First, by means of the Littlewood- Paley decomposition theory, we investigate the local well-posedness of the equation in Bp,rs with s &gt

Bull. Korean Math. Soc. 55 (2018), No. 1, pp. 267–296 https://doi.org/10.4134/BKMS.b161006

pISSN: 1015-8634 / eISSN: 2234-3016

THE CAUCHY PROBLEM FOR AN INTEGRABLE GENERALIZED CAMASSA-HOLM EQUATION

WITH CUBIC NONLINEARITY

Bin Liu and Lei Zhang

Abstract. This paper studies the Cauchy problem and blow-up phenom- ena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood- Paley decomposition theory, we investigate the local well-posedness of the equation in B_p,r^s with s > max{¹_p,¹₂, 1 −_p¹}, p, r ∈ [0, ∞]. Second, we prove that the equation is locally well-posed in B_2,r^s with the critical index s = ¹₂ by virtue of the logarithmic interpolation inequality and the Osgood’s Lemma, and it is shown that the data-to-solution mapping is H¨older continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

1. Introduction

In this paper, we consider the Cauchy problem and blow-up phenomena for the following generalized Camassa-Holm equation with cubic nonlinearity:

(1.1)







(1 − ∂²_x)ut= u²_xuxxx+ uxu²_xx+ 2uuxuxxx+ uu²_xx+ u²_xuxx

+ u²uxxx− u³_x− u²uxx− 3uu²_x− 2u²ux, x ∈ R, t ≥ 0, u(x, 0) = u₀(x), x ∈ R.

The Equ. (1.1) was recently proposed by Novikov in [38], in which it is shown that the higher symmetries of this equation are quasi-local and the first one reads

(1 + Dx)uτ = m⁻⁷(mmxx− 3m²_x− 2mmx), m = u − uxx.

Received December 20, 2016; Revised April 20, 2017; Accepted May 23, 2017.

2010 Mathematics Subject Classification. 37L05, 35G25, 35Q35, 35A10.

Key words and phrases. Cauchy problem, generalized Camassa-Holm equation, nonho- mogeneous Besov space, cubic nonlinearity, blow-up phenomena.

c

2018 Korean Mathematical Society 267

Denoting m = u − u_xx, one can transform the Equ. (1.1) into the equivalent form:

(1.2)







m_t= − u²_xm_x− 2muu_x+ m²u_x

− 2uuxmx+ mu(m − u) − mu²_x− u²mx, x ∈ R, t ≥ 0, m(x, 0) = m0(x), x ∈ R.

In [38], the author proved that the Equ. (1.2) possesses an infinite hierarchy of local higher symmetries in m, and the first such symmetry is given by

mτ = (1 − Dx)m⁻⁷(mmxx− 3m²_x− 2mmx), m = u − uxx.

It is worth pointing out that the Equ. (1.2) is a cubic integrable equation, which is a special case of the integrable non-evolutionary partial differential equations of the form

(1 − D_x²)ut= F (u, ux, uxx, uxxx, . . .), u = u(x, t), Dx= ∂

∂x

, (1.3)

where F is some function of u and its derivatives with respect to x (see Novikov [38] for more details). The Equ. (1.3) contains another two integrable equations with cubic nonlinearity, which have attracted much attention in the past few years.

The first one is the following Novikov equation:

m_t+ u²m_x+ 3uu_xm = 0, m = u − u_xx. (1.4)

After its derivation, many papers were devoted to the studying of the Novikov equation. For instance, Himonas and Holliman considered the Cauchy problem of (1.4) in Sobolev space H^s(R)(s > ³2) on both R and T = R/2πZ [25]. In [22], Grayshan investigated the non-periodic and periodic Cauchy problems for equation (1.4) in H^s(R) with s < ³2. With some sign condition on the ini- tial data, Lai etc. established the existence of global weak solution for Novikov equation in the Sobolev space H^s(R) for 1 ≤ s ≤ ³2 [32]. However, without sign condition on the initial data, Lai also obtained the existence of global weak so- lutions for the Novikov equation in the space C([0, ∞); R) ∩ L^∞([0, ∞); H¹(R)) [31]. In [37], Ni and Zhou proved the locally well-posedness for the Novikov equation in the critical Besov spaces B_2,1³² , and they also studied the well- posedness in H^s(s >³₂) by using the Kato’s semigroup theory. For s ≥ 3, it is shown that the orbit invariants can be applied to investigate the existence of periodic global strong solution [41]. In [33], Lenells and Wunsch proved that the weakly dissipative versions of the Novikov equation are equivalent to their non-dissipative counterparts up to a simple change of variables. For the other works about Equ. (1.4), see [29, 36, 44–46] and the references therein.

The second one is the cubic Fokas-Olver-Rosenau-Qiao (FORQ) equation:

mt+ (u²− u²_x)mx+ 2uxm²= 0, m = u − uxx. (1.5)

The Equ. (1.5) was introduced by Fuchssteiner [21] and Olver and Rosenau [39]

as a new generalization of integrable system by implemention a simple explicit algorithm based on the bi-Hamiltonian representation of the classical integrable system. The FORQ equation possesses the Lax pair, and it is shown that the Cauchy problem can be solved by the inverse scattering transform method. In [23], Gui etc. proved that the FORQ equation admits single peakons in the form of u(x, t) =

q3c

2e^−|x−ct|(c > 0). Moreover, it is shown in [20] that the FORQ equation does not have any nontrivial smooth traveling wave solutions. Being inspired by the approach developed in [14], the authors also established the local well-posedness of the FORQ equation in B_p,r^s (R) with s > max{⁵₂, 2 +¹_p} and p, r ∈ [1, ∞] [20]. And the well-posedness in the Besov space B_2,1^s (R) with the critical index s = ⁵₂ is investigated [20]. In [26], Himonas and Mantzavinos studied the well-posedness of (1.5) in H^s(R)(s > ⁵₂), and they proved that the solution mapping is continuous but not uniformly continuous. For more papers concerning the FORQ equation, we refer the readers to [27, 47, 48].

The most celebrated integrable member of Equ. (1.3) is the Camassa-Holm (CH) equation which has quadratic nonlinearity:

mt+ umx+ 2uxm = 0, m = u − uxx. (1.6)

It was originally derived by Camassa and Holm [4] to describe the undirec- tional propagation of shallow water waves over a flat bottom. It possesses a bi-Hamilton structure and has infinite conservation laws [4, 8]. One of the most interesting properties of the CH equation is that it has peakon solutions ce^−|x−ct| for c > 0, which describes an fundamental characteristic of the trav- eling waves of largest amplitude [13]. The local well-posedness and blow-up phenomena of the CH equation in the Sobolev and the Besov spaces are in- vestigated in [7, 9–11, 14, 40]. For the global existence of the weak and strong solutions, we refer the readers to [7, 9, 10, 12, 42]. For the global conservative and dissipative solutions to the CH equation, see for example [2, 3, 28].

One of the closest relatives of Equ. (1.6) is the Degasperis-Procesi (DP) equation [17]:

ut− uxxt= 4uux− 3uxuxx− uuxxx. (1.7)

The DP equation is similar to the CH equation in several aspects, such as the asymptotic accuracy, the completely integrability and the bi-Hamiltonian structure [16]. However, the two equations are truly different. One of the novel features of the DP equation is that it not only has peakon solutions [16]

and periodic peakon solutions [43], but also the periodic shock waves [19] and shock peakons [35]. For the other works, see for example [6, 18, 24, 34, 49] and the references therein.

To our best knowledge, the Cauchy problem and blow-up phenomena for the Equ. (1.1) (or (1.2)) has not been studied yet. In this paper, by using the Littlewood-Paley theory, we first establish the local existence and uniqueness

of the solution in B^s_p,r with s > max{¹_p,¹₂, 1 −¹_p}, p, r ∈ [0, ∞]. Second, we investigate the local well-posedness in B_2,r^s with the critical index s = ¹₂. Un- fortunately, compared with the noncritical case, it seems impossible to find any convergence subsequence of the approximation solutions in C([0, T ]; B_2,1¹² (R)) directly, because the priori estimates in noncritical case depend strongly on the inequality kf gk_Bs

p,r ≤ Ckf k_Bs

p,rkgk_Bs+1

p,r , which does not hold any more for s = −¹₂, p = 2 and r = 1. However, by taking advantage of a new Moser-type interpolation inequality (see Lemma 2.8 below), we can overcome this problem caused by the low regularity of B⁻

1 2

2,1(R), and then prove the convergence of approximation solutions in C([0, T ]; B⁻

1 2

2,∞(R)) with the help of the logarithmic interpolation inequality and the Osgood’s Lemma. Moreover, we show that the solution mapping is H¨older continuous. Finally, we give two blow-up criteria for the strong solutions by using induction and the method of characteristics.

The paper is organized as follows. In Section 2, we recall some facts on the Littwood-Paley theory and the transport theory in Besov spaces. Section 3 is devoted to the local well-posedness of the Equ. (1.1) in the Besov spaces. In Section 4, we derive two kinds of blow-up criteria for the strong solution by using mathematical induction and the characteristics method.

Notation. All function spaces are considered in R, and we shall drop them in our notation if there is no ambiguity. We denote by C the estimates that hold up to some universal constant which may change from line to line but whose meaning is clear throughout the context.

2. Preliminaries.

Unless otherwise specified, all the results presented in this section have been proved in [1, 5, 15]. Fix a function ψ(x) ∈ C₀^∞(B4/3) with B4/3= {x ∈ R; |x| ≤

4

3} and a function ϕ(x) ∈ C₀^∞(C) with C = {x ∈ R;³4 ≤ |x| ≤ ⁸₃} such that ψ(x) +X

j≥0

ϕ(2^−jx) = 1, ∀x ∈ R,

|i − j| ≥ 2 =⇒ supp ϕ(2^−j·) ∩ supp ϕ(2⁻ⁱ·) = ∅, j ≥ 1 =⇒ supp ψ(·) ∩ supp ϕ(2^−j·) = ∅.

The nonhomogeneous Littlewood-Paley decomposition operators {∆q}q≥−1are defined by

∆qu = 0, if q < −1; ∆₋₁u , ψ(D)u; ∆^qu , ϕ(2^−qD)u, if q ≥ 0, where f (D) stands for the pseudo-differential operator u →F⁻¹(fF u). Since ϕ(ξ) = ψ(^ξ₂) − ψ(ξ), we also introduce the following low frequency cut-off operator Sq:

S_qu , ϕ(2^−qD)u = X

1≤p≤q−1

∆_pu.

Definition 2.1. Let 1 ≤ p, r ≤ ∞, s ∈ R, the 1-D nonhomogeneous Besov space B^s_p,r is defined by

B_p,r^s , {u ∈ S⁰; kukB_p,r^s =

X

j∈N

2^rsjk∆juk^r_Lp

¹_r

< ∞}.

If r = ∞, B_p,r^∞ ,T

s>0B_p,r^s .

Lemma 2.2. LetC be an annulus and B a ball, there exists a constant C > 0 such that for ∀k ∈ N⁺, u ∈ L^p and (p, q) ∈ [1, ∞]²with q ≥ p ≥ 1, the following estimates hold:

Suppbu ⊆ λB ⇒ kD^kukL^q , sup

|α|≤k

k∂^αukL^q ≤ C^k+1λ^k+d(1p−1q)kukL^p, Suppbu ⊆ λC ⇒ C^−k−1λ^kkukL^p≤ kD^kukL^p≤ C^k+1λ^kkukL^p.

Definition 2.3. Let u and v be two temperate distributions. The nonhomo- geneous paraproduct of v by u is defined by

Tuv ,X

j

Sj−1u∆jv.

The nonhomogeneous remainder of u and v is defined by R(u, v) , X

|k−j|≤1

∆_ku∆_jv.

The Bony decomposition of uv is given by

uv , Tuv + T_vu + R(u, v).

Lemma 2.4. For any (s, t) ∈ R × (−∞, 0) and any (p, r¹, r2) ∈ [1, ∞]³, there exists a positive constant C such that, for r = min{1,_r¹

1 + _r¹

2}, the following estimates hold:

kTuvkB^s_p,r ≤ CkukL^∞kD^kvk_Bs−k p,r , kT_uvk_Bs+t

p,r ≤ Ckuk_Bt

∞,r1kD^kvk_Bs−k p,r2.

Lemma 2.5. Let (s1, s2) ∈ R² and (p1, p2, r1, r2) ∈ [1, ∞]⁴. Assume that 1

p = 1 p1

+ 1 p2

, 1 r = 1

r1

+ 1 r2

.

Then there exists a constant C > 0 such that the following estimates hold:

(1) If s₁+ s₂> 0, for any (u, v) in B^s_p²

1,r₁× B_p^s2

2,r₂, we have kR(u, v)k_Bs1+s2

p,r ≤ Ckuk_Bs1 p,rkvk_Bs2

p,r.

(2) If r = 1 and s1+ s2= 0, for any (u, v) in B^s_p²_1,r1× B_p^s2_2,r2, we have kR(u, v)k_B0

p,∞≤ Ckuk_Bs1 p,rkvk_Bs2

p,r.

Lemma 2.6. Let s ∈ R, 1 ≤ p, r, p1, r_i≤ ∞, i = 1, 2. Then

(1) Algebraic properties: if s > 0, B_p,r^s T L^∞ is an algebra. Furthermore, B_p,r^s is an algebra provided that s > 1/p or s = 1/p and r = 1.

(2) Embedding: If p₁ ≤ p2 and r₁ ≤ r2, then B_p^s

1,r₁ ,→ Bp^s−(1/p₂,r₂ ^1−1/p2). If s₁< s₂, the embedding B^s_p,r²

2 ,→ B_p,r^s1

1 is locally compact.

(3) Fatou’s lemma: if {un}_n∈N+ is bounded in B^s_p,r and un→ u inS⁰, then u ∈ B^s_p,r and

kukB^s_p,r ≤ lim inf

n→∞ kuⁿkB_p,r^s .

(4) If s₁≤ ¹_p < s₂ (s₂≥¹_p if r = 1) and s₁+ s₂> 0, then we have kuvk_Bs1

p,r ≤ Ckuk_Bs1 p,rkvk_Bs2

p,r.

(5) A smooth function f : R^d→ R is said to be an S^m-multiplier: if ∀α ∈ Nⁿ, there exists a constant Cα > 0 such that |∂^αf (ξ)| ≤ Cα(1 + |ξ|)^m−|α| for all ξ ∈ R^d. The operator f (D) is continuous from B_p,r^s to B_p,r^s−mfor all s ∈ R and 1 ≤ p, r ≤ ∞.

Lemma 2.7. (1) Let s₂> s₁, θ ∈ (0, 1), we have kuk_Bθs1+(1−θ)s2

p,1

≤ C

s2− s1

 1 θ+ 1

1 − θ

 kuk^θ_Bs1

p,∞kuk^1−θ_Bs2 p,∞.

(2) For ∀s ∈ R,  > 0 and 1 ≤ p ≤ ∞, there exists a constant C > 0 such that

kukB^s_p,1 ≤ C + 1

 kukB_p,∞^s 1 + logkuk_B^s+

p,∞

kuk_Bs p,∞

! .

Lemma 2.8. For any f ∈ B_2,∞⁻¹² and g ∈ B_2,∞¹² ∩ L^∞, there exists a constant C > 0 such that

kf gk

B^{− 1}_2,∞² ≤ Ckf k

B^{− 1}_2,∞²kgk

B

1 2 2,∞∩L^∞.

Lemma 2.9. Let 1 ≤ p, r ≤ ∞, s ≥ − min(¹_p, 1−¹_p), and consider the following transport equation:

∂tf + v∂xf = g, f (x, 0) = f0(x).

(2.1)

Assume that f₀∈ B_p,r^s and g ∈ L¹([0, T ]; B_p,r^s ). For any solution f ∈ L^∞([0, T ]; B_p,r^s )

of (2.1) with vx ∈ L¹([0, T ]; B_p,r^s−1) if s > 1 + ¹_p or vx ∈ L¹([0, T ]; B

1

p,rp T L^∞) otherwise.

(1) If r = 1 or s 6= 1 + ¹_p, there exists C > 0 depending only on s, p and r such that

kf k_Bs

p,r ≤ exp{CV_p(t)}kf₀k_Bs p,r+

Z t 0

exp{CV_p(t)−CV_p(s)}kg(s)k_Bs p,rds, (2.2)

with

Vp(t) ,





 Rt

0kv_x(s)k

B

1 p p,∞T L^∞

ds, if s < 1 +¹_p; Rt

0kvx(s)k_Bs−1

p,r ds, if s > 1 +¹_p or s = 1 +¹_p, r = 1.

(2) If r < ∞, then f ∈ C([0, T ]; B_p,r^s ). If r = ∞, then f ∈ C([0, T ]; B_p,1^s0 ) for all s⁰< s.

(3) If v = f and s > 0, the inequality (2.2) holds true with Vp(t) :=

Rt

0kvx(s)kL^∞ds.

Lemma 2.10. Let p, r, s, f0 and g be as in Lemma 2.9. Suppose that v ∈ L^ρ([0, T ]; B^−M_∞,∞) for some ρ > 1, M > 0 and vx∈ L¹([0, T ]; B_p,r^s−1) if s > 1 +¹_p or s = 1 +¹_p and r = 1, and v_x∈ L¹([0, T ]; B

1

p,∞p T L^∞) if s < 1 +_p¹. Then the transport equation (2.1) admits a unique solution u in the space C([0, T ]; B^s_p,r) if r < ∞ or L^∞([0, T ]; B_p,r^s )T(T_s0<sC([0, T ]; B_p,1^s0 ) if r < ∞. Moreover, the inequalities of Lemma 2.9 hold true.

Lemma 2.11. Let ρ ≥ 0 be a measurable function, γ > 0 be a locally integrable function and µ be a continuous and increasing function. For some a ≥ 0, if

ρ(t) ≤ a + Z t

t0

γ(s)µ(ρ(s))ds.

(1) If a > 0, then −M(ρ(t)) + M(a) ≤Rt

t₀γ(s)ds, where M(x) ,R1 x

1 µ(r)dr.

(2) If a = 0 and µ satisfies the condition R1 0

dr

µ(r)dr = +∞, then ρ ≡ 0.

3. Local well-posedness in the Besov spaces

In this section, we shall investigate the local well-posedness of the initial value problem (1.2) in the nonhomogeneous Besov spaces.

3.1. Local well-posedness in B^s_p,r with s > max{¹_p,¹₂, 1 −¹_p}, p, r ∈ [1, ∞].

Noting that the Equ. (1.2) can be reformulated in the form of (mt+ (u + ux)²mx= m²(u + ux) − m(u + ux)², x ∈ R, t ≥ 0,

m(x, 0) = m₀(x), x ∈ R, (3.1)

where m = u − u_xx is the momentum variable in the physical sense. To give the main result in this subsection, we introduce the following spaces for conve- nience.

Definition 3.1. For ∀T > 0, s ∈ R and 1 ≤ p ≤ +∞, we set E_p,r^s (T ) ,

(C([0, T ]; B_p,r^s ) ∩ C¹([0, T ]; B_p,r^s−1), if r < ∞, L^∞([0, T ]; B_p,∞^s ) ∩ Lip([0, T ]; B_p,∞^s−1), if r = ∞, and E^s_p,r=T

T >0E_p,r^s (T ).

The uniqueness and existence of the solution is ensured by the following theorem.

Theorem 3.2. Let m₀∈ B_p,r^s be the initial data, where s > max{_p¹,¹₂, 1 −¹_p}, p, r ∈ [1, ∞]. Then there exists a time T , T (km⁰kB^s_p,r) > 0 such that the Equ. (3.1) admits a unique solution m ∈ E_p,r^s (T ). Moreover, the solution map- ping m₀ 7→ m is continuous from B_p,r^s into C([0, T ]; B_p,r^s0 ) ∩ C¹([0, T ]; B_p,r^s0−1) for all s⁰< s if r = ∞, and s⁰= s if 1 ≤ r < ∞.

By using the Littlewood-Paley decomposition and the Plancherel’s formula, the Besov space B_2,2^s is equivalent to the Sobolev spaces H^s. Therefore, The- orem 3.2 implies the following result.

Corollary 3.3. Let m0 ∈ H^s with s > ¹₂, there exists a T , T (km⁰kH^s) > 0 such that the Equ. (3.1) admits a unique solution

m ∈ C([0, T ]; H^s) ∩ C¹([0, T ]; H^s−1).

Moreover, the solution mapping m₀→ m is continuous.

The uniqueness and continuity with respect to the initial data in some sense can be obtained by the following priori estimates.

Lemma 3.4. Assume that s > max{¹_p,¹₂, 1 − ¹_p} with p, r ∈ [1, ∞]. Let m⁽¹⁾, m⁽²⁾ ∈ L^∞([0, T ]; B_p,r^s ) ∩ C([0, T ]; S⁰) be two solutions to the Equ. (3.1) with respect to the initial datum m⁽¹⁾₀ , m⁽²⁾₀ ∈ B^s_p,r, respectively. Denoting

m⁽¹²⁾, m⁽¹⁾− m⁽²⁾ and m⁽¹²⁾₀ , m⁽¹⁾(0) − m⁽²⁾(0), then for ∀t ∈ [0, T ], the following priori estimates hold:

(1) If s > max{1 −¹_p,¹_p,¹₂} but s 6= 2 +¹_p, we have

km⁽¹²⁾(t)k_Bs−1

p,r ≤ km⁽¹²⁾₀ k_Bs−1 p,r exp

 C

Z t 0

(km⁽²⁾(s)k²_Bs

p,r+ km⁽¹⁾(s)k²_Bs p,r)ds

 . (3.2)

(2) If s = 2 + ¹_p, for any θ ∈ (0, 1), we have km⁽¹²⁾k

B^{1+ 1}_p,r^p

≤ Ckm⁽¹²⁾₀ k^θ

B^{1+ 1}_p,r^p

(km⁽¹⁾k^1−θ

B^{2+ 1}_p,r^p

+ km⁽²⁾k^1−θ

B^{2+ 1}_p,r^p

)

× exp

 Cθ

Z t 0

(km⁽²⁾(s)k²

Bp,r^{2+ 1p}

+ km⁽¹⁾(s)k²

Bp,r^{2+ 1p}

)ds

 . (3.3)

Proof. Obviously, the function m⁽¹²⁾(x, t) satisfies the following transport equa- tion:

(m⁽¹²⁾_t + (u⁽¹⁾+ u⁽¹⁾x )²m⁽¹²⁾x = F⁽¹²⁾(x, t), m⁽¹²⁾(x, 0) = m⁽¹²⁾₀ (x) , m⁽¹⁾0 − m⁽²⁾₀ , (3.4)

where F⁽¹²⁾(x, t) = [(u⁽²⁾+u⁽²⁾x )²−(u⁽¹⁾+u⁽¹⁾x )²]m⁽²⁾x +m⁽¹²⁾(m⁽¹⁾+m⁽²⁾)(u⁽¹⁾+ u⁽¹⁾x ) + (m⁽²⁾)²(u⁽¹²⁾+ u⁽¹²⁾x ) − m⁽¹²⁾(u⁽²⁾+ u⁽²⁾x )²− m⁽¹⁾(u⁽²⁾+ u⁽²⁾x + u⁽¹⁾+ u⁽¹⁾x )(u⁽¹²⁾+ u⁽¹²⁾x ).

For s > max{¹_p,¹₂, 1 −¹_p} but s 6= 2 +¹_p, by applying Lemma 2.9 to the first equation of (3.1), we deduce that

km⁽¹²⁾(t)k_Bs−1

p,r ≤ km⁽¹²⁾₀ k_Bs−1 p,r +

Z t 0

kF⁽¹²⁾(·, t)k_Bs−1 p,r ds (3.5)

+ C Z t

0

V_p⁰(s)km⁽¹²⁾(s)k_Bs−1 p,r ds, where

V_p(t) , Z t

0

(k∂_x(u⁽¹⁾+ u⁽¹⁾_x )²k

B

1 p p,r∩L^∞

+ k∂_x(u⁽¹⁾+ u⁽¹⁾_x )²k_Bs−2 p,r )dτ.

Noting that (1 − ∂_x²)⁻¹ is a multiplier of degree −2, it follows from the (5) of Lemma 2.6 that

C1ku⁽ⁱ⁾k_Bs+2

p,r . km⁽ⁱ⁾kB_p,r^s . C²ku⁽ⁱ⁾k_Bs+2

p,r , for ∀s ∈ R, i = 12, 1, 2, (3.6)

where C1 and C2are positive constants independent of the index i.

Since s > max{1 −¹_p,¹_p,¹₂}, the Besov space B^s_p,r is a Banach algebra, so we have

V_p⁰(t) = k∂_x(u⁽¹⁾+ u⁽¹⁾_x )²k

B

1 p p,r∩L^∞

+ k∂_x(u⁽¹⁾+ u⁽¹⁾_x )²k_Bs−2 p,r

≤ Ck∂x(u⁽¹⁾+ u⁽¹⁾_x )²kB_p,r^s + k(u⁽¹⁾+ u⁽¹⁾_x )²kB_p,r^s

≤ C(ku⁽¹⁾k_Bs+1

p,r + ku⁽¹⁾_x k_Bs+1

p,r )²≤ Ckm⁽¹⁾k²_Bs p,r. (3.7)

Next, let us estimate the term kF⁽¹²⁾(·, t)k_Bs−1

p,r in (3.5). If max{¹_p,¹₂} < s ≤ 1 + ¹_p, it is obvious that s − 1 ≤ ¹_p ≤ s and (s − 1) + s > 0, by using the interpolation inequality ((4) of Lemma 2.6) and the fact that B_p,r^s is a Banach algebra, we have

k[(u⁽²⁾+ u⁽²⁾_x )²− (u⁽¹⁾+ u⁽¹⁾_x )²]m⁽²⁾_x k_Bs−1

(3.8) p,r

≤ Ck(u⁽²⁾+ u⁽²⁾_x + u⁽¹⁾+ u⁽¹⁾_x )(u⁽¹²⁾+ u⁽¹²⁾_x )kB_p,r^s km⁽²⁾_x k_Bs−1 p,r

≤ Cku⁽²⁾+ u⁽²⁾_x + u⁽¹⁾+ u⁽¹⁾_x kB^s_p,rku⁽¹²⁾+ u⁽¹²⁾_x kB^s_p,rkm⁽²⁾kB_p,r^s

≤ C(km⁽¹⁾kB_p,r^s + km⁽²⁾kB_p,r^s )km⁽²⁾kB_p,r^s ku⁽¹²⁾k_Bs+1 p,r

≤ C(km⁽¹⁾k²_Bs

p,r+ km⁽²⁾k²_Bs

p,r)km⁽¹²⁾k_Bs−1 p,r .

By taking the similar argument, we can also estimate that km⁽¹²⁾(m⁽¹⁾+ m⁽²⁾)(u⁽¹⁾+ u⁽¹⁾_x )k_Bs−1

(3.9) p,r

≤ Ckm⁽¹²⁾k_Bs−1

p,r k(m⁽¹⁾+ m⁽²⁾)(u⁽¹⁾+ u⁽¹⁾_x )kB_p,r^s

≤ Ckm⁽¹²⁾k_Bs−1

p,r (km⁽¹⁾k_Bs

p,r+ km⁽²⁾k_Bs

p,r)(ku⁽¹⁾k_Bs

p,r+ ku⁽¹⁾k_Bs+1 p,r)

≤ Ckm⁽¹²⁾k_Bs−1

p,r (km⁽¹⁾k²_Bs

p,r+ km⁽²⁾k²_Bs p,r),

k(m⁽²⁾)²(u⁽¹²⁾+ u⁽¹²⁾_x ) − m⁽¹²⁾(u⁽²⁾+ u⁽²⁾_x )²k_Bs−1

(3.10) p,r

≤ C(km⁽²⁾k²_Bs

p,rku⁽¹²⁾+u⁽¹²⁾_x k_Bs−1

p,r +km⁽¹²⁾k_Bs−1

p,r k(u⁽²⁾+u⁽²⁾_x )²k_Bs−1 p,r )

≤ C(km⁽²⁾k²_Bs

p,rku⁽¹²⁾kB^s_p,r+ km⁽¹²⁾k_Bs−1

p,r ku⁽²⁾+ u⁽²⁾_x k²_Bs p,r)

≤ Ckm⁽¹²⁾k_Bs−1

p,r km⁽²⁾k²_Bs p,r,

km⁽¹⁾(u⁽²⁾+ u⁽²⁾_x + u⁽¹⁾+ u⁽¹⁾_x )(u⁽¹²⁾+ u⁽¹²⁾_x )k_Bs−1

(3.11) p,r

≤ Ckm⁽¹⁾(u⁽²⁾+ u⁽²⁾_x + u⁽¹⁾+ u⁽¹⁾_x )kB^s_p,rku⁽¹²⁾+ u⁽¹²⁾_x k_Bs−1 p,r

≤ Ckm⁽¹⁾kB^s_p,r(ku⁽²⁾k_Bs+1

p,r + ku⁽¹⁾k_Bs+1

p,r)ku⁽¹²⁾k_Bs+1 p,r

≤ Ckm⁽¹²⁾k_Bs−1

p,r (km⁽²⁾k²_Bs

p,r+ km⁽¹⁾k²_Bs p,r).

Putting the estimates (3.8)-(3.11) together, we obtain kF⁽¹²⁾(·, t)k_Bs−1

p,r ≤ C(km⁽²⁾k²_Bs

p,r+ km⁽¹⁾k²_Bs

p,r)km⁽¹²⁾k_Bs−1 p,r . (3.12)

Therefore, it follows from (3.5), (3.7) and (3.12) that km⁽¹²⁾(t)k_Bs−1

p,r ≤ km⁽¹²⁾₀ k_Bs−1 p,r

+ C Z t

0

km⁽¹²⁾(τ )k_Bs−1

p,r (km⁽²⁾k²_Bs

p,r+ km⁽¹⁾k²_Bs p,r)dτ + C

Z t 0

km⁽¹⁾k²_Bs

p,rkm⁽¹²⁾(τ )k_Bs−1 p,r dτ

≤ km⁽¹²⁾₀ k_Bs−1 p,r

+ C Z t

0

km⁽¹²⁾(τ )k_Bs−1

p,r (km⁽²⁾k²_Bs

p,r+ km⁽¹⁾k²_Bs p,r)dτ.

By applying the Gronwall inequality, we get

km⁽¹²⁾(t)k_Bs−1

p,r ≤ km⁽¹²⁾₀ k_Bs−1 p,r exp

 C

Z t 0

(km⁽²⁾(τ )k²_Bs

p,r+ km⁽¹⁾(τ )k²_Bs p,r)dτ

 . (3.13)

On the other hand, for s > 1 + ¹_p but s 6= 2 + ¹_p, the Besov space B_p,r^s−1 is a Banach algebra, by virtue of the property of (3.6), it is easier to obtain the desired inequality (3.2), and we shall omit the details here. Hence we have proved (1).

To deal with the criteria case s = 2 + ¹_p, we use the interpolation method.

Indeed, by choosing θ = ¹₂(1 − _2p¹) ∈ (0, 1), it is clear that s − 1 = 1 + ¹_p =

1

2pθ + (2 +_2p¹)(1 − θ). By utilizing the interpolation inequality ((4) of Lemma

2.6), we deduce that km⁽¹²⁾k_Bs−1

p,r ≤ Ckm⁽¹²⁾k^θ

B

1 p,r2p

km⁽¹²⁾k^1−θ

B^{2+ 1}p,r^2p

≤ Ckm⁽¹²⁾₀ k^θ

B

2p1 p,r

exp

 Cθ

Z t

0

(km⁽²⁾(τ )k²

Bp,r^{1+ 12p}

+ km⁽¹⁾(τ )k²

B^{1+ 1}p,r^2p

)dτ



km⁽¹²⁾k^1−θ

B^{2+ 1}p,r^2p

≤ Ckm⁽¹²⁾₀ k^θ

B

2p1 p,r

exp

 Cθ

Z t

0

(km⁽²⁾(τ )k²

Bp,r^{1+ 12p}

+ km⁽¹⁾(τ )k²

B^{1+ 1}p,r^2p

)dτ



× (km⁽¹⁾k^1−θ

B_p,r^{2+ 12p}

+ km⁽²⁾k^1−θ

B_p,r^{2+ 12p}

)

≤ Ckm⁽¹²⁾₀ k^θ_Bs−1

p,r (km⁽¹⁾k^1−θ_Bs

p,r+ km⁽²⁾k^1−θ_Bs p,r)

× exp

 Cθ

Z t 0

(km⁽²⁾(τ )k²_Bs

p,r+ km⁽¹⁾(τ )k²_Bs p,r)dτ

 . (3.14)

This completes the proof of Lemma 3.4. 

Next, we shall construct the approximation solution to the Equ. (3.1) by means of the classical Friedrich’s regularization method.

Lemma 3.5. Let p, r be the same as in the statement Lemma 3.4. Let s >

max{¹_p,¹₂, 1 −¹_p} and m0 ∈ B_p,r^s be the initial data. Assume that m⁽⁰⁾ = 0, then

(1) there exists a sequence of smooth functions {m⁽ⁿ⁾}_n∈N+∈ C([0, ∞); B^∞_p,r) which solves the following linear transport equation:

(m⁽ⁿ⁺¹⁾_t +(u⁽ⁿ⁾+u⁽ⁿ⁾x )²m⁽ⁿ⁺¹⁾x = (m⁽ⁿ⁾)²(u⁽ⁿ⁾+u⁽ⁿ⁾x )−m⁽ⁿ⁾(u⁽ⁿ⁾+u⁽ⁿ⁾x )², m⁽ⁿ⁺¹⁾(x, 0) = m⁽ⁿ⁺¹⁾₀ (x) , Sⁿ⁺¹m0,

(3.15)

where Sn+1is the low frequency cut-off of m0given by Sn+1m0=P

q≥−1∆qm0. (2) there exists a T , T (km0kB^s_p,r) > 0 such that the sequence {m⁽ⁿ⁾}_n∈N⁺ is uniformly bounded in E_p,r^s (T ), and it is also a Cauchy sequence in

C([0, T ]; B_p,r^s−1) and converges to some function m ∈ C([0, T ]; B_p,r^s−1).

Proof. Noting that Sn+1m0∈ B_p,r^∞ ,T

s∈RB_p,r^s , by virtue of Lemma 2.10 and induction with respect to the index n, we deduce that the Equ. (3.15) admits a unique global solution m⁽ⁿ⁺¹⁾∈ C([0, T ]; B_p,r^∞) for any T > 0. It remains to prove (2).

To this end, let us first show that the sequence {m⁽ⁿ⁾}_n∈N+ is uniformly bounded in C([0, T ]; B_p,r^s ) for some T > 0 which depends on the initial data.

By applying Lemma 2.9 to (3.15), we get km⁽ⁿ⁺¹⁾(t)k_Bs

p,r ≤ exp{CV (t)}kS_n+1m₀k_Bs p,r+

Z t 0

exp{CV (t) − CV (τ )}

×

k(m⁽ⁿ⁾)²(u⁽ⁿ⁾+u⁽ⁿ⁾_x )kB^s_p,r+km⁽ⁿ⁾(u⁽ⁿ⁾+u⁽ⁿ⁾_x )²kB^s_p,r

 dτ, (3.16)

where the V (t) is defined as follows V (t) ,

Z t 0

(k∂_x(u⁽ⁿ⁾+ u⁽ⁿ⁾_x )²k_Bs−1

p,r + k∂_x(u⁽ⁿ⁾+ u⁽ⁿ⁾_x )²k

B

1 p p,r∩L^∞

)dτ

≤ C Z t

0

ku⁽ⁿ⁾+ u⁽ⁿ⁾_x k²_Bs

p,rdτ ≤ C Z t

0

km⁽ⁿ⁾(τ )k²_Bs p,rdτ.

(3.17)

Since s > max{¹_p,¹₂, 1 −¹_p}, the Besov space B_p,r^s is a Banach algebra, we can estimate that

k(m⁽ⁿ⁾)²(u⁽ⁿ⁾+ u⁽ⁿ⁾_x )k_Bs

p,r+ km⁽ⁿ⁾(u⁽ⁿ⁾+ u⁽ⁿ⁾_x )²k_Bs p,r

≤ Ckm⁽ⁿ⁾k²_Bs

p,rku⁽ⁿ⁾+ u⁽ⁿ⁾_x kB_p,r^s + Ckm⁽ⁿ⁾kB_p,r^s ku⁽ⁿ⁾+ u⁽ⁿ⁾_x k²_Bs p,r

≤ Ckm⁽ⁿ⁾k²_Bs

p,r(ku⁽ⁿ⁾kB^s_p,r+ ku⁽ⁿ⁾k_Bs+1 p,r) + Ckm⁽ⁿ⁾k_Bs

p,r(ku⁽ⁿ⁾k²_Bs

p,r+ ku⁽ⁿ⁾_x k²_Bs p,r)

≤ Ckm⁽ⁿ⁾k³_Bs

p,r+ Ckm⁽ⁿ⁾kB_p,r^s (km⁽ⁿ⁾k²_Bs−2

p,r + km⁽ⁿ⁾k²_Bs−1 p,r )

≤ Ckm⁽ⁿ⁾k³_Bs p,r. (3.18)

Plugging the estimates (3.17)-(3.18) into (3.16), we obtain km⁽ⁿ⁺¹⁾(t)kB^s_p,r ≤ C



exp{CV (t)}km0kB_p,r^s

+ Z t

0

exp{CV (t) − CV (τ )}km⁽ⁿ⁾(τ )k³_Bs p,rdτ



≤ C

 exp

 C

Z t 0

km⁽ⁿ⁾(τ )k²_Bs p,rdτ

 km₀k_Bs

p,r

+ Z t

0

exp

 C

Z t τ

km⁽ⁿ⁾(τ⁰)k²_Bs p,rdτ⁰



km⁽ⁿ⁾(τ )k³_Bs p,rdτ

 . (3.19)

Here we have used the fact that kS_n+1m₀k_Bs

p,r≤ Ckm₀k_Bs

p,r, where the positive constant C is independent of the index n.

If km⁽ⁿ⁾kB^s_p,r < 1, it follows from (3.19) that km⁽ⁿ⁺¹⁾kB_p,r^s < C(km0kB_p,r^s + t), which implies that the sequence {m⁽ⁿ⁾}_n∈N+ is uniformly bounded. On the other hand, if km⁽ⁿ⁾kB^s_p,r ≥ 1, let us fix a T > 0 such that 4T C³km0k²_Bs

p,r < 1, and

km⁽ⁿ⁾(t)kB_p,r^s ≤ Ckm₀kB_p,r^s

q1 − 4C³km0k²_Bs p,rt (3.20)

≤ Ckm₀kB^s_p,r

q1 − 4C³km0k²_Bs

p,rT , H for ∀t ∈ [0, T ].

In view of the definition of V (t), we can deduce from (3.20) that

exp{CV (t) − CV (τ )} = exp

 C

Z t τ

km⁽ⁿ⁾(τ⁰)k²_Bs p,rdτ⁰



≤ exp

 Z t τ

C³km0k²_Bs p,r

1 − 4C³km0k²_Bs p,rtdτ⁰



= ⁴ v u u t

1 − 4C³km0k²_Bs p,rτ 1 − 4C³km0k²_Bs

p,rt. (3.21)

Especially, by taking τ = 0, we have

exp{CV (t)} ≤ 1

4

q1 − 4C³km0k²_Bs p,rt

for ∀t ∈ [0, T ].

(3.22)

By (3.20)-(3.22), the inequality (3.19) is equivalent to

km⁽ⁿ⁺¹⁾(t)kB_p,r^s

≤ C

 km0kB_p,r^s

4

q1 − 4C³km0k²_Bs p,rt

+ Z t

0

4

s1 − 4C³km0kB^s_p,rτ 1 − 4C³km₀k_Bs

p,rtkm⁽ⁿ⁾(τ )k³_Bs p,rdτ



≤ C

 km0kB_p,r^s

4

q1 − 4C³km0k²_Bs p,rt

+ km0kB^s_p,r

4

q1 − 4C³km0k²_Bs p,rt

Z t 0

C³km0k²_Bs p,rdτ (1 − 4C³km0k²_Bs

p,rt)⁵⁴



= Ckm₀kB_p,r^s

q1 − 4C³km0k²_Bs p,rt

for ∀t ∈ [0, T ]. Hence we obtain that the sequence {m⁽ⁿ⁾}_n∈N+ is uniformly bounded in the case of km⁽ⁿ⁾kB^s_p,r ≥ 1 in the interval [0, T ]. On this base, utilizing the Equ. (3.15), it is not difficult to see that {∂_tm⁽ⁿ⁾}_n∈N⁺ is uni- formly bounded in C([0, T ]; B_p,r^s−1). Therefore, we have proved the uniformly boundedness of the sequence {m⁽ⁿ⁾}_n∈N+ in E_p,r^s (T ).