The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space

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Ann. I. H. Poincaré – AN 32 (2015) 443–469

www.elsevier.com/locate/anihpc

The Cauchy problem for the modified two-component Camassa–Holm system in critical Besov space

Wei Yan

, Yongsheng Li

aCollege of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, PR China bDepartment of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, PR China

Received 20 April 2013; accepted 10 January 2014 Available online 28 January 2014

Abstract

In this paper, we are concerned with the Cauchy problem for the modified two-component Camassa–Holm system in the Besov space with data having critical regularity. The key elements in our paper are the real interpolations and logarithmic interpolation among inhomogeneous Besov space and Lemma 5.2.1 of[7]which is also called Osgood Lemma and the Fatou Lemma. The new ingredient that we introduce in this paper can be seen on pages453–457.

©2014 Elsevier Masson SAS. All rights reserved.

MSC:35G25; 35L05; 35R25

Keywords:Cauchy problem; Modified two-component Camassa–Holm system; Critical Besov space; Osgood Lemma

1. Introduction

In this paper, we consider the Cauchy problem for the modified two-component Camassa–Holm equation (MCH2)

m_t+um_x+2uxm= −ρρ_x, t >0, x∈R, (1.1)

ρ_t+(uρ)_x=0, t >0, x∈R, (1.2)

m(x,0)=m₀(x), x∈R, (1.3)

ρ(x,0)=ρ₀(x), x∈R, (1.4)

wherem(x, t )=u(x, t )−u_xx(x, t),ρ(x)=(1−∂_x²)(ρ−ρ₀).(1.1)–(1.2)are proposed by Holm et al. in [33]to find a model that describes the motion of shallow water waves other than Camassa–Holm (CH) or two-component Camassa–Holm (CH2) and has both similar and different dynamics of singular solutions compared with CH or CH2.

These equations allow a dependence on not only the pointwise densityρbut also the average densityρ0. When the evolution of the density is ignored, i.e.ρ=0, the MCH2 reduces to the CH equation

m_t+um_x+2uxm=0, m=u−u_xx, (1.5)

* Corresponding author.

E-mail addresses:yanwei19821115@sina.cn(W. Yan),yshli@scut.edu.cn(Y. Li).

0294-1449/$ – see front matter ©2014 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2014.01.003

which models the unidirectional propagation of shallow water waves over a flat bottom, whereu(x, t )represents the fluid velocity at time t in the spatial x direction[6,18].(1.5)possesses the bi-Hamiltonian structures and is com- pletely integrable[6,8,9,25]and has attracted the attention of many researchers, e.g.[2,6,10–17,19–21,31,32,36–40].

(1.5)possesses two remarkable properties. One is that(1.5)possesses the peaked solitary wavesu(x, t )=ce^−|x−ct|, c=0 which is orbitally stable[15,16]. The other is breaking waves. More precisely, the solution remains bounded while its slope becomes unbounded in finite time[10–12]. The existence and uniqueness of the weak solutions to the Cauchy problem for(1.5)has been studied in[13,14,28,43,44]. The results of[2]and[32]implied thats=³₂ is the critical Sobolev index for the well-posedness inH^s in the sense of Hadamard. The local well-posedness of(1.5)in Besov spaceB_2,1^3/2has been proved by Danchin[20]. Bressan and Constantin[3,4]showed that after wave breaking, the solutions can be continued uniquely as either global conservative or global dissipative solutions. Recently, the initial-boundary value problem for(1.5)has been studied by Escher and Yin[23,24]. Very recently, the new and direct proof for Mckean’s theorem[41]on wave-breaking of the Camassa–Holm equation has been given by Jiang et al.[34].

Withm=u−u_xx,ρ=γ−γ_xx andγ=ρ−ρ0, we can rewrite(1.1)–(1.4)

m_t+um_x+2mux= −ργ_x, (1.6)

ρ_t+(uρ)_x=0, (1.7)

m(x,0)=m0(x)=u0x−u0xx, (1.8)

ρ(x,0)=γ0(x)−γ0xx. (1.9)

LetG(x)=¹₂e^−|x|,x∈R, then(1−∂_x²)⁻¹f =G∗f for allf ∈L²(R)andg∗m=u, where∗denotes the spatial convolution. DefiningP1(D)= −∂x(1−∂_x²)⁻¹andP2(D)= −(1−∂_x²)⁻¹, we can rewrite(1.6)–(1.9)equivalently as follows:

u_t+uu_x=P₁(D)

u²+1 2u²_x+1

2γ²−1 2γ_x²

, t >0, x∈R, (1.10)

γ_t+uγ_x=P₂(D)

(u_xγ_x)_x+u_xγ

, t >0, x∈R, (1.11)

u(x,0)=u₀(x), x∈R (1.12)

γ (x,0)=γ₀(x), x∈R. (1.13)

The mathematical properties of(1.10)–(1.13)have been investigated in many works, e.g.[27,29,30,45]. Guan and Yin[27]proved that the system(1.10)–(1.13)is locally well-posed inH^s(R)×H^s−1(R)withs > ⁵₂ and presented some blow-up results. By using Helly theorem and some a priori one-sided upper bound and higher integrability space–time estimates on the first-order derivatives of approximation solutions, Guan and Yin [30]obtained the ex- istence of global-in-time weak solutions. Guo and Zhu[29]established sufficient conditions on the initial data to guarantee blow-up solutions. Recently, by using the transport equation theory and the inhomogeneous Besov spaces, Yan et al.[45]established the local well-posedness inB_p,r^s ×B_p,r^s withs >max{1+¹_p,³₂}, 1p, r∞.

In this paper we will show that the system (1.10)–(1.13)is locally well-posed inB_2,1^3/2×B_2,1^3/2 via the iterative method and give a blow-up criterion. Now we give the outline of the well-posedness and blow-up of the system (1.10)–(1.13)in our paper. Firstly, we construct a sequence of the approximate solution(u⁽ⁿ⁾, γ⁽ⁿ⁾)via the iterative method. Secondly, by using the real interpolations and logarithmic interpolation among inhomogeneous Besov space, we prove that(u⁽ⁿ⁾, γⁿ)is a Cauchy sequence inC([0, T];B_2,^1/2_∞)×C([0, T];B_2,^1/2_∞). This is the key step to prove that(u⁽ⁿ⁾, γ⁽ⁿ⁾)is a Cauchy sequence inC([0, T];B_2,1^1/2)×C([0, T];B_2,1^1/2). The limit is the solution to(1.10)–(1.13).

Thirdly, we use the Osgood Lemma and the logarithmic interpolation and among the Besov spaces to establish the uniqueness of the solution to the system(1.10)–(1.13). Finally, we give a criterion for the blow-up of the solution.

However, we notice that we cannot obtain the solution simply extracting a convergent subsequence since(u⁽ⁿ⁾, γ⁽ⁿ⁾) is an iterative sequence.

Fors∈R, we introduce

X_s=B_2,1^s ×B_2,1^s , Y_s=B_2,^s_∞×B_2,^s_∞, E_2,1^s (T )=C

[0, T];B_2,1^s

∩C¹

[0, T];B_2,1^s−1 (u, v) ,

Xs = uB^s_2,1+ vB_2,1^s .

In this paper,C denotes the generic positive constant which may vary from line to line andC(θ )=_{θ (1}^C_{−θ )}, where θ∈(0,1). In this paper, we denote byLipthe space of bounded functions with bounded first derivatives.

The main results of this paper are as follows:

Theorem 1.1. Let (u0, γ0)∈X3/2. Then the system(1.10)–(1.13) is locally well-posed. More precisely, for any (u₀, γ₀)∈X_3/2and anyT >0, there exists a unique solution inE_2,1^3/2(T )×E^3/2_2,1(T ). The solution map which maps (u0, γ0)to(u, γ )is Hölder continuous fromX3/2toE_2,1^3/2(T )×E_2,1^3/2(T ).

Moreover, u(t )

B_2,1^3/2+γ (t)

B^3/2_2,1 u_0B^3/2

2,1 + γ_0B^3/2

2,1

1−C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)t fort∈ [0, T].

Remark.We say that X_3/2 is the critical Besov space for the well-posedness of modified Camassa–Holm system based on the following two facts.

(1) The Cauchy problem for the CH equation is locally well-posedH^s(R)withs >3/2[2,40] and is not locally well-posed inH^s(R)withs <3/2[32].

(2) Danchin[20] proved that the Cauchy problem for the CH equation is not locally well-posed in B_2,^3/2_∞ in the following sense.

There exist two solutionsu, v∈L^∞(0, T;B_2,^3/2_∞)such that for any >0 u(0)−v(0)

B_2,^3/2_∞ and u−v_L∞(0,T;B^3/2

2,∞)1.

These imply that the exponents=3/2 is a critical regularity exponent of Besov spacesB_2,r^s ,r∈ [1,+∞].

Theorem 1.2. Let (u₀, γ₀)∈X_3/2 as in Theorem1.1 and (u, γ ) be the corresponding solution to (1.10)–(1.13).

Assume thatT^∗is the maximal existence time. IfT^∗<∞, then

T^∗

0

u_xL^∞+ γ_xL^∞

dτ= +∞.

Moreover,T_C(u0 ¹

B3/2 2,1

+γ0_B3/2 2,1

).

The remainder of this paper is organized as follows. In Section2, we give some preliminaries. In Section3, we prove the existence of the solution to the problem(1.10)–(1.13)with the initial data(u₀, γ₀)∈X_3/2. In Section4, we establish the uniqueness of the solution inX_3/2. In Section5, we obtain continuity of the solution inC([0, T];X_3/2) with the initial data(u₀, γ₀)∈X_3/2. In Section6, we prove the blow-up criterion.

2. Preliminaries

In this section, we will state some preliminaries. The proof ofLemmas 2.1, 2.3–2.5can be seen in[5,19–22,42].

Let(χ , φ) be two smooth radial functions, 0(χ , φ)1, such thatχ is supported in the ball B= {ξ ∈Rⁿ,

|ξ|⁴₃}andφis supported in the ringC= {ξ∈Rⁿ, ³₄|ξ|⁸₃}. Moreover, χ (ξ )+

∞ j=0

φ 2^−jξ

=1, ∀ξ∈Rⁿ and

suppφ 2^−j·

∩suppφ 2^−j·

=∅, ifj−j 2, suppχ (·)∩suppφ

2^−j·

=∅, ifj 1.

Foru∈S (R), define the nonhomogeneous dyadic block operators _ju=0, ifj−2,

₋1u=χ (D)u=Fx⁻¹χFxu, ju=φ

2^−jD

=Fx⁻¹φ 2^−jξ

Fxu, ∀j∈N, ifj0.

Lemma 2.1(Littlewood–Paley decomposition).

(i) Foru∈S (R), u= ^∞

j=−1

_ju converges inS (R).

(ii) Foru∈H^s(R), u= ^∞

j=−1

_ju converges inH^s(R).

Remark.The low frequency cut-offS_j is defined by S_ju=

j−1

p=−1

_pu=χ 2^−jD

u=F_x⁻¹χ 2^−jξ

Fxu, ∀j∈N.

Obviously,∀u, v∈S (R)

_iju≡0, if|i−j|2, j(Si−1uiv)≡0, if|i−j|5, _juL^puL^p, ∀u∈L^p(R), S_juL^pCuL^p, ∀u∈L^p(R) whereCis a positive constant independent ofj.

Definition 2.2(Besov spaces).Lets∈R, 1p+∞. The nonhomogeneous Besov spaceB_p,r^s (R⁾is defined by B_p,r^s =B_p,r^s (R)=

f∈S (R): fB^s_p,r<∞ wherefB_p,r^s = (2^qsqfL^p)_q−1l^r.

In particular,B_p,r^∞ =

s∈RB_p,r^s .

Lemma 2.3.Lets∈R,1p, r, pj, rj∞,j=1,2, then

(1) B_p,r^s is a Banach space and is continuously embedded inS (R).

(2) B_p^s1_1,r1→B_p^s2_2,r2, ifp₁p₂andr₁r₂ands₂=s₁−n(_p¹

1 −_p¹₂), B_p,r^s1

2→B_p,r^s2

1 locally compact ifs₂< s₁.

(3) ∀s >0,B_p,r^s ∩L^∞is a Banach algebra.B_p,r^s is a Banach algebra iffB_p,r^s →L^∞and iffs >¹_p or(s_p¹ and r=1).

(4) (i) Fors >0,

f gB^s_p,rC

fB_p,r^s gL∞+ fL∞gB_p,r^s

, ∀f, g∈B_p,r^s ∩L^∞.

(ii) ∀s1_p¹ < s2(s2¹_p ifr=1)ands1+s2>0,

f g_Bp,r^s1 Cf_Bp,r^s1 g_B^s_p,r² , ∀f ∈B_p,r^s1 , g∈B_p,r^s2 . (5) ∀θ∈ [0,1]ands=θ s₁+(1−θ )s₂,

fB_p,r^s Cf^θ_B^s1

p,rf^1−θ

Bp,r^s2

, ∀f ∈B_p,r^s1 ∩B_p,r^s2 .

(6) ∀θ∈(0,1),s₁> s₂,s=θ s₁+(1−θ )s₂, there exists a constantCsuch that uB_p,1^s C(θ )

s₁−s₂u^θ_B^s1

p,∞u^1−θ

Bp,^s2∞, ∀u∈B_p,^s1_∞.

(7) If(u_n)_n∈Nis bounded inB_p,r^s andu_n→uinS (R), thenu∈B_p,r^s and uB_p,r^s lim inf

n→∞ unB_p,r^s .

(8) Letm∈RandΨ be anS^m-multiplier. Then the operatorΨ (D)is continuous fromB_p,r^s intoB_p,r^s−m. (9) The multiplication is continuous fromB_2,1^−1/2×(B_2,^1/2_∞∩L^∞)toB_2,⁻_∞^1/2.

(10) There exists a constantC >0such that for alls∈R, >0and1p∞, fB_p,1^s C1+

fB_p,∞^s ln

e+f_Bp,∞^s+

fB_p,^s_∞

, ∀f ∈B_p,^s+_∞.

Remark.

(i) From (3) and (4) we see thatB_2,1^1/2is continuously embedded inB_2,^1/2_∞∩L^∞. (ii) A special case of (6) that we shall frequently use is that for any 0< θ <1,

u_B^1/2

2,1 u

B

3 2−θ 2,1

C(θ )u^θ

B_2,∞^1/2u^1−θ

B_2,^3/2_∞. (2.1)

(iii) (8) is the lower semi-continuity of the norm ofB_p,r^s . It is also often called the Fatou Lemma.

(iv) We recall that a smooth functionΨ is said to be anS^m-multiplier if∀α∈Nⁿ, there exists a constantC_α>0 s.t.

|∂_αΨ (ξ )|C_α(1+ |ξ|)^m−|α|for allξ∈Rⁿ.

(v) For anym0,S_j is anS^m-multiplier. That is to say, for anym0, S_j is continuous fromB_p,r^s into B_p,r^s+m, P₂(D)is anS⁻²-multiplier. That is,P₂(D)is continuous fromB_p,r^s intoB_p,r^s+2.

Below are somea prioriestimates in Besov spaces of transport equation.

Lemma 2.4.Let1p, r∞ands >−min{_p¹,1−_p¹}. Assume thatf₀∈B_p,r^s ,F ∈L¹(0, T;B_p,r^s )and∂_xvbelongs toL¹(0, T;B_p,r^s−1)ifs >1+_p¹or toL¹(0, T;Bp,r^1/p∩L^∞)otherwise. Iff ∈L^∞(0, T;B_p,r^s )∩C([0, T];S (R))solves the following1-D linear transport equation:

f_t+vf_x=F, (2.2)

f (x,0)=f₀, (2.3)

then there exists a constantCdepending only ons, p, rsuch that the following statements hold:

(1) Ifr=1ors=1+_p¹, fB_p,r^s e^{CV (t )}

f0B_p,r^s +

t

0

e^{−CV (τ )}F (τ )

B_p,r^s dτ

(2.4)

where

V (t )=

t

0

vx(τ )

Bdτ (2.5)

withB=B_p,r^1/p∩L^∞ifs <1+_p¹ andB=B_p,r^s−1else.

(2) Ifs1+_p¹,f₀∈L^∞andfx∈L^∞((0, T )×R)andFx∈L¹(0, T;L^∞), then f (t )

B^s_p,r+f_x(t)

L^∞

e^{CV (t )}

f0B^s_p,r+f₀

L^∞+

t

0

e^{−CV (τ )}F (τ )

B_p,r^s +Fx(τ )

L^∞

dτ

whereV (t )is defined by(2.5)withB=B_p,r^1/p∩L^∞.

(3) Ifv=f, then∀s >0,(2.4)holds withV (t )being as in(2.5)andB=L^∞.

(4) Ifr <∞, thenf ∈C([0, T];B_p,r^s ). Ifr= ∞, thenf ∈C([0, T];B_p,1^s )for alls < s.

Lemma 2.5 (Existence and uniqueness). Let p, r, s, f0 and F be as in the statement of Lemma 2.4. Assume that v∈L^ρ(0, T;B_∞^−M_,∞)for someρ >1andM >0andvx∈L¹(0, T;B_p,r^s−1)ifs >1+_p¹ ors=1+_p¹ andr=1and v_x∈L¹(0, T;B_p,^1/p_∞∩L^∞)ifs <1+¹_p. Then the problem(2.1)–(2.2)has a unique solutionf ∈L^∞(0, T;B_p,r^s )∩ (

s<sC([0, T];B_p,1^s ))and the inequalities of Lemma2.4are true. Moreover, ifr <∞, thenf∈C([0, T];B_p,r^s ).

Below we state some basic properties with respect to logarithm function that we shall use in the sequel. The proofs are easy and thus are omitted.

Lemma 2.6.

(1) Letf (x)=x(1−lnx),x∈(0,1], thenf (x)is a monotonic increasing function forx∈(0,1].

(2) Forx∈(0,1]andα >0, we have ln

e+α

x

ln(e+α)(1−lnx). (2.6)

(3) Forx >0andα >0, we have that f (x)=xln

e+α

x

(2.7) is a monotonic increasing function inx >0.

Lemma 2.7.Letρbe a measurable, nonnegative function,γa positive, locally integrable function andμa continuous, increasing function.a0is a real number. Assume thatρsatisfies

ρ(t )a+

t

t₀

γ (s)μ ρ(s)

ds. (2.8)

Ifa >0, then we have

−Ω ρ(t )

+Ω(a)

t

t0

γ (s) ds, (2.9)

where

Ω(x)=

1

x

dr

μ(r). (2.10)

Ifa=0and ifμsatisfies

1

0

dr

μ(r) = +∞, (2.11)

then the functionρ≡0.

Lemma 2.7can be seen in Lemma 5.2.1 of[7]and[26].Lemma 2.7is also called Osgood Lemma.

Remark.InLemma 2.7, when 0a, ρR, where R >0 is real constant andμ(r)=rln(e+^C_r),r >0,C >0 which is a real constant, we claim that

ρ(t )

eR

a eR

exp(−ln(e+^C_R)t t0γ (τ ) dτ )

. (2.12)

Now we prove the claim. Whena=0, since

1

0

dr

rln(e+^C_r)= +∞, byLemma 2.7, we know thatρ≡0.

Now we consider the casea >0.

From(2.9), we derive that

ρ(t)

a

dr rln(e+^C_r)

t

t₀

γ (τ ) dτ. (2.13)

Combining(2.6)with(2.13), we have that

ρ(t)

a

dr

ln(e+^C_R)r(1−ln_R^r)

ρ(t)

a

dr rln(e+ ^CRr

R

)

=

ρ(t)

a

dr rln(e+^C_r)

t

t₀

γ (τ ) dτ. (2.14)

By(2.14), we derive that

ρ(t)

a

dr

r(1−ln_R^r)ln

e+C R

^t

t₀

γ (τ ) dτ. (2.15)

Solving(2.15)yields ρ(t )

eR

a eR

exp(−ln(e+^C_R)t t0γ (τ ) dτ )

. (2.16)

We have completed the proof of the claim.

3. Existence of solution with data inX3/2

In this section, we prove the existence of solution to(1.10)–(1.13)with data inX_3/2with the following four steps.

3.1. Approximate solution

Let (u₀, γ₀)∈X_3/2, via the iterative method, we will construct a solution. Starting from (u⁽⁰⁾, γ⁽⁰⁾)=(0,0), we define inductively a sequence of smooth functions {(u⁽ⁿ⁾, γ⁽ⁿ⁾)}n∈N by solving the following linear transport equations:

∂t+u⁽ⁿ⁾∂x

u⁽ⁿ⁺¹⁾=P1(D) u⁽ⁿ⁾2

+1 2

u⁽ⁿ⁾_x 2

+1 2

γ⁽ⁿ⁾2

−1 2

γ_x⁽ⁿ⁾2

, (3.1)

∂_t+u⁽ⁿ⁾∂_x

γ⁽ⁿ⁺¹⁾=P2(D)

u⁽ⁿ⁾_x γ_x⁽ⁿ⁾

x+u⁽ⁿ⁾_x γ⁽ⁿ⁾

, (3.2)

u⁽ⁿ⁺¹⁾(x,0)=u⁽ⁿ₀⁺¹⁾(x)=S_n+1u₀(x), (3.3)

γ⁽ⁿ⁺¹⁾(x,0)=γ₀⁽ⁿ⁺¹⁾(x)=S_n+1γ₀(x). (3.4)

By the remark afterLemma 2.3, we have that(S_n+1u₀, S_n+1γ₀)∈

s∈RX_s. FromLemma 2.4, for alln∈N, we can show by induction that the above system has a global solution(u⁽ⁿ⁺¹⁾, γ⁽ⁿ⁺¹⁾)∈

s∈RC(R⁺, X_s).

3.2. Uniform bounds

For alln∈N, letH⁽ⁿ⁾(t)= u⁽ⁿ⁾(t)_B^3/2

2,1 + γ⁽ⁿ⁾(t)_B^3/2

2,1

andH (0)= u0_B^3/2

2,1 + γ0_B^3/2

2,1

. We claim that

H⁽ⁿ⁺¹⁾(t)exp

CUⁿ(t)

H (0)+C

t

0

exp

−CUⁿ(τ )

H⁽ⁿ⁾(τ )2

dτ

, (3.5)

withUⁿ=t

0u⁽ⁿ⁾_B^3/2

2,1

dτ.

Applying(2.4)ofLemma 2.4to(3.1), we derive u⁽ⁿ⁺¹⁾(t)

B_2,1^3/2exp

C

t

0

u⁽ⁿ⁾ t

B^3/2_2,1 dt

u_0B^3/2

2,1

+

t

0

exp

C

t

τ

uⁿ t

B_2,1^3/2dt F₁

u⁽ⁿ⁾, u⁽ⁿ⁾_x , γ⁽ⁿ⁾, γ_x⁽ⁿ⁾

B^3/2_2,1 dτ, (3.6)

where F1

u⁽ⁿ⁾, u⁽ⁿ⁾_x , γ⁽ⁿ⁾, γ_x⁽ⁿ⁾

=P1(D) u⁽ⁿ⁾2

+1 2

u⁽ⁿ⁾_x 2

+1 2

γ⁽ⁿ⁾2

−1 2

γ_x⁽ⁿ⁾2 .

SinceP₁(D)is continuous fromB_p,r^s intoB_p,r^s+1andB

1 2

2,1is a Banach algebra,B_2,1^3/2→B_2,1^1/2, we have F1

u⁽ⁿ⁾, u⁽ⁿ⁾_x , γ⁽ⁿ⁾, γ_x⁽ⁿ⁾

B_2,1^3/2

C

u⁽ⁿ⁾2

+1 2

u⁽ⁿ⁾_x 2

+1 2

γ⁽ⁿ⁾2

−1 2

γ_x⁽ⁿ⁾2 B

1 2 2,1

Cu⁽ⁿ⁾²

B^1/2_2,1 +Cu⁽ⁿ⁾_x ²

B_2,1^1/2+Cγ⁽ⁿ⁾²

B_2,1^1/2+Cγ_x⁽ⁿ⁾²

B_2,1^1/2

Cu⁽ⁿ⁾²

B^3/2_2,1 +Cγ⁽ⁿ⁾²

B^3/2_2,1. (3.7)

Similarly, applying(2.4)ofLemma 2.4to(3.2), we have

γ⁽ⁿ⁺¹⁾(t)

B_2,1^3/2exp

C

t

0

u⁽ⁿ⁾ t

B_2,1^3/2dt

γ0_B^3/2

2,1

+

t

0

exp

C

t

τ

u⁽ⁿ⁾ t

B_2,1^3/2dt

P₂(D)

u⁽ⁿ⁾_x γ_x⁽ⁿ⁾

x+u⁽ⁿ⁾_x γ⁽ⁿ⁾

B

3 2 2,1

dτ

exp

C

t

0

u⁽ⁿ⁾ t

B_2,1^3/2dt

γ0_B^3/2

2,1

+

t

0

exp

C

t

τ

u⁽ⁿ⁾ t

B_2,1^3/2dt

u⁽ⁿ⁾

B_2,1^3/2γ⁽ⁿ⁾

B^3/2_2,1 dτ. (3.8)

Combining(3.6),(3.7)with(3.8), we derive(3.5). Thus we prove the claim.

Fix aT >0 such that

T 1

4C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

) (3.9)

and assume that∀t∈ [0, T] u⁽ⁿ⁾

B_2,1^3/2+γ⁽ⁿ⁾

B_2,1^3/2

(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t. (3.10)

Recalling thatUⁿ=t

0u⁽ⁿ⁾_B^3/2

2,1

dτ, by using(3.10), we have exp

CUⁿ(t)−CUⁿ(τ )

=exp

C

t

τ

u⁽ⁿ⁾

B^3/2_2,1 dτ

exp

t τ

C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)t dt

=

1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)τ 1−2C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)t 1/2

(3.11) and

exp

CUⁿ(t)

=exp

C

t

0

u⁽ⁿ⁾

B_2,1^3/2dτ

1 1−2C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)t 1/2

. (3.12)

Inserting(3.10)–(3.12)into(3.5), we have Hⁿ⁺¹(t) u0_B^3/2

2,1 + γ0_B^3/2

2,1

[1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t]^1/2

+ 1

[1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t]^1/2

t

0

C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)² [1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)τ]^3/2dτ

u0_B^3/2

2,1 + γ0_B^3/2

2,1

1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t. (3.13)

Consequently, we derive that{(u⁽ⁿ⁾, γ⁽ⁿ⁾)}n∈Nis uniformly bounded inC([0, T];X_3/2). Noting thatB_2,1^1/2is a Banach algebra andB_2,1^3/2→B_2,1^1/2, we have

u⁽ⁿ⁾u⁽ⁿ_x⁺¹⁾

B_2,1^1/2Cu⁽ⁿ⁾

B_2,1^1/2u⁽ⁿ_x⁺¹⁾

B_2,1^1/2Cu⁽ⁿ⁾

B_2,1^3/2u⁽ⁿ⁺¹⁾

B_2,1^3/2

C

(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t 2

. (3.14)

Thus, from(3.7),(3.8)and(3.14), we have u⁽ⁿ_t ⁺¹⁾

B_2,1^1/2u⁽ⁿ⁾u⁽ⁿ_x⁺¹⁾

B_2,1^1/2+F1

u⁽ⁿ⁾, u⁽ⁿ⁾_x , γ⁽ⁿ⁾, γ_x⁽ⁿ⁾

B^1/2_2,1

C

(u0_B^3/2

2,1 + γ0_B^3/2

2,1

) 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t 2

. (3.15)

Similarly, from(3.2), we have γ_t⁽ⁿ⁺¹⁾

B_2,1^1/2=P₂(D)

u⁽ⁿ⁾_x γ_x⁽ⁿ⁾

x+u⁽ⁿ⁾_x γ⁽ⁿ⁾

B_2,1^1/2+u⁽ⁿ⁾∂_xγ⁽ⁿ⁺¹⁾

B_2,1^1/2

P₁(D)

u⁽ⁿ⁾_x γ_x⁽ⁿ⁾

B_2,1^1/2+P₂(D)

u⁽ⁿ⁾_x γ⁽ⁿ⁾

B^1/2_2,1 +Cu⁽ⁿ⁾

B_2,1^1/2∂_xγ⁽ⁿ⁺¹⁾

B_2,1^1/2

Cu⁽ⁿ⁾

B_2,1^3/2γ⁽ⁿ⁾

B_2,1^3/2

C

(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u0

B

3 2 2,1

+ γ0

B

3 2 2,1

)t 2

Consequently, for∀n∈N, we have u⁽ⁿ⁾, γ⁽ⁿ⁾

∈E_2,1^3/2(T )×E_2,1^3/2(T ). (3.16)

Remark.Fort∈ [0, T]and∀(n, k)∈N², from(3.13)and(3.9), we have that exp

C

t

0

u⁽ⁿ⁾

B_2,1^3/2dτ

1 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t 1/2

1 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)T 1/2

2 (3.17)

and u^(n+k)−u⁽ⁿ⁾ (t)

B_2,∞^3/2 +γ^(n+k)−γ⁽ⁿ⁾ (t)

B^3/2_2,∞

u^(n+k)−u⁽ⁿ⁾ (t)

B_2,1^3/2+γ^(n+k)−γ⁽ⁿ⁾ (t)

B^3/2_2,1

u^(n+k)(t)

B^3/2_2,1 +u⁽ⁿ⁾(t)

B_2,1^3/2+γ^(n+k)(t)

B^3/2_2,1 +γ⁽ⁿ⁾(t)

B_2,1^3/2

2(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u_0B^3/2

2,1 + γ_0B^3/2

2,1

)t

2(u_0B^3/2

2,1 + γ_0B^3/2

2,1

) 1−2C(u0_B^3/2

2,1 + γ0_B^3/2

2,1

)T

4

u_0B^3/2

2,1 + γ_0B^3/2

2,1

(3.18)

sinceB_2,1^3/2→B_2,^3/2_∞. We define

M=4

u_0B^3/2

2,1 + γ_0B^3/2

2,1

. (3.19)