Smoothing properties for a Hirota-Satsuma systems

Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 61, 1-30;http://www.math.u-szeged.hu/ejqtde/

Smoothing properties for a Hirota-Satsuma systems

Salvattore Jimenez

& Octavio Paulo Vera Villagr´ an

Abstract

We study local existence and smoothing properties for the initial value problem associated to Hirota- Satsuma systems that describes an interaction of two long waves with different dispersion relations.

Keywords and phrases: Evolution equations, weighted Sobolev space, gain in regularity.

Mathematics Subject Classification: 35Q53, 47J35

1 Introduction

This paper is concerned with gain in regularity of solutions of the Hirota-Satsuma system

ut−a uxxx+ 6u ux= 2b v vx (1.1)

vt+vxxx+ 3u vx= 0 (1.2)

u(x,0) =u0(x) (1.3)

v(x,0) =v0(x) (1.4)

where x∈R, t∈Rand u=u(x, t), v=v(x, t) are real unknown functions. aand b are real constants with b >0.In equation (1.1), 2b v vx acts as a force term on the Korteweg-de Vries(KdV) wave system with the linear dispersion relation ω=a κ³.This system was introduced by Hirota and Satsuma [19] to describe and interaction of two long waves with different dispersion relations. If there is no effect of one of the long waves on the other, the latter obeys the ordinary KdV equation. They showed that for all values ofaanb this system possesses three conservation laws. Indeed

I1=u, I2= 1 2 Z

R

u²+2

3 b v²

dx I3=

Z

R

1

2(1 +a)u²_x+b v_x²−(1 +a)u³−b u v²

dx.

(1.5)

They further showed that for all values ofb,but onlya=¹₂,the system possesses two further conservation laws

I4 = u⁴−2u u²_x+1 5u²_xx +4

5

u²v²+2

3u v vxx+8

3u v_x²−13 18v_xx²

+ 4

15b²v⁴ (1.6)

I5=u⁵−5u²u²_x+u u²_xx− 1 14u²_xxx + 1

21b 20u³v²−10u²_xv²−+20u²v vxx+ 40u²v_x²+ 4u v vxxxx+ 56u vxvxxx+ 12u v²_xx+ 8v²_xxx +20

63b²(u v⁴−4v²v_x²). (1.7)

∗Universidade Bolivariana, Los Angeles, Chile. e-mail: sjimenez@ubolivariana.cl

†Departamento de Matem´atica, Universidad del B´ıo-B´ıo, Collao 1202, Casilla 5-C, Concepci´on, Chile.

e-mails: octaviovera49@gmail.com and overa@ubiobio.cl

The system (1.1)-(1.4) has been studied by several authors, see [8, 19, 20] and the references there. In 1986, N. Hayashi et al. [16] showed that for the nonlinear Schr¨odinger equation (NLS): i ut+uxx = λ|u|^p−1u, (x, t) ∈ R×R with initial conditionu(x,0) = u0(x), x ∈ R and a certain assumption on λ and p, all solutions of finite energy are smooth for t 6= 0 provided the initial functions in H¹(R)(or on L²(R)) decay sufficiently rapidly as |x| → ∞. The main tool is the operator J defined by Ju = e^{i x2/4t}(2i t)∂x(e^{−i x2/4t}u) = (x+ 2i t ∂x)uwhich has the remarkable property that it commutes with the operatorLdefined by L= (i ∂t+∂²_x),namelyLJ−JL= [L, J] = 0.

For the Korteweg-de Vries type equation (KdV), Saut and Temam [29] remarked that a solutionucannot gain or lose regularity. They showed that if u(x,0) = u0(x) ∈H^s(R) fors≥2, thenu(·, t) ∈H^s(R) for all t > 0. The same result was obtained independently by Bona and Scott [4] though a different method. For the KdV equation on the line, Kato [22] motivated by work of Cohen [9] showed that if u(x,0) = u0(x) ∈ L²_b ≡ H²(R)∩L²(e^bxdx)(b > 0) then the solution u(x, t) of the KdV equation becomes C^∞ for all t > 0. A main ingredient in the proof was the fact that formally the semi-group S(t) =e^−∂3x inL²_b(R) is equivalent toSb(t) =e^{−t(∂x−b)3} inL²(R) whent >0.One would be inclined to believe that this was a special property of the KdV equation. This is not however the case. The effect is due to the dispersive nature of the linear part of the equation. Kruzkov and Faminskii [26] proved that u(x,0) =u0(x)∈L²(R) such thatx^αu0(x)∈L²((0,+∞)) the weak solution of the KdV equation has l-continuous space derivatives for allt >0 ifl <2α.The proof of this result is based on the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV equation which was found in [22, 26]. While the proof of Kato appears to depend on special a priori estimates, some of this mystery has been resolved by the result of local gain of finite regularity for various others linear and nonlinear dispersive equation due to Constantin and Saut [13], Ginibre and Velo [15] and others. However, all of them require growth conditions on the nonlinear term. In 1992, W. Craiget al. [12] proved for fully nonlinear KdV equationut+f(uxxx, uxx, ux, u, x, t) = 0 and certain additional assumption overf that C^∞ solutionsu(x, t) are obtained for allt >0 if the initial datau0(x) decays faster than polynomially onR⁺={x∈R: x >0}and has certain initial Sobolev regularity. Following with this idea, in 2001, O.

Vera and G. Perla Menzala [33, 34] proved that the solutions of the initial value problem (P) are locally smooth due to the dispersive of the coupled system of equations of Korteweg - de Vries type

(P)









ut+uxxx+a3vxxx+u ux+a1v vx+a2(u v)x= 0 x∈R, t≥0 b1vt+vxxx+b2a3uxxx+v vx+b2a2u ux+b2a1(u v)x= 0

u(x,0) =u0(x) v(x,0) =v0(x)

whereu=u(x, t), v=v(x, t) are real-valued functions of the variablesxandtanda1, a2, a3, b1, b2are real constants withb1>0 andb2>0.The original coupled system is

(Pb)









ut+uxxx+a3vxxx+u^pux+a1v^pvx+a2(u^pv)x= 0 in − ∞< x <+∞, t≥0 b1vt+vxxx+b2a3uxxx+v^pvx+b2a2u^pux+b2a1(u v^p)x= 0

u(x,0) =u0(x) v(x,0) =v0(x)

whereu=u(x, t), v=v(x, t) are real-valued functions of the variablesxandtanda1, a2, a3, b1, b2are real constants withb1>0 andb2>0.The powerpis an integer larger than or equal to one. The system (Pb) has the structure of a pair of Korteweg - de Vries equations coupled through both dispersive and nonlinear effects. In the case p= 1, system (Pb) was derived by Gear and Grimshaw in 1984 [14] as a model to describe the strong interaction of weakly nonlinear, long waves. Mathematical results on the system (Pb) were given by J. Bona et al. [3]. They proved that (P) is globally well posed inb H^s(R)×H^s(R) for any s≥1 provided |a3| <1/√

b2.The system (Pb) has been intensively studied by several authors.

See [2, 3] and the references therein. We have the following conservation laws φ1(u) =

Z

R

u dx, φ2(v) = Z

R

v dx, φ3(u, v) = Z

R

(b2u²+b1v²)dx (1.8) The time-invariance of the functionals φ1 and φ2 expresses the property that the mass of each mode

for the system of two models taken as a whole. Solutions of (Pb) satisfy an additional conservation law which is revealed by the time-invariance of the functional

φ4= Z

R

b2u²_x+v²_x+ 2b2a3uxvx−b2

u³

3 −b2a2u²v−b2a2u²v−b2a1u v²−v³ 3

dx (1.9) The functional φ4 is a Hamiltonian for the system (Pb) and ifb2a²₃ <1, φ4 will be seen to provide an a priori estimate for the solutions (u, v) of (Pb) in the spaceH¹(R)×H¹(R).Furthermore, the linearization of (Pb) about the rest state can be reduced to two, linear Korteweg - de Vries equations by a process of diagonalization. Using this remark and the smoothing properties (in both the temporal and spatial variables) for the linear Korteweg - de Vries derived by Kato [22, 23], Kenig, Ponce and Vega [24, 25]

it will be shown that (Pb) is locally well-posed in H^s(R)×H^s(R) for any s ≥1 whenever √

b2a3 6= 1.

Indeed, all this appears in the following Theorem:

Theorem 1.1 (See [3]). Let s≥1 and (u0(x), v0(x))∈H^s(R)×H^s(R). Consider the system (Pb)to- gether with these initial conditions. Let p≥1, p be an integer andaj, bk (real) constants√

b2a3<1, b1>

0, b2>0 (j= 1, 2,3 ;k= 1,2).Then, there existsT0=T0(k(u0(·), v0(·))kYm, p)>0 and a unique solu- tion (u(x, t), v(x, t))∈Xs(T0)×Xs(T0), of(Pb)with initial data(ϕ(x), ψ(x))whereXs(T0) =C(0, T0: H^s(R))∩ C¹(0, T0: H^s−3(R)). Moreover, the pair (u, v) depends continuously on (u0(x), v0(x))in the sense that the map (u0(x), v0(x))→(u, v)is continuous fromYsinto the space Xs(T0)×Xs(T0), where sis a real number, Ys=H^s(R)×H^s(R)with the normk(u, v)k²Ys=kuk²H^s(R)+kvk²H^s(R).

This result was improved by J. Marshall et al.[1] They proved that the system (Pb)(withp= 1), is globally well-posed inL²(R)×L²(R) provided that |a3| 6= 1/√

b2. This kind of dispersive problem exhibits the interesting phenomenon of dispersive smoothing, that is, If the initial data belong to a certain Sobolev space and has a good behavior as|x| →+∞,then the solutions in any timet6= 0 are smoother than the initial data.

Our aim in this paper, is to study gain in regularity for the equation (1.1)-(1.4). Specifically, we prove conditions on (1.1)-(1.4) for which initial data (u0, v0) possessing sufficient decay at infinity and minimal amount of regularity will lead to a unique solution (u(t), v(t))∈C^∞(R)×C^∞(R) for 0< t < T,where T is the existence time of the solution. This paper is organized as follows: Section 2 outlines briefly the notation and terminology to be used subsequently. Section 3 we prove the main inequality. Section 4 we prove an important a priori estimate. Section 5 we prove a basic-local-in-time existence and uniqueness theorem. Section 6 we develop a series of estimates for solutions of equations (1.1)-(1.4) in weighted Sobolev norms. These provide a starting point for the a priori gain of regularity. In section 7 we prove the following theorem:

Theorem 1.2 (Main Theorem) Let T > 0, a < 0 and (u, v) be a solution of (1.1)-(1.4) in the region R×[0, T]such that

(u, v)∈L^∞([0, T] : H³(W0L0))×L^∞([0, T] : H³(W0L0)) (1.10) for someL≥2.Then

u∈L^∞([0, T] : H^3+l(Wσ, L−l, l))∩L²([0, T] : H^4+l(Wσ, L−l−1, l)) v∈L^∞([0, T] : H^3+l(Wσ, L−l, l))∩L²([0, T] : H^4+l(Wσ, L−l−1, l)) for all0≤l≤L−1 and allσ >0 where the weight classes will be defined in Section 2.

2 Preliminaries

We consider the initial value problem

ut−a uxxx+ 6u ux= 2b v vx (2.1)

vt+vxxx+ 3u vx= 0 (2.2)

u(x,0) =u0(x) (2.3)

v(x,0) =v0(x) (2.4)

where x∈R, t∈Rand u=u(x, t), v=v(x, t) are real unknown functions. b anda are real constants withb >0.

Notation 2.1 We write∂=∂/∂x, ∂t=∂/∂t and we abbreviate uj =∂^ju.

Definition 2.2 A function ξ=ξ(x, t)belongs to the weight classWσ i k if it is a positiveC^∞ function onR×[0, T], ∂ξ >0 and there is a constant Cj,0≤j≤5 such that

0< C1≤t^−ke^{−σ x}ξ(x, t)≤C2 ∀x <−1, 0< t < T. (2.5) 0< C3≤t^−kx⁻ⁱξ(x, t)≤C4 ∀x >1, 0< t < T. (2.6)

t|∂tξ|+|∂^jξ|

/ξ≤C5 ∀(x, t)∈R×[0, T], ∀j ∈N. (2.7) Remark 2.3 We shall always take σ≥0, i≥1 andk≥0.

Example 2.4 Let

ξ(x) =

1 +e^−1/x for x >0

1 for x≤0

thenξ∈W0i0.

Definition 2.5 Let N be a positive integer. By H^N(Wσ, i, k) we denote the Sobolev space on R with a weight; that is, with the norm

kvk²H^N(Wσ, i, k)= XN j=0

Z

R

|∂^jv(x)|²ξ(x, t)dx <+∞

for anyξ∈Wσ i k and0< t < T.Even though the norm depends onξ, all such choices lead to equivalent norms.

Remark 2.6 H^s(Wσ i k) depends ont (becauseξ=ξ(x, t)).

Lemma 2.7 (See [7]). For ξ ∈Wσ i0 and σ≥ 0, i ≥0, there exists a constant C > 0 such that, for u∈H¹(Wσ i0)

sup

x∈R|ξ u²| ≤C Z +∞

−∞ |u|²+|∂u|² ξ dx.

Lemma 2.8 (The Gagliardo-Nirenberg inequality). Letq, rbe any real numbers satisfying 1≤q, r≤ ∞ and letj andm be a nonnegative integers such thatj ≤m. Then

k∂^jukL^p(R)≤Ck∂^muk^aL^r(R)kuk^1−aL^q(R)

where ¹_p = j+a ¹_r−m

+ ^(1−a)_q for all a in the interval _m^j ≤ a ≤ 1, and M is a positive constant depending onlym, j, q, r anda.

Definition 2.9 By L²([0, T] : H^N(Wσ i k)) we denote the space of functions v(x, t)with the norm(N positive integer)

kvk²L²([0, T]:H^N(Wσ i k))= Z T

kv(·, t)k²H^N(Wσ i k)dt <+∞.

Remark 2.10 The usual Sobolev space is H^N(R) =H^N(W0 0 0) without a weight.

Remark 2.11 We shall derive the a priori estimates assuming that the solution is C^∞, bounded as x→ − ∞,and rapidly decreasing as x→+∞,together with all of its derivatives.

Considering the above notation, the Hirota-Satsuma system can be written as

ut−a u3+ 6u u1= 2b v v1 (2.8)

vt+v3+ 3u v1= 0 (2.9)

u(x,0) =u0(x) (2.10)

v(x,0) =v0(x) (2.11)

where x∈R, t∈Rand u=u(x, t), v=v(x, t) are real unknown functions. b anda are real constants withb >0.

Throughout this paperC is a generic constant, not necessarily the same at each occasion(it will change from line to line), which depend in an increasing way on the indicated quantities. In this part we only consider the caset >0.The caset <0 can be treated analogously.

3 Main Inequality

Lemma 3.1 Let(u, v)be a solution to (2.8)-(2.9)with enough Sobolev regularity(for instance, (u, v)∈ H^N(R)×H^N(R), N≥3), then

1 4b ∂t

Z

R

ξ u²_αdx+1 6 ∂t

Z

R

ξ v_α²dx+ Z

R

µ1u²_α+1dx+ Z

R

µ2v_α+1² dx Z

R

θ1u²_αdx+ Z

R

θ2v²_αdx+ Z

R

Rαdx= 0 (3.1)

where

µ1 = −3a

4b ∂ξ f or a <0 (3.2)

µ2 = 3

2 ∂ξ (3.3)

θ1 = − 1

4b[ξt−a ∂³ξ+ 6∂(ξ u) ] (3.4)

θ2 = − 1

6[ξt+∂³ξ] (3.5)

Rα = 1 3b

Xα β=1

α!

β! (α−β)!ξ uαuβuα+1−β−

α−1X

β=0

α!

β! (α−β)! ξ uαvβvα+1−β

+

α−1X

β=0

α!

β! (α−β)! ξ vαuβvα+1−β (3.6)

Proof. Differentiating (2.8)α-times(forα≥0) over x∈Rleads to

∂tuα−a uα+3+ 6 (u u1)α= 2b(v v1)α (3.7) Letξ=ξ(x, t),then multiplying (3.7) by 2ξ uαwe have

2 Z

R

ξ uα∂tuα −2a Z

R

ξ uαuα+3dx+ 12 Z

R

ξ uα(u u1)αdx= 4b Z

R

ξ uα(v v1)αdx. (3.8) Each term is calculated separately, integrating by parts

2 Z

R

ξ uα∂tuαdx= d dt

Z

R

ξ u²_αdx− Z

R

ξtu²_αdx

−2a Z

R

ξ uαuα+3dx=a Z

R

∂³ξ u²_αdx−3a Z

R

∂ξ u²_α+1dx.

Using Leibniz’s Formula, we have (u u1)α =

Xα β=0

α!

β! (α−β)! uβuα+1−β=u uα+1+ Xα β=1

α!

β! (α−β)! uβuα+1−β

(v v1)α = Xα β=0

α!

β! (α−β)! vβvα+1−β

then 12

Z

R

ξ uα(u u1)αdx = 12 Z

R

ξ u uαuα+1dx+ 12 Xα β=1

α!

β! (α−β)!

Z

R

ξ uαuβuα+1−βdx

= −6 Z

R

∂(ξ u)u²_αdx+ 12 Xα β=1

α!

β! (α−β)!

Z

R

ξ uαuβuα+1−βdx,

4b Z

R

ξ uα(v v1)αdx= 4b Xα β=0

α!

β! (α−β)!

Z

R

ξ uαvβvα+1−βdx Hence replacing in (3.8) and performing straightforward calculus we have

1 4b

d dt

Z

R

ξ u²_αdx−3a 4b Z

R

∂ξ u²_α+1dx+ 1 4b

Z

R

[−ξt+a ∂³ξ−6∂(ξ u) ]u²_αdx 1

3b Xα β=1

α!

β! (α−β)!

Z

R

ξ uαuβuα+1−βdx= Xα β=0

α!

β! (α−β)!

Z

R

ξ uαvβvα+1−βdx. (3.9) Differentiating (2.9) α-times of (forα≥0) over x∈Rleads to

∂tvα+vα+3=−3 (u v1)α (3.10)

Multiply this equation by 2ξ vα and integrate overx∈Rwe have 2

Z

R

ξ vα∂tvαdx+ 2 Z

R

ξ vαvα+3dx=−6 Z

R

ξ vα(u v1)αdx (3.11) Performing straightforward calculations as above we obtain

1 6

d dt

Z

R

ξ v_α²dx+3 2

Z

R

∂ξ v_α+1² dx+1 6

Z

R

[−ξt−∂³ξ]v_α²dx

=− Xα β=0

α!

β! (α−β)!

Z

R

ξ vαuβvα+1−βdx. (3.12)

Adding (3.9) and (3.12) we have 1

4b d dt

Z

R

ξ u²_αdx+1 6

d dt

Z

R

ξ v_α²dx−3a 4b Z

R

∂ξ u²_α+1dx+3 2

Z

R

∂ξ v²_α+1dx + 1

4b Z

R

[−ξt+a ∂³ξ−6∂(ξ u)]u²_αdx+1 6

Z

R

[−ξt−∂³ξ]v_α²dx 1

3b Xα β=1

α!

β! (α−β)!

Z

R

ξ uαuβuα+1−βdx

=

Xα α!

β! (α−β)!

Z

ξ uαvβvα+1−βdx−

Xα α!

β! (α−β)!

Z

ξ vαuβvα+1−βdx. (3.13)

We takeβ =αin (3.14) we obtain 1

4b d dt

Z

R

ξ u²_αdx+1 6

d dt

Z

R

ξ v_α²dx−3a 4b Z

R

∂ξ u²_α+1dx+3 2 Z

R

∂ξ v_α+1² dx + 1

4b Z

R

[−ξt+a ∂³ξ−6∂(ξ u)]u²_αdx+1 6

Z

R

[−ξt−∂³ξ]v²_αdx 1

3b Xα β=1

α!

β! (α−β)!

Z

R

ξ uαuβuα+1−βdx

=

α−1X

β=0

α!

β! (α−β)!

Z

R

ξ uαvβvα+1−βdx−

α−1X

β=0

α!

β! (α−β)!

Z

R

ξ vαuβvα+1−βdx (3.14) the lemma follows.

Lemma 3.2 Forµ1, µ2∈Wσ i k an arbitrary weight functions and a <0, there exist ξ1, ξ2 ∈Wσ i+1k

respectively such that

µ1=− 3a

4b ∂ξ1 and µ2=−3

2 ∂ξ2. (3.15)

Indeed, we have

ξ1=−4b 3a

Z x

−∞

µ1(y, t)dy and ξ2= 2 3

Z x

−∞

µ2(y, t)dy. (3.16)

Lemma 3.3 The expressionRαin the inequality of Lemma 3.1 is a sum of terms of the form

ξ uν1uν2uα, ξ vν1vν2uα, ξ uν1vν2vα (3.17) where1≤ν1≤ν2≤αand

ν1+ν2=α+ 1. (3.18)

Proof. The result follows using (3.6).

4 An a priori estimate

We show now a fundamental a priori estimate used for a basic local-in-time existence theorem. We con- struct a mapping Z : L^∞([0, T] : H^s(R))7−→L^∞([0, T] : H^s(R)) with the following property: Given u⁽ⁿ⁾=Z(u⁽ⁿ⁻¹⁾) and esssup_{t∈[0, T]}ku⁽ⁿ⁻¹⁾k^s≤C0then esssup_{t∈[0, T]}ku⁽ⁿ⁾k^s≤C0,wheresandC0>0 are constants. This property tells us thatZ:B_C

0(0)7−→B_C

0(0) where B_C

0(0) ={v(x, t) : kv(·, t)k^s≤C0} is a ball inL^∞([0, T] : H^s(R)).To guarantee this property, we will appeal to an a priori estimate which is the main object of this section.

Differentiating (2.8) and (2.9) respectively two times leads to

∂tu2−a u5+ 6u u3+ 18u1u2= 2b v v3+ 6b v1v2 (4.1)

∂tv2+v5+ 3u v3+ 6u1v2+ 3u2v1= 0. (4.2) Let u = ∧w and v = ∧z where ∧ = (I −∂²)⁻¹. Then u = (I −∂²)⁻¹w then u−u2 = w where

∂tu2 =−wt+ut, and in a similar way,∂tv2 =−zt+vt. Replacing on (4.1) and (4.2) respectively we obtain

−wt−a∧w5+ 6∧w∧w3+ 18∧w1∧w2−[−a∧w3+ 6∧w∧w1] + 2b∧z∧z1

= 2b∧z∧z3+ 6b∧z1∧z2 (4.3)

−zt+∧z5+ 3∧w∧z3+ 6∧w1∧z2+ 3∧w2∧z1−[∧z3+ 3∧w∧z1] = 0 (4.4) The equations (4.3), (4.4) are linearized equations by substituting a new variable θandφin each coeffi- cient:

wt = −a∧w5+ 6∧θ∧w3+ 18∧θ1∧w2−[−a∧w3+ 6∧θ∧w1] + 2b∧φ∧z1

+ 2b∧φ∧z3+ 6b∧φ₁∧z2 (4.5)

zt = ∧z5+ 3∧θ∧z3+ 6∧θ1∧z2+ 3∧θ2∧z1−[∧z3+ 3∧θ∧z1] (4.6) Equations (4.5) and (4.6) are linear equations at each iteration which can be solved in any interval of time in which the coefficients are defined. These equations have the form

∂tw=−a∧w5+h⁽²⁾∧w3+h⁽¹⁾∧z3+h⁽⁰⁾ (4.7)

∂tz=∧z5+k⁽²⁾∧w3+k⁽¹⁾∧z3+k⁽⁰⁾ (4.8) We consider the following Lemma to help us to set up the iteration scheme.

Lemma 4.1. Given initial data (u0(x), v0(x)) ∈ T

k≥0H^k(W0i0)×H^k(W0i0) and a < 0. Then there exists a unique solution of (4.7), (4.8) where h⁽²⁾=h⁽²⁾(∧θ), h⁽¹⁾=h⁽¹⁾(∧φ),

h⁽⁰⁾=h⁽⁰⁾(∧θ2,∧θ1,∧θ,∧φ2,∧φ1,∧φ), and k⁽²⁾=k⁽²⁾(∧θ),

k⁽¹⁾ =k⁽¹⁾(∧φ), k⁽⁰⁾ =k⁽⁰⁾(∧θ2,∧θ1,∧θ,∧φ2,∧φ1,∧φ).The solution is defined in any time interval in which the coefficients are defined.

Proof. From equations (4.7)-(4.8) we have

Wt=A∧W5+B^1,2∧W3+C⁽⁰⁾ (4.9)

where

A=

−a 0

0 1

, B^1,2=

h⁽²⁾ h⁽¹⁾ k⁽²⁾ k⁽¹⁾

, C⁽⁰⁾= h⁽⁰⁾

k⁽⁰⁾

, W =

w z

.

LetT >0 be arbitrary andM >0 a constant. DefineL= 2ξ(∂t−A∧∂⁵−B^1,2∧∂³).Then in (4.9) we haveLW = 2ξ C⁽⁰⁾.We consider the bilinear form

B:D × D −→ R

B(U1, U2) = hU1, U2i= Z T

0

Z

R

e^{−M t}(u1u2+v1v2)dx dt

whereD={U = (u, v)∈C₀^∞(R×[0, T])×C₀^∞(R×[0, T]) :u(x,0) = 0 andv(x,0) = 0} and U1=

u1

v1

, U2= u2

v2

. We have

Z

R

LU ·U dx = Z

R

2ξ[w wt+a w∧w5−h⁽²⁾w∧w3−h⁽¹⁾w∧z3

+z zt−z∧z5−k⁽²⁾z∧w3−k⁽¹⁾z∧z3]dx.

Each term is treated separately integrating by parts. The first two terms we have 2

Z

R

ξ w wtdx = ∂t

Z

R

ξ w²dx− Z

R

ξtw²dx.

2a Z

R

ξ w∧w5dx = 2a Z

R

ξ∧(I−∂²)w∧w5dx

= 2a Z

R

ξ∧w∧w5dx−2a Z

R

ξ∧w2∧w5dx

= −a Z

R

∂⁵ξ(∧w)²dx+ 5a Z

R

∂³ξ(∧w1)²dx−a Z

R

(5∂ξ−∂³ξ) (∧w2)²dx

−3a Z

R

∂ξ(∧w1)²dx.

The other terms are calculated the same form

−2 Z

R

ξ h⁽²⁾w∧w3dx = Z

R

∂³(ξ h⁽²⁾) (∧w)²dx−3 Z

R

∂(ξ h⁽²⁾) (∧w1)²dx

− Z

R

∂(ξ h⁽²⁾) (∧w2)²dx.

−2 Z

R

ξ h⁽¹⁾w∧z3dx = −2 Z

R

∂²(ξ h⁽¹⁾)∧w∧z1dx−2 Z

R

∂(ξ h⁽¹⁾)∧w1∧z1dx + 2

Z

R

ξ h⁽¹⁾∧w1∧z2dx+ 2 Z

R

ξ h⁽¹⁾∧w2∧z3dx.

Using that∧wn= (I−(I−∂²))∧wn−2=∧wn−2−wn−2 (fornpositive integer) and standard estimates follow that

Z

R

LU · U dx≥∂t

Z

R

ξ w²dx+∂t

Z

R

ξ z²dx−c Z

R

ξ w²dx−c Z

R

ξ z²dx. (4.10) Multiplying (4.10) fore^{−M t},and integrate in timet fort∈[0, T] andU = (w, z)∈ D.

Z T 0

Z

R

e^{−M t}LU · U dx dt ≥ Z T

0

e^{−M t}

∂t

Z

R

ξ w²dx

dt+ Z T

0

e^{−M t}

∂t

Z

R

ξ z²dx

dt

−c Z T

0

Z

R

ξ e^{−M t}w²dx dt−c Z T

0

Z

R

ξ e^{−M t}z²dx dt

= e^{−M t} Z

R

ξ w²(x, t)dx|^T0 +M Z T

0

Z

R

ξ e^{−M t}w²dx dt +e^{−M t}

Z

R

ξ z²(x, t)dx|^T0 +M Z T

0

Z

R

ξ e^{−M t}z²dx dt

−c Z T

0

Z

R

ξ w²dx dt−c Z T

0

Z

R

ξ z²dx dt.

Hence Z T

0

Z

R

e^{−M t}LU · U dx dt≥e^{−M t} Z

R

ξ(x, T)w²(x, T)dx+e^{−M t} Z

R

ξ(x, T)z²(x, T)dx + (M−c)

Z T 0

Z

R

ξ e^{−M t}w²dx dt+ (M −c) Z T

0

Z

R

ξ e^{−M t}z²dx dt

≥ Z T

0

Z

R

ξe^{−M t}w²dx dt+ Z T

0

Z

R

ξ e^{−M t}z²dx dt= Z T

0

Z

R

ξ e^{−M t}(w²+z²)dx dt providedM is chosen large enough. Then

hLU, Ui ≥ hU, Ui, ∀ U ∈ D.

LetL^∗ = 2ξ(−∂t+A∧∂⁵+B^1,2∧∂³) the formal adjoint ofL.LetD^∗ such that D^∗={W = (w, z)∈ C₀^∞(R×[0, T])×C₀^∞(R×[0, T]) : w(x, T) = 0 andz(x, T) = 0}.

The same form for L^∗ the formal adjoint ofLwe show that

hL^∗W, Wi ≥ hW, Wi ∀ W ∈ D^∗. (4.11) From (4.11) we have thatL^∗is one to one. ThereforehL^∗W,L^∗Wiis an inner product onD^∗.Denote by X the completion ofD^∗ with respect to this inner product. By the Riesz representation Theorem, there exists a unique solutionV ∈X,such that for anyW ∈ D^∗

hξ C⁽⁰⁾, Wi=hL^∗V,L^∗Wi where we used thatξ C⁽⁰⁾ ∈X.Then ifZ =L^∗V we have

hZ,L^∗Wi=hξ C⁽⁰⁾, Wi or hL^∗W, Zi=hW, ξ C⁽⁰⁾i

henceZ =L^∗V is a weak solution ofLZ =ξ C⁽⁰⁾ withZ∈L²(R×[0, T])×L²(R×[0, T])≃L²([0, T] : L²(R))×L²([0, T] : L²(R)).

Remark 4.1 To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product. It is a standard approximation procedure to obtain a result for general initial data.

The following estimate is related to the existence of solutions theorem.

Lemma 4.2. Let θ, φ, w, z ∈ C^k([0,+∞) : H^N(W0i0)) for all k, N which satisfy (4.5), (4.6) and a <0. For eachαthere exist positive, nondecreasing functions G, E and M such that for all t≥0

∂t

Z

R

ξ w²αdx+∂t

Z

R

ξ zα²dx ≤ G(kθk^λ, kφk^λ) (kwk²α+kzk²α) (4.12) +E(kθkλ, kφkλ) (kθk²α+kφk²α) +M(kθkα, kφkα)

wherek · k^α is the norm inH^α(W0i0)andλ= max{1, α}. Proof. We begin by applying∂ to (4.5), our equation become

∂tw1 = −a∧w6+a∧w4+ 6∧θ∧w4+ 2b∧φ∧z4+ 24∧θ1∧w3+ 8b∧φ₁∧z3

+ 18∧θ2∧w2+ 6b∧φ₂∧z2−6∧θ∧w2+ 2b∧φ∧z2−6∧θ1∧w1+ 2b∧φ₁∧z1

follow that

∂tw1 = −a∧w6+a∧w4+ 6∧θ∧w4+ 2b∧φ∧z4+ 24∧θ1∧w3+ 8b∧φ₁∧z3

+p1(∧θ2,∧φ₂, . . . ,∧θ,∧φ) where

p1 = 18∧θ2∧w2+ 6b∧φ₂∧z2−6∧θ∧w2+ 2b∧φ∧z2−6∧θ1 ∧w1+ 2b∧φ₁∧z1. The similar form applying∂² to (4.5), follow that

∂tw2 = −a∧w7+a∧w5+ 6∧θ∧w5+ 2b∧φ∧z5+ 30∧θ1∧w4+ 10b∧φ₁∧z4

+p2(∧θ3,∧φ₃, . . . ,∧θ,∧φ) where

p2 = 42∧θ2∧w3+ 14b∧φ₂∧z3−6∧θ∧w3+ 2b∧φ∧z3+ 18∧θ3 ∧w2+ 2b∧φ₃∧z2

−6∧θ2∧w1+ 2b∧φ₂∧z1−12∧θ1∧w2+ 4b∧φ₁∧z2.

Applying∂^αto (4.5), our equation become

∂twα = −a∧wα+5+

α+3X

j=3

h^(j)₁ ∧wj+r1∧θα+1+r2(∧θα,∧θα−1, . . . ,∧θ)

+

α+3X

j=3

h^(j)₂ ∧zj+s1∧φ_α+1+s2(∧φ_α, ∧φ_α−1, . . . ,∧φ) (4.13)

where h^(j)₁ and h^(j)₂ are smooth functions depending on ∧θi,∧θi−1, . . . , ∧θ;∧φ_i, ∧φ_i−1, . . . ,∧φ with i= 3 +α−j.

Multiplying (4.13) by 2ξ wα,and integrate overx∈R,as follows 2

Z

R

ξ wα∂twαdx=−2a Z

R

ξ wα∧wα+5dx+ 2

α+3X

j=3

Z

R

ξ h^(j)₁ wα∧wjdx

+ 2 Z

R

ξ r1wα∧θα+1dx+ 2 Z

R

ξ wαr2(∧θα,∧θα−1, . . . , ∧θ)dx+

α+3X

j=3

Z

R

ξ h^(j)₂ wα∧zjdx + 2

Z

R

ξ s1wα∧φ_α+1dx+ 2 Z

R

ξ wαs2(∧φ_α,∧φ_α−1, . . . ,∧φ)dx. (4.14) Each term in (4.14) is treated separately. The first two terms yield

2 Z

R

ξ wα∂twαdx=∂t

Z

R

ξ w²_αdx− Z

R

ξtw²_αdx.

−2a Z

R

ξ wα∧wα+5dx = −2a Z

R

ξ∧(I−∂²)wα∧wα+5dx

= −2a Z

R

ξ∧wα∧wα+5dx+ 2a Z

R

ξ∧wα+2∧wα+5dx

= a

Z

R

∂⁵ξ(∧wα)²dx−5a Z

R

∂³ξ(∧wα+1)²dx+ 5a Z

R

∂ξ(∧wα+2)²dx

−a Z

R

∂³ξ(∧wα+2)²dx+ 3a Z

R

∂ξ(∧wα+3)²dx

= 3a Z

R

∂ξ(∧wα+3)²dx−a Z

R

(∂³ξ−5∂ξ) (∧wα+2)²dx

−5a Z

R

∂³ξ(∧wα+1)²dx+a Z

R

∂⁵ξ(∧wα)²dx.

The other terms in (4.14) are treated the similar form, using integrating by parts.

∂t

Z

R

ξ wα2dx− Z

R

ξtwα2dx+ 3a Z

R

∂ξ(∧wα+3)²dx−a Z

R

(∂⁵ξ−5∂ξ) (∧wα+2)²dx

−5a Z

R

∂³ξ(∧wα+1)²dx+a Z

R

∂⁵ξ(∧wα)²dx

− Z

R

∂³(ξ h^(α+3)₁ ) (∧wα)²dx+ 3 Z

R

∂(ξ h^(α+3)₁ ) (∧wα+1)²dx Z

R

∂(ξ h^(α+3)₁ ) (∧wα+2)²dx+ Z

R

∂²(ξ h^(α+2)₁ )(∧wα)²dx

−2 Z

R

ξ h^(α+2)₁ (∧wα+1)²dx−2 Z

R

ξ h^(α+2)₁ (∧wα+2)²dx + 2

α+1X

j=3

Z

R

ξ h^(j)₁ wα∧wjdx+ 2 Z

R

ξ r1wα∧θα+1dx+ 2 Z

R

ξ r2wαdx

−2 Z

R

∂(ξ h^(α+3)₂ wα)∧zα+2dx−2 Z

R

∂(ξ h^(α+2)₂ wα)∧zα+1dx + 2

α+1X

j=3

Z

R

ξ h^(j)₂ wα∧zjdx+ 2 Z

R

ξ s1wα∧φ_α+1dx+ 2 Z

R

ξ s2wαdx= 0. (4.15) Performing similar calculations to (4.6), our equation become

∂tzα = ∧zα+5+

α+3X

j=3

k^(j)₁ ∧wj+m1∧θα+1+m2(∧θα,∧θα−1, . . . , ∧θ)

+

α+3X

j=3

k^(j)₂ ∧zj+n1∧φ_α+1+n2(∧φ_α,∧φ_α−1, . . . ,∧φ) (4.16)

where k^(j)₁ and k^(j)₂ are smooth functions depending on∧θi,∧θi−1, . . . , ∧θ, ∧φ_i, ∧φ_i−1, . . . ,∧φwith i= 3 +α−j.

We now multiply (4.16) by 2ξ zα,integrate overx∈Rand performing calculations we obtain

−∂t

Z

R

ξ z²_αdx+ Z

R

ξtz²_αdx−3 Z

R

∂ξ(∧zα+3)²dx+ Z

R

(∂⁵ξ−5∂ξ) (∧zα+2)²dx + 5

Z

R

∂³ξ(∧zα+1)²dx− Z

R

∂⁵ξ(∧zα)²dx−2 Z

R

∂(ξ k₁^(α+3)zα)∧wα+2dx

−2 Z

R

∂(ξ k₁^(α+2)zα)∧wα+1dx+ 2

α+1X

j=3

Z

R

ξ k^(j)₁ zα∧wjdx+ 2 Z

R

ξ m1zα∧θα+1dx+ 2 Z

R

ξ m2zαdx

− Z

R

∂³(ξ k₂^(α+3)) (∧zα)²dx+ 3 Z

R

∂(ξ k^(α+3)₂ ) (∧zα+1)²dx+ Z

R

∂(ξ k₂^(α+3)) (∧zα+2)²dx +

Z

R

∂²(ξ k₂^(α+2)) (∧zα)²dx−2 Z

R

ξ k₂^(α+2)(∧zα+1)²dx−2 Z

R

ξ k₂^(α+2)(∧zα+2)²dx + 2

α+1X

j=3

Z

R

ξ k₂^(j)zα∧zjdx+ 2 Z

R

ξ n1zα∧φ_α+1dx+ 2 Z

R

ξ n2zαdx= 0. (4.17)

Adding (4.15) with (4.17) we have the following identity

−∂t

Z

R

ξ w²_αdx−∂t

Z

R

ξ z_α²dx+ Z

R

ξtw²_αdx+ Z

R

ξtz_α²dx+ 3a Z

R

∂ξ(∧wα+3)²dx

−3 Z

R

∂ξ(∧zα+3)²dx−a Z

R

(∂⁵ξ−5∂ξ) (∧wα+2)²dx+ Z

R

(∂⁵ξ−5∂ξ) (∧zα+2)²dx

−5a Z

R

∂³ξ(∧wα+1)²dx+ 5 Z

R

∂³ξ(∧zα+1)²dx+a Z

R

∂⁵ξ(∧wα)²dx− Z

R

∂⁵ξ(∧zα)²dx

− Z

R

∂³(ξ h^(α+3)₁ ) (∧wα)²dx+ 3 Z

R

∂(ξ h^(α+3)₁ ) (∧wα+1)²dx+ Z

R

∂(ξ h^(α+3)₁ ) (∧wα+2)²dx

− Z

R

∂³(ξ k₂^(α+3)) (∧zα)²dx+ 3 Z

R

∂(ξ k^(α+3)₂ ) (∧zα+1)²dx+ Z

R

∂(ξ k₂^(α+3)) (∧zα+2)²dx +

Z

R

∂²(ξ h^(α+2)₁ ) (∧wα)²dx−2 Z

R

ξ h^(α+2)₁ (∧wα+1)²dx−2 Z

R

ξ h^(α+2)₁ (∧wα+2)²dx +

Z

R

∂²(ξ k₂^(α+2)) (∧zα)²dx−2 Z

R

ξ k^(α+2)₂ (∧zα+1)²dx−2 Z

R

ξ k₂^(α+2)(∧zα+2)²dx

−2 Z

R

∂(ξ h^(α+3)₂ wα)∧zα+2dx−2 Z

R

∂(ξ h^(α+2)₂ wα)∧zα+1dx

−2 Z

R

∂(ξ k₁^(α+3)zα)∧wα+2dx−2 Z

R

∂(ξ k^(α+2)₁ zα)∧wα+1dx + 2

α+1X

j=3

Z

R

ξ h^(j)₁ wα∧wjdx+ 2

α+1X

j=3

Z

R

ξ k₂^(j)zα∧zjdx

+ 2

α+1X

j=3

Z

R

ξ h^(j)₂ wα∧zjdx+ 2

α+1X

j=3

Z

R

ξ k₁^(j)zα∧wjdx + 2

Z

R

ξ r1wα∧θα+1dx+ 2 Z

R

ξ m1zα∧θα+1dx+ 2 Z

R

ξ s1wα∧φ_α+1dx+ 2 Z

R

ξ n1zα∧φ_α+1dx + 2b2

Z

R

ξ r2wαdx+ 2 Z

R

ξ m2zαdx+ 2 Z

R

ξ s2wαdx+ 2 Z

R

ξ n2zαdx= 0.

where

∂t

Z

R

ξ w_α²dx+∂t

Z

R

ξ z_α²dx= Z

R

ξtw²_αdx+ Z

R

ξtz²_αdx+ 3a Z

R

∂ξ(∧wα+3)²dx

−3 Z

R

∂ξ(∧zα+3)²dx−a Z

R

(∂⁵ξ−5∂ξ) (∧wα+2)²dx+ Z

R

(∂⁵ξ−5∂ξ) (∧zα+2)²dx

−5a Z

R

∂³ξ(∧wα+1)²dx+ 5 Z

R

∂³ξ(∧zα+1)²dx+a Z

R

∂⁵ξ(∧wα)²dx− Z

R

∂⁵ξ(∧zα)²dx

− Z

R

∂³(ξ h^(α+3)₁ ) (∧wα)²dx+ 3 Z

R

∂(ξ h^(α+3)₁ ) (∧wα+1)²dx+ Z

R

∂(ξ h^(α+3)₁ ) (∧wα+2)²dx

− Z

R

∂³(ξ k₂^(α+3)) (∧zα)²dx+ 3 Z

R

∂(ξ k^(α+3)₂ ) (∧zα+1)²dx+ Z

R

∂(ξ k₂^(α+3)) (∧zα+2)²dx +

Z

R

∂²(ξ h^(α+2)₁ ) (∧wα)²dx−2 Z

R

ξ h^(α+2)₁ (∧wα+1)²dx−2 Z

R

ξ h^(α+2)₁ (∧wα+2)²dx +

Z

R

∂²(ξ k₂^(α+2)) (∧zα)²dx−2 Z

R

ξ k^(α+2)₂ (∧zα+1)²dx−2 Z

R

ξ k₂^(α+2)(∧zα+2)²dx

−2 Z

R

∂(ξ h^(α+3)₂ wα)∧zα+2dx−2 Z

R

∂(ξ h^(α+2)₂ wα)∧zα+1dx

−2 Z

R

∂(ξ k₁^(α+3)zα)∧wα+2dx−2 Z

R

∂(ξ k^(α+2)₁ zα)∧wα+1dx + 2

α+1X

j=3

Z

R

ξ h^(j)₁ wα∧wjdx+ 2

α+1X

j=3

Z

R

ξ k₂^(j)zα∧zjdx

+ 2

α+1X

j=3

Z

R

ξ h^(j)₂ wα∧zjdx+ 2

α+1X

j=3

Z

R

ξ k₁^(j)zα∧wjdx + 2

Z

R

ξ r1wα∧θα+1dx+ 2 Z

R

ξ m1zα∧θα+1dx+ 2 Z

R

ξ s1wα∧φ_α+1dx+ 2 Z

R

ξ n1zα∧φ_α+1dx + 2b2

Z

R

ξ r2wαdx+ 2 Z

R

ξ m2zαdx+ 2 Z

R

ξ s2wαdx+ 2 Z

R

ξ n2zαdx.