WELL-POSEDNESS OF HIGHER-ORDER CAMASSA-HOLM EQUATIONS

Pure Mathematics No. 15 ISSN 0806–2439 October 2006

WELL-POSEDNESS OF

HIGHER-ORDER CAMASSA–HOLM EQUATIONS

G. M. COCLITE, H. HOLDEN, AND K. H. KARLSEN

Abstract. We consider higher-order Camassa–Holm equations describing ex- ponential curves of the manifold of smooth orientation preserving diffeomor- phisms of the unit circle in the plane. We establish the existence of a strongly continuous semigroup of global weak solutions. We also present some invariant spaces under the action of that semigroup. Moreover, we prove a “weak equals strong” uniqueness result.

1. Introduction

Consider the unit circle S¹ in the plane and the manifold D of the smooth orientation-preserving diffeomorphisms ofS¹. Following [14] we study the equation for the exponential curves on D using the Riemannian structure induced by the Sobolev inner product (·,·)_Hk(R), k ∈ N (where we identify H⁰(R) and L²(R)).

Letk∈Nand

Γ :t≥07→u(t,·)∈ D

be a curve. It is a an exponential curve if it satisfies the following equation [14, (3.7)]

(1.1) ∂tu=Bk(u, u), t >0, x∈R, where (see [14, (3.2), (3.3), and Proof of Theorem 2])

Bk(u, u) :=A⁻¹_k Ck(u)−u∂xu, A_k(u) :=

k

X

j=0

(−1)^j∂_x^2ju, C_k(u) :=−uA_k ∂_xu

+A_k u∂_xu

−2∂_xuA_k(u).

In the casesk= 0 andk= 1, (1.1) becomes the inviscid Burgers equation [24]

(1.2) ∂tu+ 3u∂xu= 0,

and the Camassa–Holm equation [2, 9]

(1.3) ∂tu−∂t∂²_xu+ 3u∂xu= 2∂xu∂_x²u+u∂_x³u,

respectively (see [14, Examples 1 and 2]). This infinite sequence of higher-order Camassa–Holm equations is distinct from what is normally called the Camassa–

Holm hierarchy, where the equations beyond the Camassa–Holm equation itself are non-local and all equations are completely integrable in the sense that one can find

Date: October 24, 2006.

1991Mathematics Subject Classification. 35G25, 35L05, 35A05.

Key words and phrases. Higher order Camassa–Holm equation, weak solutions, existence, uniqueness, geometric interpretation.

The research is supported in part by the Research Council of Norway. The research of KHK is supported by an Outstanding Young Investigators Award from the Research Council of Norway.

The current address of GMC is Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy.

1

a zero-curvature formulation for each equation in the hierarchy. Indeed, that is the main mechanism behind their construction. For details about the Camassa–Holm hierarchy, see [20, 21] and references therein.

In this paper we study the wellposedness of the equation (1.1). In particular, we show that it possesses a globally defined weak solution uin C([0,∞);C^k−1(R))∩ L^∞ [0,∞);H^k(R)

when the initial data u0 ∈ H^k(R), ∂_x^ku0 ∈ L^p(R), for some 2< p <∞, see Definition 2.3 and Theorem 2.4. Furthermore, we show the existence of a semigroup Stassociated with the problem in the sense that u=St(u0) solves the equation (1.1) with initial datau0. The semigroup is continuous in the following sense: If u0,n → u0 in H^k(R), then S(u0,n) → S(u0) in L^∞([0, T];H^k(R)), see Theorem 2.4.

Similar results holds for the Camassa–Holm equation (1.3) in the casek= 1 [5].

This equation models the propagation of unidirectional shallow water waves on a flat bottom, andu(t, x) represents the fluid velocity at timetin the horizontal di- rectionx[2, 23]. The Camassa–Holm equation possesses a bi-Hamiltonian structure (and thus an infinite number of conservation laws) [19, 2] and is completely inte- grable [2, 1, 12, 8]. Moreover, it has an infinite number of solitary wave solutions, called peakonsdue to the discontinuity of their first derivatives at the wave peak, interacting like solitons: u(t, x) =ce^−|x−ct|, c∈R. From a mathematical point of view the Camassa–Holm equation is well studied. Local well-posedness results are proved in [10, 22, 25, 27]. It is also known that there exist global solutions for a particular class of initial data and also solutions that blow up in finite time for a large class of initial data [7, 10, 9]. Here blow up means that the slope of the solu- tion becomes unbounded while the solution itself stays bounded. More relevant for the present paper, we recall that existence and uniqueness results for global weak solutions of (1.3) are proved in [11, 13, 29, 30, 15, 16], see also [5].

On the other hand we recall that the solutions of the Burgers equation (1.2) in the case k= 0 experience shock formation and indeed it is well-posed in the space L^∞([0,∞);BV(R)). Let us mention the Degasperis–Procesi equation [17, 18]

(1.4) ∂_tu−∂_t∂²_xu+ 4u∂_xu= 3∂_xu∂_x²u+u∂_x³u.

It appears to be similar to the Camassa–Holm equation (1.3), but its solutions are in general discontinuous, see [6] and the references cited therein.

To keep the presentation short, details are presented for the case k= 2 only. In Appendices A and B we show how to extend the theory to general k >2.

The paper is organized as follows. In Section 2 we introduce the equations and state the main result. The existence result is obtained as a singular limit of a viscous regularization. The necessary a priori estimates are treated in Section 3.

In Section 4 stability with respect to the viscous regularization is proved using a homotopy argument. The necessary compactness arguments as well as regularity of the solution is obtained in Section 6. In Section 7 we prove a “weak equals strong”

uniqueness result. Appendices A and B deal with the general casek >2.

2. The governing equations and the main theorem

We construct a family of higher-order Camassa–Holm equations as follows. Let k∈N⁰:=N∪ {0}. Consider the equation

(2.1) ∂tu=Bk(u, u)

where u=u(t, x) : [0,∞)×R→Ris the unknown function and Bk(u, u) :=A⁻¹_k Ck(u)−u∂xu,

Ak(u) :=

k

X

j=0

(−1)^j∂_x^2ju, (2.2)

Ck(u) :=−uAk ∂xu

+Ak u∂xu

−2∂xuAk(u).

It turns out that the operator C_k(u) is a total derivative, that is, there exists a differential polynomial in udenoted by Fk such that

Ck(u) =−∂xFk(u).

One can see this as follows

−Fk(u) = Z x

−∞

Ck(u(ξ))dξ

= Z x

−∞

−uAk(∂xu) +Ak(u∂xu)−2∂xuAk(u) dξ

=

k

X

j=0

(−1)^j Z x

−∞

−u∂_x^2j∂_xu+∂_x^2j(u∂_xu)−2∂_xu∂_x^2ju dξ

=1 2

k

X

j=0

(−1)^j∂_x^2j(u²)−

k

X

j=0

(−1)^j Z x

−∞

u∂_x^2j∂xu+ 2∂xu∂_x^2ju dξ.

Lemma A.2 shows that indeed the integrand in each term is a total derivative, makingFk(u) a differential polynomial inu.

Remark 2.1. The operatorA⁻¹_k has a convolution structure, more precisely (2.3) A⁻¹_k (f)(x) =

Z

R

Gk(x−y)f(y)dy, x∈R, where G_k has Fourier transformGb_k given by

Gb_k(ζ) = 1

1 +ζ²+· · ·+ζ^2k, ζ∈R. We also have

(2.4) Gk≥0, kGkk_W2k−1,1(R),kGkk_W2k−1,∞(R)≤C0, for some constant C0>0. In the special casek= 1we find

G1(x) =1 2e^−|x|. We will repeatedly use that

∂_x^jA_k(u) =A_k(∂_x^ju) as well as

Z

R

vA_k(w)dx= Z

R

A_k(v)w dx.

Example 2.2. (2.1)reads in the casesk= 0,1,2,3 as follows[14].

(i) For k= 0 we find the following:

∂_tu+u∂_xu=−∂x(u²) or ∂_tu+ 3u∂_xu= 0, which constitutes the inviscid Burgers equation, and

A0(u) =u, C0(u) =−2u∂xu, F0(u) =u². (ii) For k= 1 we obtain the following equation:

∂_tu+u∂_xu=−∂xA⁻¹₁ (u²+1

2(∂_xu²))

or

∂_tu−∂_t∂²_xu+ 3u∂_xu= 2∂_xu∂_x²u+u∂_x³u, which is the Camassa–Holm equation. Furthermore,

A1(u) =u−∂_x²u, C1(u) =−2u∂xu−∂xu∂_x²u, F1(u) =u²+1 2(∂xu)². (iii) For k= 2, equation (2.1)becomes

(2.5) ∂tu+u∂xu=−A⁻¹₂ ∂x

u²+1

2 ∂xu2

−1 2 ∂_x²u2

−3∂x ∂xu∂_x²u

, or equivalently

(2.6) ∂tu−∂t∂_x²u+∂t∂⁴_xu+ 3u∂xu−2∂xu∂²_xu−u∂_x³u+ 2∂xu∂_x⁴u+u∂_x⁵u= 0.

In particular,

A2(u) =∂_x⁴u−∂_x²u+u, C2(u) =−uA2 ∂xu

+A2 u∂xu

−2∂xuA2(u)

=−∂xu∂_x²u+ 10∂_x²u∂_x³u+ 3∂xu∂_x⁴u−2u∂xu, F2(u) =−

Z x

−∞

C2(u)dx

=u²+1 2 ∂_xu2

−7 2 ∂_x²u2

−3∂_xu∂_x³u

=u²+1 2 ∂_xu²

−1 2 ∂_x²u²

−3∂_x ∂_xu∂²_xu . (iv) Fork= 3, equation (2.1)becomes

∂_tu+u∂_xu=−A⁻¹₂ ∂_x

u²+1 2 ∂_xu2

−7 2 ∂_x²u²

−3∂_xu∂_x³u (2.7)

+5∂_xu∂_x⁵u+ 16∂_x²u∂_x⁴u+19 2 ∂_x³u²

, or equivalently

∂tu−∂t∂²_xu+∂t∂_x⁴u−∂t∂⁶_xu+ 3u∂xu−2∂xu∂²_xu−u∂_x³u

+ 2∂xu∂_x⁴u+u∂_x⁵u−2∂xu∂_x⁶u−u∂_x⁷u= 0.

In particular,

A3(u) =−∂_x⁶u+∂_x⁴u−∂²_xu+u=A2(u)−∂_x⁶u, C3(u) =−uA3 ∂xu

+A3(u∂xu)−2∂xuA3(u)

=C2(u)−35∂_x³u∂_x⁴u−21∂_x²u∂_x⁵u−5∂xu∂_x⁶u, F3(u) =−

Z x

−∞

C3(u)dx

=u²+1 2 ∂xu2

−7 2 ∂_x²u2

−3∂xu∂_x³u + 5∂xu∂⁵_xu+ 16∂_x²u∂_x⁴u+19

2 ∂_x³u2

. We are interested in the Cauchy problem

(2.8)

(∂tu=Bk(u, u), (t, x)∈(0,∞)×R, u(0, x) =u₀(x), x∈R

in the casek≥2. We will assume

(2.9) u0∈H^k(R), ∂_x^ku0∈L^p(R) for some 2< p <∞.

For the definition of weak solutions of (2.8) we reformulate the equation as a system of an hyperbolic equation and an higher order elliptic one, namely

(2.10)

(∂_tu+u∂_xu+∂_xP = 0, A_k(P) =Fk(u).

This formulation is formally equivalent to (2.8).

Definition 2.3. We call a functionu: [0,∞)×R→Ra weak solution of (2.8)if (i) u∈C([0,∞);C^k−1(R))∩L^∞ [0,∞);H^k(R)

; (ii) usatisfies (2.10)in the sense of distributions;

(iii) u(0, x) =u0(x)for everyx∈R;

(iv) ku(t,·)k_Hk(R)≤ ku0k_Hk(R), for eacht >0.

Our main result is the following.

Theorem 2.4. Let 2 < p < ∞. There exists a strongly continuous semigroup of solutions

S: [0,∞)× Hk,p→C([0,∞);C^k−1(R))∩L^∞ [0,∞);H^k(R) associated with the Cauchy problem (2.8), where

Hk,p:=n

f ∈H^k(R)|∂_x^kf ∈L^p(R)o . More precisely, we have

(j) for each u0∈ Hk,p the map u(t, x) =St(u0)(x)is a weak solution of (2.8) according to Definition 2.3;

(jj) the semigroup is stable with respect to the initial condition:

(2.11) u_0,n→u₀ inH^k(R)impliesS(u_0,n)→S(u₀)in L^∞([0, T];H^k(R)), for every {u0,n}_n∈N⊂ Hk,p,u0∈ Hk,p,T >0.

Moreover, the spacesH^k+1(R)andHk,r,2≤r <∞are invariant under the action of S, i.e.,

S([0, T]×H^k+1(R))⊂L^∞([0, T];H^k+1(R)), (2.12)

S([0, T]× Hk,r)⊂L^∞([0, T];Hk,r), 2≤r <∞, (2.13)

for each T >0.

Moreover we show the uniqueness of the solution of the Cauchy problem (2.8) within the class of the maps with bounded second spatial derivative. A similar result was proved in [30], in the casek= 1, for the Camassa–Holm equation. More precisely, we prove the following “weak equals strong” uniqueness principle.

Theorem 2.5. Assume k = 2. Let ube a weak solution of the Cauchy problem (2.8) in the sense of Definition 2.3. If there exists a map b ∈ L¹([0, T]), T > 0, such that

∂_x²u(t,·) _L∞(

R)≤b(t), t≥0,

then, uis unique within the class of the maps satisfying such a condition.

In particular, here we assumeb∈L¹([0, T]), T >0,and in [30] whenk= 1, the authors assumedb∈L²([0, T]), T >0.

One should observe that the behavior of the Camassa–Holm equation (k= 1) is quite different from the behaviour of (2.8). Indeed the equation forq=∂_x²u, which is a relevant quantity for (2.8), is

∂tq+u∂xq+Pe= 0,

where Pe is a given function that will be defined later on. On the other hand, ifu solves the Camassa–Holm equation (1.3), thenq=∂xu, which is the corresponding relevant quantity, satisfies (P is a another given function)

∂_tq+u∂_xq+1

2q²−u²+P = 0, which now contains the nonlinear termq².

We apply the following singular perturbation approach. Letε >0, and consider the system

(2.14)











∂_tu_ε+u_ε∂_xu_ε+∂_xP_ε=ε∂_x²u_ε, A_k(P_ε) =F_k(u_ε),

u_ε(0, x) =u_0,ε(x).

We call the solution uε =uε(t, x) of (2.14) aviscous approximant to the solution u=u(t, x) of (2.8). Furthermore, we shall assume

(2.15) u0,ε∈H^k+1(R), ku0,εk_Hk(R)≤ ku0k_Hk(R), u0,ε→u0 in H^k(R).

Example 2.6. The equations (2.10) and (2.14) read in the special cases k = 0,1,2,3 as follows.

(i) For k= 0 we find the following:

∂tu+u∂xu+∂xP= 0, P=u², and

∂tuε+uε∂xuε+∂xPε=ε∂_x²uε, Pε=u²_ε. (ii) For k= 1 we obtain the following equations

∂_tu+u∂_xu+∂_xP = 0, P−∂_x²P =u²+1 2(∂_xu)², and

∂tuε+uε∂xuε+∂xPε=ε∂_x²uε, Pε−∂_x²Pε=u²_ε+1

2(∂xuε)². (iii) For k= 2 we find

∂tu+u∂xu+∂xP = 0, (2.16)

∂_x⁴P−∂²_xP+P =u²+1 2 ∂_xu²

−1 2 ∂_x²u²

−3∂_x ∂_xu∂_x²u , and

∂tuε+uε∂xuε+∂xPε=ε∂_x²uε, (2.17)

∂_x⁴Pε−∂_x²Pε+Pε=u²_ε+1

2 ∂xuε²

−7

2 ∂²_xuε²

−3∂xuε∂_x³uε, or equivalently

(2.18) ∂_tu_ε−∂_t∂_x²u_ε+∂_t∂_x⁴u_ε+ 3u_ε∂_xu_ε−2∂_xu_ε∂²_xu_ε

−u∂_x³u_ε+ 2∂_xu_ε∂_x⁴u_ε+u∂_x⁵u_ε=ε∂²_xu_ε−ε∂_x⁴u_ε+ε∂_x⁶u_ε. (iv) Fork= 3we find

∂tu+u∂xu+∂xP= 0, (2.19)

−∂_x⁶P+∂⁴_xP−∂_x²P+P=u²+1 2 ∂_xu2

−7 2 ∂_x²u²

−3∂_xu∂_x³u + 5∂_x² ∂_xu∂_x³u

+ 6∂_x ∂_x²u∂_x³u

−3 2 ∂_x³u²

.

Remark 2.7. Introducing the quantity

m:=A_k(u), m_ε:=A_k(u_ε), we have, see [14], that equations (2.10)and (2.14)equal

∂tm+u∂xm+ 2m∂xu= 0, (2.20)

and

∂tmε+uε∂xmε+ 2mε∂xuε=ε∂²_xmε, (2.21)

respectively.

3. Viscous approximants: Global existence and energy estimate We begin with the existence of the viscous approximants to (2.8).

Lemma 3.1. Assume (2.9) and (2.15). Let ε > 0. Then there exists a unique global smooth solution u_ε = u_ε(t, x) of the Cauchy problem (2.14) belonging to C([0,∞);H^k+1(R)).

Proof. The proof of this statement is similar to the one of [4, Theorem 2.3], and is

therefore omitted.

Lemma 3.2 (Energy estimate). Assume (2.9)and (2.15). The identity (3.1) kuε(t,·)k²_Hk(R)+ 2ε

Z t 0

∂xuε(τ,·)

2

H^k(R)dτ =ku0,εk²_Hk(R)

holds for each t≥0 andε >0. In addition, (3.2) ku_εk_L∞([0,∞)×R), . . . ,

∂_x^k−1u_{ε L∞}

([0,∞)×R)≤ 1

√2ku₀k_Hk(R), for each ε >0.

Proof. Fixt >0. Multiplying the first equation of (2.14) byA_k(u_ε) and integrating overR, we get

Z

R

∂_tu_εA_k(u_ε)dx−ε Z

R

∂_x²u_εA_k(u_ε)dx (3.3)

=− Z

R

u_ε∂_xu_εA_k(u_ε)dx− Z

R

∂_xP_εA_k(u_ε)dx.

Integrating by parts we have for the left-hand side, Z

R

∂tuεAk(uε)dx−ε Z

R

∂_x²uεAk(uε)dx (3.4)

=1 2

d

dtku_ε(t,·)k²_Hk(R)+ε

∂_xu_ε(t,·)

2 H^k(R), and, using the second equation of (2.14), we have for the right-hand side,

− Z

R

u_ε∂_xu_εA_k(u_ε)dx− Z

R

∂_xP_εA_k(u_ε)dx (3.5)

=− Z

R

u_ε∂_xu_εA_k(u_ε)dx− Z

R

∂_x(A_k(P_ε))u_εdx

=− Z

R

u_ε∂_xu_εA_k(u_ε)dx+ Z

R

C_k(u_ε)u_εdx

=−3 Z

R

u_ε∂_xu_εA_k(u_ε)dx− Z

R

u²_εA_k ∂_xu_ε +

Z

R

u_εA_k u_ε∂_xu_ε dx

=−3 Z

R

uε∂xuεAk(uε)dx+ Z

R

∂x(u²_ε)Ak(uε) + Z

R

Ak(uε)uε∂xuεdx= 0.

Substituting (3.4) and (3.5) in (3.3), d

dtkuε(t,·)k²_Hk(R)+ 2ε

∂_xu_ε(t,·)

2

H^k(R)= 0.

Integrating over [0, t], we get (3.1). Finally, (3.2) is direct consequence of [26,

Theorem 8.5], equations (2.15) and (3.1).

4. Bounds on the source termPε and invariance properties with k= 2 From now on we assume k = 2. We show in Appendix B how to extend the proofs to the general case k >2.

Using Remark 2.1, we may write

(4.1) Pε=P1,ε+P2,ε,

where

P1,ε(t, x) :=

Z

R

G2(x−y)

u²_ε(t, y) +1

2(∂xuε(t, y))²−1

2 ∂_x²uε(t, y)2 dy, P_2,ε(t, x) :=−3

Z

R

G₂(x−y)h

∂_x²u_ε(t, y)²

+∂_xu_ε(t, y)∂_x³u_ε(t, y)i dy.

Moreover, since G₂is the Green’s function of the operatorA₂, we have

∂_x³P_2,ε(t, x) =−3 Z

R

G⁰⁰⁰₂(x−y)h

∂²_xu_ε(t, y)²

+∂_xu_ε(t, y)∂_x³u_ε(t, y)i dy

=−3∂xuε(t, x)∂_x²uε(t, x)

−3 Z

R

(G⁰⁰₂(x−y)−G₂(x−y))∂_xu_ε(t, y)∂_x²u_ε(t, y)dy.

Hence

(4.2) ∂_x³Pε=−3∂xuε∂_x²uε+∂_x³P1,ε+P3,ε, where

P_3,ε(t, x) :=−3 Z

R

(G⁰⁰₂(x−y)−G₂(x−y))∂_xu_ε(t, y)∂_x²u_ε(t, y)dy, for eachε >0, t≥0, x∈R.

Lemma 4.1. Assumek= 2,(2.9)and (2.15). The following inequalities hold kPε(t,·)kW^2,1(R),kPε(t,·)kW^2,∞(R)≤4C₀ku0k²_H2(R),

(4.3)

kP1,ε(t,·)k_W4,1(R),kP1,ε(t,·)k_W^4,∞_(R)≤(6C0+ 1)ku0k²_H2(R), (4.4)

kP2,ε(t,·)kW^2,1(R),kP2,ε(t,·)kW^2,∞(R)≤2C0ku0k²_H2(R), (4.5)

k∂_x³Pε(t,·)k_L1(R)≤(7C0+ 3)ku0k²_H2(R), (4.6)

kP3,ε(t,·)k_W1,1(R),kP3,ε(t,·)k_W^1,∞_(R)≤12C0ku0k²_H2(R), (4.7)

for each t≥0 andε >0.

Proof. Fixt >0. We begin by proving (4.4). Observing that,

∂_xⁱP_1,ε(t, x) = Z

R

dⁱG₂ dxⁱ (x−y)

u²_ε+1

2(∂_xu_ε(t, y))²−1

2 ∂_x²u_ε(t, y)²

dy, from (2.4) and (3.2),

k∂ⁱ_xP1,ε(t,·)k_Lp(R)≤

dⁱG2

dxⁱ L^p(R)

Z

R

u²_ε+1

2 ∂xuε² +1

2 ∂_x²uε²

dy (4.8)

≤C0ku(t,·)k²_H2(R)≤C0ku0k²_H2(R),

for each p∈ {1,∞}, i∈ {0, 1,2,3}.Recalling that G₂ is the Green’s function of the operatorA2(see Remark 2.1), we find

(4.9) ∂⁴_xP_1,ε=∂_x²P_1,ε−P_1,ε+u²_ε+1

2 ∂_xu_ε2

−1

2 ∂_x²u_ε2

, hence, (4.4) is a direct consequence of (3.1), (4.8), and (4.9).

We continue by proving (4.5). Observing that,

∂_x^jP2,ε(t, x) =−3 Z

R

d^jG2

dx^j (x−y)h

∂_x²uε(t, y)2

+∂xuε(t, y)∂_x³uε(t, y)i dy

=−3 Z

R

d^j+1G2

dx^j+1 (x−y)∂xuε(t, y)∂_x²uε(t, y)dy, we conclude, using the H¨older inequality, (2.4) and (3.2), that

k∂_x^jP_2,ε(t,·)kL^p(R)≤

d^j+1G₂ dx^j+1

_Lp(

R)

Z

R

|∂xu_ε∂_x²u_ε|dy

≤C0k∂xuε(t,·)k_L2(R)k∂_x²uε(t,·)k_L2(R)

≤C0ku(t,·)k²_H2(R)≤C0ku0k²_H2(R),

for each p∈ {1,∞}, j ∈ {0, 1,2}. This proves (4.5). Clearly, estimates (4.4) and (4.5) imply (4.3).

Finally, using the H¨older inequality, (2.4) and (3.2), we obtain Z

R

|∂xu_ε∂_x²u_ε|dx≤ k∂xu_ε(t,·)kL²(R)k∂_x²u_ε(t,·)kL²(R)

(4.10)

≤ kuε(t,·)k²_H2(R)≤ ku0k²_H2(R), k∂_xⁱP3,ε(t,·)k_Lp(R)≤3

d²⁺ⁱG2

dx²⁺ⁱ _Lp(

R)

+

dⁱG2

dxⁱ _Lp(

R)

Z

R

|∂xuε∂_x²uε|dx (4.11)

≤6C₀ku₀k²_H2(R),

for p∈ {1,∞}and i∈ {0,1}. The estimates (4.4), (4.10), and (4.11) imply (4.6)

and (4.7).

Next we turn to estimates of time derivatives. Introduce the notation ΠT := [0, T]×R,

forT positive.

Lemma 4.2. Assumek= 2,(2.9)and (2.15). The following inequalities hold k∂tuε(t,·)kL²(R)≤ 1

√

2ku0k²_H2(R)+ 4C0ku0k²_H2(R)+ku0kH²(R), (4.12)

k∂_t∂_xu_εk_L2(Π_T)≤√

2Tku₀k²_H2(R)+ 4C₀ku₀k²_H2(R)

√ T+ 1

√2ku₀k_H2(R), (4.13)

for each T, t >0 and0< ε <1.

Proof. LetT, t >0 and 0< ε <1. From (2.17) and Lemma 3.2 and 4.1, k∂tuε(t,·)kL²(R)≤ kuε(t,·)∂xuε(t,·)kL²(R)+k∂xPε(t,·)kL²(R)

+εk∂²_xuε(t,·)kL²(R)

≤ kuεk_L^∞_([0,∞)×R)k∂xuε(t,·)k_L2(R)+k∂xPε(t,·)k_L2(R)

+εk∂²_xuε(t,·)k_L2(R)

≤ 1

√

2ku0k²_H2(R)+ 4C0ku0k²_H2(R)+εku0k_H2(R), this proves (4.12).

Moreover, differentiating (2.17) with respect tox, we get (4.14) ∂t∂xuε+ ∂xuε²

+uε∂²_xuε+∂_x²Pε=ε∂³_xuε, then, from Lemmas 3.2 and 4.1 we find that

k∂t∂xuεkL²(Π_T)

≤ k∂xuεk²_L4(ΠT)+kuε∂_x²uεkL²(Π_T)+k∂_x²PεkL²(Π_T)+εk∂_x³uεkL²(Π_T)

≤ 1

√2ku0k²_H2(R)

√

T+kuεkL^∞k∂²_xuεk_L2(ΠT)+k∂_x²Pεk_L2(ΠT)+εk∂_x³uεk_L2(ΠT)

≤√

2Tku0k²_H2(R)+ 4C0ku0k²_H2(R)

√ T+ 1

√

2ku0kH²(R),

which proves (4.13).

Lemma 4.3. Assume k = 2, (2.9) and (2.15). Let T > 0. There exists two positive constantsK_1,T, K_2,T depending only onku₀k_H2(R)andT and independent of ε, such that

k∂t∂_x³P1,εk_L1(ΠT),k∂t∂_x³P1,εk_L^∞_(ΠT)≤K1,T, (4.15)

k∂_tP_3,εk_L1(Π_T),k∂_tP_3,εk_L∞(Π_T)≤K_2,T, (4.16)

for each 0< ε <1.

Proof. Fix 0< ε <1 andT >0. We begin by proving (4.15). Observe that (4.17) ∂t∂_x²uε+ 3∂_x²uε∂xuε+uε∂_x³uε+∂_x³Pε=ε∂_x⁴uε,

and, from (4.2),

(4.18) ∂t∂_x²uε+uε∂_x³uε+∂_x³P1,ε+P3,ε=ε∂⁴_xuε.

Hence, since G₂ is the Green’s function of the operator A₂ (see Remark 2.1), we find from the definition of P_1,ε and (4.18) that

∂_t∂_x³P_1,ε(t, x) = Z

R

G⁰⁰⁰₂(x−y)

∂_xu_ε∂_t∂_xu_ε+ 2u_ε∂_tu_ε−∂_x²u_ε∂_t∂_x²u_ε dy (4.19)

= Z

R

G⁰⁰⁰₂(x−y) [∂_xu_ε∂_t∂_xu_ε+ 2u_ε∂_tu_ε]dy +

Z

R

G⁰⁰⁰₂(x−y)

∂²_xu_ε∂_x³P_1,ε+∂_x²u_εP_3,ε dy +

Z

R

G⁰⁰⁰₂(x−y)

uε∂²_xuε∂_x³uε−ε∂⁴_xuε∂_x²uε dy

= Z

R

G⁰⁰⁰₂(x−y) [∂xuε∂t∂xuε+ 2uε∂tuε]dy +

Z

R

G⁰⁰⁰₂(x−y)

∂²_xuε∂_x³P1,ε+∂_x²uεP3,ε dy +1

2u_ε ∂²_xu_ε²

−ε∂_x³u_ε∂_x²u_ε +

Z

R

(G⁰⁰₂−G₂) (x−y) 1

2u_ε ∂²_xu_ε²

−ε∂_x³u_ε∂_x²u_ε

dy

− Z

R

G⁰⁰⁰₂(x−y) 1

2∂xuε ∂²_xuε²

−ε ∂_x³uε²

dy.

Using the H¨older inequality, k∂t∂_x³P1,εkL¹(Π_T)≤C0

Z

Π_T

h|∂xuε∂t∂xuε|+ 2|uε∂tuε|

+|∂_x²uε∂_x³P1,ε|+|∂_x²uεP3,ε|+|uε| ∂²_xuε² (4.20)

+ 2ε|∂_x³uε∂_x²uε|+1

2|∂xuε| ∂²_xuε²

+ε ∂_x³uε²i dtdx +

Z

Π_T

1

2|uε| ∂_x²uε

2

+ε|∂³_xuε∂_x²uε|

dtdx

≤C0

hk∂xuεk_L2k∂t∂xuεk_L2+ 2kuεk_L2k∂tuεk_L2

+k∂²_xuεkL²k∂_x³P1,εkL²+k∂_x²uεkL²kP3,εkL²

+kuεkL^∞k∂_x²uεk²_L2+ 2εk∂_x³uεkL²k∂_x²uεkL²

+1

2k∂xu_εkL^∞k∂_x²u_εk²_L2+εk∂_x³u_εk²_L2

i

+1

2kuεkL^∞k∂_x²u_εk²_L2+εk∂_x³u_εkL²k∂_x²u_εkL².

Then, the estimate (4.15) is consequence of (2.4), (3.1), (3.2), (4.4), (4.7), (4.12), and (4.13).

We continue by proving (4.16). Observing that, P3,ε(t, x) =−3

2 Z

R

(G⁰⁰⁰₂(x−y)−G⁰₂(x−y)) (∂xuε(t, y))²dy,

∂tP3,ε(t, x) =−3 Z

R

(G⁰⁰⁰₂(x−y)−G⁰₂(x−y))∂xuε(t, y)∂t∂xuε(t, y)dy, we have

k∂tP3,εkL^p(Π_T)≤3 kG⁰⁰⁰₂kL^p(R)+kG⁰₂kL^p(R)

Z

Π_T

|∂xuε∂t∂xuε|dx

≤3 kG⁰⁰⁰₂kL^p(R)+kG⁰₂kL^p(R)

k∂xu_εkL²(Π_T)k∂t∂_xu_εkL²(Π_T), (4.21)

forp∈ {1,∞}. Hence, the estimate (4.16) follows from (2.4), (3.1) and (4.13).

Now we look for invariance properties of the problem (2.17).

Lemma 4.4. Assumek= 2,(2.9)and (2.15). The following estimate holds (4.22) k∂²_xuε(t,·)k_Lp(R)≤ k∂_x²u0,εk_Lp(R)e^K1t+K2

e^K1t−1 K₁ , for each t≥0,2≤p <∞andε >0, where

K₁:= 1 p√

2ku₀k_H2(R), K₂:= (18C₀+ 1)²ku₀k⁶_H2(R). Proof. Let 2≤p <∞. Denote

q_ε:=∂_x²u_ε, there results

(4.23) ∂tqε+ 3qε∂xuε+uε∂xqε+∂³_xPε=ε∂_x²qε, and, from (4.2),

(4.24) ∂_tq_ε+u_ε∂_xq_ε+Pe_ε=ε∂²_xq_ε, where

(4.25) Peε:=∂_x³P1,ε+P3,ε. Multiplying (4.24) bypq_ε|qε|^p−2 there results

∂t(|qε|^p) +uε∂x(|qε|^p) +pPeεqε|qε|^p−2=pεqε|qε|^p−2∂_x²qε

=ε∂_x²(|q_ε²|)−εp(p−1)(∂xqε)². (4.26)

By (3.2), (4.4), and (4.7), pkq_ε(t,·)k^p−1_Lp(R)

d

dtkq_ε(t,·)k_Lp(R)= d dt

Z

R

|q_ε|^pdx

≤ Z

R

∂xuε|qε|^pdx+p Z

R

|Peε||qε|^p−1dx

≤K1

Z

R

|qε|^pdx+p

Peε(t,·) _Lp(

R)

kqε(t,·)k^p−1_Lp(R)

≤K1kqε(t,·)k^p_Lp(R)+pK2kqε(t,·)k^p−1_Lp(R), hence

d

dtkqε(t,·)k_Lp(R)≤K₁

p kqε(t,·)k_Lp(R)+K₂.

The claim is a direct consequence of the Gronwall inequality.

Lemma 4.5. Assumek= 2,(2.9)and (2.15). The following estimate holds k∂_x³uε(t,·)k²_L2(R)+ 2ε

Z t 0

e^K3(t−τ)k∂_x⁴uε(τ,·)k²_L2(R)dτ (4.27)

≤ k∂³_xu_0,εk²_L2(R)e^K3t+K₄e^K3t−1 K3

, for each t≥0 andε >0, where

K₃:= 1

√2ku₀k_H2(R)+7

2, K₄:=

3

4 + 16C₀²

ku₀k⁴_H2(R). Proof. Using the notation from the proof of Lemma 4.4, we have (4.28) ∂t∂xqε+∂xuε∂xqε−1

2q²_ε+uε∂_x²qε+1 2 ∂xuε

2

+u²_ε+∂_x²Pε−Pε=ε∂_x³qε. By (4.28), (3.2), and (4.3),

1 2

d dt

Z

R

(∂xqε)²dx= Z

R

∂xqε∂t∂xqεdx

=ε Z

R

∂_x³qε∂xqεdx− Z

R

∂xuε(∂xqε)²dx +1

2 Z

R

q_ε²∂xqεdx− Z

R

uε∂²_xqε∂xqεdx

−1 2 Z

R

∂xuε

2

∂xqεdx− Z

R

u²_ε∂xqεdx

− Z

R

∂_x²Pε∂xqεdx+ Z

R

Pε∂xqεdx

=−ε Z

R

∂²_xqε

² dx−1

2 Z

R

∂xuε(∂xqε)²dx

−1 2 Z

R

∂_xq_ε ∂_xu_ε2

dx− Z

R

u²_ε∂_xq_εdx

− Z

R

∂_x²P_ε∂_xq_εdx+ Z

R

P_ε∂_xq_εdx

≤ −ε Z

R

∂²_xqε

2

dx+ 1

2k∂xuεkL^∞([0,∞)×R)+7 4

Z

R

(∂xqε)²dx +1

4 Z

R

∂_xu_ε⁴ dx+1

2 Z

R

u⁴_εdx+1 2

Z

R

∂_x²P_ε² dx+1

2 Z

R

P_ε²dx

≤ −ε Z

R

∂²_xqε²

dx+K3

2 Z

R

(∂xqε)²dx+K4

2 ,