . (11) (6)Fatou lemma: If(𝑢_𝑛)_𝑛∈Nis bounded in𝐵^𝑠_𝑝,𝑟and𝑢_𝑛 →

𝑢inS^󸀠, then𝑢 ∈ 𝐵^𝑠_𝑝,𝑟and

‖𝑢‖_𝐵^𝑠_𝑝,𝑟≤lim inf

𝑛 → ∞󵄩󵄩󵄩󵄩𝑢^𝑛󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟. (12) (7)Let𝑚 ∈Rand f be an𝑆^𝑚-multiplier (i.e.,𝑓 :R^𝑑 → R is smooth and satisfies that for all𝛼 ∈N^𝑑, there exists a constant𝐶_𝛼, s.t.|𝜕^𝛼𝑓(𝜉)| ≤ 𝐶_𝛼(1 + |𝜉|^{𝑚−|𝛼|})for all 𝜉 ∈ R^𝑑). Then the operator𝑓(𝐷)is continuous from 𝐵^𝑠_𝑝,𝑟to𝐵^𝑠−𝑚_𝑝,𝑟 .

Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later.

Lemma 7 (see [26,28]). Suppose that(𝑝, 𝑟) ∈ [1, +∞]²and 𝑠 > −(𝑑/𝑝). LetVbe a vector field such that∇Vbelongs to 𝐿¹([0, 𝑇]; 𝐵^𝑠−1_𝑝,𝑟)if𝑠 > 1 + (𝑑/𝑝)or to𝐿¹([0, 𝑇]; 𝐵_𝑝,𝑟^𝑑/𝑝∩ 𝐿^∞) otherwise. Suppose also that𝑓₀ ∈ 𝐵^𝑠_𝑝,𝑟,𝐹 ∈ 𝐿¹([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟) and that𝑓 ∈ 𝐿^∞(𝐿¹([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟) ∩ 𝐶([0, 𝑇];S^󸀠)solves the𝑑- dimensional linear transport equations

𝜕_𝑡𝑓 +V⋅ ∇𝑓 = 𝐹,

𝑓󵄨󵄨󵄨󵄨𝑡=0= 𝑓₀. (𝑇)

Then there exists a constant𝐶depending only on𝑠, 𝑝, and𝑑 such that the following statements hold.

(1)If𝑟 = 1or𝑠 ̸= 1 + (𝑑/𝑝), then

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟≤ 󵄩󵄩󵄩󵄩𝑓⁰󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ ∫^𝑡

0‖𝐹 (𝜏)‖_𝐵^𝑠_𝑝,𝑟𝑑𝜏 + 𝐶 ∫^𝑡

0𝑉^󸀠(𝜏) ‖𝐹 (𝜏)‖_𝐵^𝑠_𝑝,𝑟𝑑𝜏,

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or

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟≤ 𝑒^{𝐶𝑉(𝑡)}𝐶 (󵄩󵄩󵄩󵄩𝑓⁰󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ ∫^𝑡

0𝑒^{−𝐶𝑉(𝜏)}‖𝐹 (𝜏)‖_𝐵^𝑠_𝑝,𝑟𝑑𝜏) (14)

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hold, where𝑉(𝑡) = ∫₀^𝑡‖∇V(𝜏)‖_𝐵^𝑑/𝑝

𝑝,𝑟∩𝐿^∞𝑑𝜏if𝑠 < 1+(𝑑/𝑝) and𝑉(𝑡) = ∫₀^𝑡‖∇V(𝜏)‖_𝐵^𝑠−1_𝑝,𝑟𝑑𝜏else.

(2)If𝑠 ≤ 1 + (𝑑/𝑝)and∇𝑓₀∈ 𝐿^∞,∇𝑓 ∈ 𝐿^∞([0, 𝑇] ×R^𝑑) and∇𝐹 ∈ 𝐿¹([0, 𝑇]; 𝐿^∞), then

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩∇𝑓󵄩󵄩󵄩󵄩𝐿^∞

≤ 𝑒^{𝐶𝑉(𝑡)}(󵄩󵄩󵄩󵄩𝑓⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩∇𝑓⁰󵄩󵄩󵄩󵄩𝐿^∞

+ ∫^𝑡

0𝑒^{−𝐶𝑉(𝜏)}‖𝐹 (𝜏)‖_𝐵^𝑠_𝑝,𝑟+‖∇𝐹 (𝜏)‖_𝐿^∞𝑑𝜏) (15)

with𝑉(𝑡) = ∫₀^𝑡‖∇V(𝜏)‖_𝐵^𝑑/𝑝

𝑝,𝑟∩𝐿^∞𝑑𝜏.

(3)If𝑓 =V, then for all𝑠 > 0, the estimate(14)holds with 𝑉(𝑡) = ∫₀^𝑡‖∇V(𝜏)‖_𝐵^𝑠−1_𝑝,𝑟𝑑𝜏.

(4)If𝑟 < +∞, then𝑓 ∈ 𝐶([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟). If𝑟 = +∞, then 𝑓 ∈ 𝐶([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟^󸀠 )for all𝑠^󸀠< 𝑠.

Lemma 8 (existence and uniqueness see [26, 28]). Let (𝑝, 𝑝₁, 𝑟) ∈ [1, +∞]³and𝑠 > −𝑑min{1/𝑝₁, 1/𝑝^󸀠}with𝑝^󸀠 ≐ (1−(1/𝑝))⁻¹. Assume that𝑓₀∈ 𝐵^𝑠_𝑝,𝑟,𝐹 ∈ 𝐿¹([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟). LetV be a time-dependent vector field such thatV∈ 𝐿^𝜌([0, 𝑇]; 𝐵^−𝑀_∞,∞) for some𝜌 > 1,𝑀 > 0and∇V ∈ 𝐿¹([0, 𝑇]; 𝐵^𝑑/𝑝_𝑝,𝑟 ∩ 𝐿^∞)if 𝑠 < 1 + (𝑑/𝑝₁)and∇V∈ 𝐿¹([0, 𝑇]; 𝐵^𝑠−1_𝑝₁_,𝑟)if𝑠 > 1 + (𝑑/𝑝)or 𝑠 = 1+(𝑑/𝑝₁)and𝑟 = 1. Then the transport equations(𝑇)have a unique solution𝑓 ∈ 𝐿^∞([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟) ∩ (∩_𝑠^󸀠_<𝑠𝐶[0, 𝑇]; 𝐵^𝑠_𝑝,1^󸀠 ) and the inequalities inLemma 7hold true. Moreover,𝑟 < ∞, then one has𝑓 ∈ 𝐶[0, 𝑇]; 𝐵^𝑠_𝑝,1).

Lemma 9 (1-𝐷Morse-type estimates [26,28]). Assume that 1 ≤ 𝑝,𝑟 ≤ +∞, the following estimates hold.

(i)For𝑠 > 0,

󵄩󵄩󵄩󵄩𝑓𝑔󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟≤ 𝐶 (󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞+ 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐿^∞) . (16) (ii)For all𝑠₁≤ 1/𝑝 < 𝑠₂(𝑠₂≥ 1/𝑝if𝑟 = 1) and𝑠₁+𝑠₂> 0,

one has

󵄩󵄩󵄩󵄩𝑓𝑔󵄩󵄩󵄩󵄩𝐵^𝑠1_𝑝,𝑟≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐵^𝑠1_𝑝,𝑟󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐵^𝑠2_𝑝,𝑟. (17) (iii)In Sobolev spaces𝐻^𝑠= 𝐵^𝑠_2,2, one has for𝑠 > 0,

󵄩󵄩󵄩󵄩𝑓𝜕^𝑥𝑔󵄩󵄩󵄩󵄩𝐻^𝑠≤ 𝐶 (󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐻^𝑠+1󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩𝐿^∞+ 󵄩󵄩󵄩󵄩𝜕^𝑥𝑔󵄩󵄩󵄩󵄩𝐻^𝑠󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝐿^∞) , (18) where𝐶is a positive constant independent of𝑓and𝑔.

Definition 10. For𝑇 > 0,𝑠 ∈Rand1 ≤ 𝑝 ≤ +∞, we set 𝐸^𝑠_𝑝,𝑟(𝑇) ≜ 𝐶 ([0, 𝑇] ; 𝐵^𝑠_𝑝,𝑟) ∩ 𝐶¹([0, 𝑇] ; 𝐵^𝑠−1_𝑝,𝑟) if𝑟 < +∞,

𝐸^𝑠_𝑝,∞(𝑇) ≜ 𝐿^∞([0, 𝑇] ; 𝐵^𝑠_𝑝,∞) ∩lip¹([0, 𝑇] ; 𝐵^𝑠−1_𝑝,∞) , 𝐸^𝑠_𝑝,𝑟≜ ∩_𝑇>0𝐸^𝑠_𝑝,𝑟(𝑇) .

(19)

In the following, we denote 𝐶 > 0a generic constant only depending on𝑝,𝑟, 𝑠. Uniqueness and continuity with respect to the initial data are an immediate consequence of the following result.

Proposition 11. Let1 ≤ 𝑝,𝑟 ≤ +∞and𝑠 > max{5/2, 2 + (1/𝑝)}. Suppose that(𝑢^(𝑖);V^(𝑖)) ∈ {𝐿^∞([0, 𝑇]; 𝐵^𝑠_𝑝,𝑟) ∩ 𝐶([0, 𝑇];

S^󸀠)}² (𝑖 = 1, 2) be two given solutions of the initial-value problem(1)with the initial data(𝑢^(𝑖)₀ ;V^(𝑖)₀ ) ∈ 𝐵^𝑠_𝑝,𝑟× 𝐵^𝑠_𝑝,𝑟 (𝑖 = 1; 2). Then for every𝑡 ∈ [0; 𝑇], one has

󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾(𝑡) − 𝑢⁽²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩V⁽¹⁾(𝑡) −V⁽²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟

≤ (󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾⁰ − 𝑢⁽²⁾₀ 󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩V⁽¹⁾₀ −V₀⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟)

×exp{𝐶 ∫^𝑡

0(󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾(𝜏)󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾(𝜏)󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟

+󵄩󵄩󵄩󵄩󵄩V⁽¹⁾(𝜏)󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽²⁾(𝜏)󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠𝑝,𝑟) 𝑑𝜏} . (20) Proof. Denote𝑢⁽¹²⁾ = 𝑢⁽²⁾− 𝑢⁽¹⁾,V⁽¹²⁾ = V⁽²⁾−V⁽¹⁾,𝑚⁽¹²⁾ = 𝑚⁽²⁾− 𝑚⁽¹⁾and𝑛⁽¹²⁾= 𝑛⁽²⁾− 𝑛⁽¹⁾. It is obvious that

𝑢⁽¹²⁾,V⁽¹²⁾∈ 𝐿^∞([0, 𝑇] ; 𝐵^𝑠_𝑝,𝑟) ∩ 𝐶 ([0, 𝑇] ;S^󸀠) , (21) which implies that 𝑢⁽¹²⁾,V⁽¹²⁾ ∈ 𝐶([0, 𝑇]; 𝐵^𝑠−1_𝑝,𝑟) and (𝑢⁽¹²⁾, V⁽¹²⁾, 𝑚⁽¹²⁾, 𝑚⁽¹²⁾)solves the transport equations

𝑚⁽¹²⁾_𝑡 + 𝑢⁽¹⁾V⁽¹⁾𝑚⁽¹²⁾_𝑥 = 𝐹,

𝑛⁽¹²⁾_𝑡 + 𝑢⁽¹⁾V⁽¹⁾𝑛⁽¹²⁾_𝑥 = 𝐺, (22) with

𝐹 = − (𝑢⁽²⁾V⁽¹²⁾+ 𝑢⁽¹²⁾V⁽¹⁾) 𝑚⁽²⁾_𝑥

− 3 (𝑢⁽²⁾_𝑥 V⁽¹²⁾𝑚⁽²⁾+ 𝑢⁽²⁾_𝑥 V⁽¹⁾𝑚⁽¹²⁾+ 𝑢⁽¹²⁾_𝑥 V⁽¹⁾𝑚⁽¹⁾) , 𝐺 = − (V⁽²⁾𝑢⁽¹²⁾+V⁽¹²⁾𝑢⁽¹⁾) 𝑛⁽²⁾_𝑥

− 3 (V⁽²⁾_𝑥 𝑢⁽¹²⁾𝑛⁽²⁾+V⁽²⁾_𝑥 𝑢⁽¹⁾𝑛⁽¹²⁾+V⁽¹²⁾_𝑥 𝑢⁽¹⁾𝑛⁽¹⁾) . (23)

According toLemma 7, we have 𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠󵄩󵄩󵄩󵄩󵄩𝑚⁽¹²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩󵄩𝑚⁽¹²⁾⁰ 󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

+ 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠(‖𝐹‖_𝐵^𝑠−3_𝑝,𝑟) 𝑑𝜏, 𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠󵄩󵄩󵄩󵄩󵄩𝑛⁽¹²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩󵄩𝑛⁽¹²⁾⁰ 󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3𝑝,𝑟

+ 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠(‖𝐺‖_𝐵^𝑠−3_𝑝,𝑟) 𝑑𝜏, (24)

(5)

For𝑠 >max{5/2, 2 + (1/𝑝)}, byLemma 9, we have

‖𝐹‖_𝐵^𝑠−3_𝑝,𝑟

= 󵄩󵄩󵄩󵄩󵄩−(𝑢⁽²⁾V⁽¹²⁾+ 𝑢⁽¹²⁾V⁽¹⁾) 𝑚⁽²⁾_𝑥

+3 (𝑢⁽²⁾_𝑥 V⁽¹²⁾𝑚⁽²⁾+ 𝑢⁽²⁾_𝑥 V⁽¹⁾𝑚⁽¹²⁾+ 𝑢⁽¹²⁾_𝑥 V⁽¹⁾𝑚⁽¹⁾)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

≤ 𝐶󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾V⁽¹²⁾+ 𝑢⁽¹²⁾V⁽¹⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑚⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠−2󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾𝑚⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠−2󵄩󵄩󵄩󵄩󵄩V⁽¹⁾𝑚⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2𝑝,𝑟󵄩󵄩󵄩󵄩󵄩V⁽¹⁾𝑚⁽¹⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3𝑝,𝑟

≤ 𝐶 (󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟)

× (󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟) ,

‖𝐺‖_𝐵^𝑠−3_𝑝,𝑟

= 󵄩󵄩󵄩󵄩󵄩−(V⁽²⁾𝑢⁽¹²⁾+V⁽¹²⁾𝑢⁽¹⁾) 𝑛⁽²⁾_𝑥

+3 (V⁽²⁾_𝑥 𝑢⁽¹²⁾𝑛⁽²⁾+V⁽²⁾_𝑥 𝑢⁽¹⁾𝑛⁽¹²⁾+V⁽¹²⁾_𝑥 𝑢⁽¹⁾𝑛⁽¹⁾)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

≤ 𝐶󵄩󵄩󵄩󵄩󵄩V⁽²⁾𝑢⁽¹²⁾+V⁽¹²⁾𝑢⁽¹⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑛⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩V⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾𝑚⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3_𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩V⁽²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾𝑛⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3𝑝,𝑟

+ 𝐶󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾𝑛⁽¹⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−3𝑝,𝑟

≤ 𝐶 (󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟)

× (󵄩󵄩󵄩󵄩󵄩V⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟) . (25)

Therefore, inserting the above estimates to (24), we obtain

𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠

× (󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟)

≤ 󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾⁰ 󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾₀ 󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1𝑝,𝑟

+ 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−2^𝑝,𝑟^𝑑𝜏^󸀠

× (󵄩󵄩󵄩󵄩󵄩V⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟 + 󵄩󵄩󵄩󵄩󵄩𝑢⁽¹²⁾󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟)

× (󵄩󵄩󵄩󵄩󵄩V⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩𝑢⁽²⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟) 𝑑𝜏.

(26) Hence, thanks to

󵄩󵄩󵄩󵄩󵄩𝜕^𝑥(𝑢⁽¹⁾V⁽¹⁾)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟 ≤ 𝐶 (󵄩󵄩󵄩󵄩󵄩𝑢⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V⁽¹⁾󵄩󵄩󵄩󵄩󵄩²𝐵^𝑠_𝑝,𝑟) , (27) and then applying the Gronwall’s inequality, we reach (20).

Now let us start the proof ofTheorem 1, which is motivated by the proof of local existence theorem about the Camassa-Holm equation in [26]. Firstly, we will use the classical Friedrichs’ regularization method to construct the approximate solutions to the Cauchy problem (14).

Lemma 12. Assume that𝑢⁽⁰⁾ =V⁽⁰⁾= 0. Let1 ≤ 𝑝,𝑟 ≤ +∞, 𝑠 > max{5/2, 2 + (1/𝑝)}and𝑢₀,V₀ ∈ 𝐵^𝑠_𝑝,𝑟. Then there exists a sequence of smooth functions (𝑢^(𝑙),V^(𝑙))_𝑙∈N ∈ 𝐶(𝑅⁺; 𝐵^∞_𝑝,𝑟)² solving the following linear transport equation by induction:

(𝜕_𝑡+ (𝑢^(𝑙)V^(𝑙)) 𝜕_𝑥) 𝑚^(𝑙+1)

= −3V^(𝑙)𝑢^(𝑙)_𝑥 𝑚^(𝑙), 𝑡 > 0, 𝑥 ∈R (𝜕_𝑡+ (𝑢^(𝑙)V^(𝑙)) 𝜕_𝑥) 𝑛^(𝑙+1)

= −3𝑢^(𝑙)V^(𝑙)_𝑥𝑛^(𝑙), 𝑡 > 0, 𝑥 ∈R 𝑢^(𝑙+1)(𝑥, 0) = 𝑢^(𝑙+1)₀ (𝑥) = 𝑆_𝑙+1𝑢₀, 𝑥 ∈R,

V^(𝑙+1)(𝑥, 0) =V^(𝑙+1)₀ (𝑥) = 𝑆_𝑙+1V₀, 𝑥 ∈R.

(28)

Moreover, there is a positive𝑇such that the solutions satisfying the following properties:

(i)(𝑢^(𝑙),V^(𝑙))_𝑙∈Nis uniformly bounded in𝐸^𝑠_𝑝,𝑟(𝑇)×𝐸^𝑠_𝑝,𝑟(𝑇), (ii)(𝑢^(𝑙),V^(𝑙))_𝑙∈Nis a Cauchy sequence in𝐶([0, 𝑇]; 𝐵^𝑠−1_𝑝,𝑟) ×

𝐶([0, 𝑇]; 𝐵^𝑠−1_𝑝,𝑟).

Proof. Since all the data𝑆_𝑛+1𝑢₀ and 𝑆_𝑛+1V₀ belong to 𝐵^∞_𝑝,𝑟, Lemma 8enables us to show by induction that for all 𝑙 ∈ N, (28) has a global solution which belongs to𝐶(𝑅⁺; 𝐵^∞_𝑝,𝑟)². Thanks toLemma 7and the proof ofProposition 11, we have the following inequality for all𝑙 ∈N:

𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢^(𝑙)^V^(𝑙)^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠󵄩󵄩󵄩󵄩󵄩𝑚^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩𝑚⁰󵄩󵄩󵄩󵄩𝐵^𝑠−2𝑝,𝑟 + 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠

× 󵄩󵄩󵄩󵄩󵄩3V^(𝑙)𝑢^(𝑙)_𝑥 𝑚^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟𝑑𝜏,

(6)

𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢^(𝑙)^V^(𝑙)^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠󵄩󵄩󵄩󵄩󵄩𝑛^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩𝑚⁰󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠−2+ 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠

× 󵄩󵄩󵄩󵄩󵄩3𝑢^(𝑙)V^(𝑙)_𝑥𝑛^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵𝑝,𝑟^𝑠−2𝑑𝜏.

(29)

Thanks to𝑠 >max{5/2, 2 + (1/𝑝)}, we find𝐵^𝑠−2_𝑝,𝑟 is an algebra.

From this, one obtains

󵄩󵄩󵄩󵄩󵄩V^(𝑙)𝑢^(𝑙)_𝑥𝑚^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟 ≤ 𝐶󵄩󵄩󵄩󵄩󵄩V^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑚^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑢^𝑥^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟

≤ 𝐶(󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)³,

󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)V^(𝑙)_𝑥𝑛^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟 ≤ 𝐶󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩𝑛^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟󵄩󵄩󵄩󵄩󵄩V^(𝑙)_𝑥󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−2_𝑝,𝑟

≤ (󵄩󵄩󵄩󵄩󵄩V^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠 + 󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)³,

(30)

which along with the above inequality leads to

𝑒^{−𝐶 ∫}⁰^𝑡^‖𝜕^𝑥^(𝑢^(𝑙)^V^(𝑙)^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠

× (󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)

≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟

+ 𝐶 ∫^𝑡

0𝑒^{−𝐶 ∫}⁰^𝜏^‖𝜕^𝑥^(𝑢⁽¹⁾^V⁽¹⁾^)(𝜏^󸀠^)‖^𝐵𝑠−1^𝑝,𝑟^𝑑𝜏^󸀠

× (󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)³𝑑𝜏.

(31)

Let us choose a𝑇 > 0such that4𝐶(‖ 𝑢₀‖_𝐵^𝑠_𝑝,𝑟+ ‖V₀‖_𝐵_𝑝,𝑟^𝑠 )²𝑇

< 1, and suppose by induction that for all𝑡 ∈ [0, 𝑇]

󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟

(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝑡)^1/2

. (32)

Indeed, since𝐵^𝑠−1_𝑝,𝑟 is an algebra, one obtains from (32) that for any0 < 𝜏 < 𝑡

𝐶 ∫^𝑡

𝜏󵄩󵄩󵄩󵄩󵄩𝜕^𝑥(𝑢^(𝑙)V^(𝑙)) (𝜏^󸀠)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠−1_𝑝,𝑟𝑑𝜏^󸀠

≤ 𝐶 ∫^𝑡

𝜏(󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟)²𝑑𝜏

≤ 𝐶 ∫^𝑡

𝜏

(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟)² 1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝑡𝑑𝜏

=1

4ln(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟)²𝜏)

−1

4ln(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝑡) .

(33)

And then inserting (33) and (32) into (31) leads to

󵄩󵄩󵄩󵄩󵄩𝑢^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩󵄩V^(𝑙+1)(𝑡)󵄩󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟

≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵𝑝,𝑟^𝑠 + 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟

(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟)²𝑡)^1/4

+ 𝐶

(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟)²𝑡)^1/4

× ∫^𝑡

0(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝜏)^1/4

× (󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)² (1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝜏)^3/2

𝑑𝜏

≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩^𝐵^𝑠𝑝,𝑟

(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠𝑝,𝑟)²𝑡)^1/4

× (1 + 𝐶 ∫^𝑡

0

(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)² (1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠 + 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝑡)^5/4

𝑑𝜏)

= 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵_𝑝,𝑟^𝑠 + 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟

(1 − 4𝐶(󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟+ 󵄩󵄩󵄩󵄩V₀󵄩󵄩󵄩󵄩𝐵^𝑠_𝑝,𝑟)²𝑡)^1/2 .

(34)