The Cauchy problem for a 4-parameter family of equations with peakon traveling waves

Nonlinear Analysis 133 (2016) 161–199

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Nonlinear Analysis www.elsevier.com/locate/na

The Cauchy problem for a 4-parameter family of equations with peakon traveling waves A. Alexandrou Himonas a , Dionyssios Mantzavinos b,∗ a b

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, United States

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Article history: Received 6 December 2015 Accepted 14 December 2015 Communicated by Enzo Mitidieri MSC: primary 35Q53 37K10 35C07

abstract The initial value problem for a novel 4-parameter family of evolution equations, which are nonlinear and nonlocal and possess peakon traveling wave solutions, is studied on both the line and the circle. It is proved that this family of equations is well-posed in the sense of Hadamard when the initial data belong to the Sobolev spaces H s with s > 5/2. Also, it is shown that the data-to-solution map is not uniformly continuous. However, if H s , s > 5/2, is equipped with a weaker H r norm, 0 6 r < s, then the solution map becomes H¨ older continuous. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Fokas–Olver–Rosenau–Qiao equation Peakon traveling waves Novikov equation Integrable equations Camassa–Holm equation Degasperis–Procesi equation Cauchy problem Well-posedness in Sobolev spaces Non-uniform dependence on initial data H¨ older continuity Approximate solutions Commutator estimate Conserved quantities

1. Introduction For k a positive integer with k > 2 and a, b, c real numbers, we consider the initial value problem for the following k-abc family of equations (k-abc-equation)   b ut + uk ux − auk−2 u3x + D−2 ∂x uk+1 + c uk−1 u2x − a (k − 2) uk−3 u4x k+1     (1.1) + D−2 k (k + 2) − 8a − b − c (k + 1) uk−2 u3x − 3a (k − 2) uk−3 u3x uxx = 0, ∗ Corresponding author. E-mail addresses: [email protected] (A. Alexandrou Himonas), [email protected] (D. Mantzavinos).

http://dx.doi.org/10.1016/j.na.2015.12.012 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

. where D−2 = (1 − ∂x2 )−1 , and prove its well-posedness in Sobolev spaces H s , s > 52 , on both the line R . and the circle T = R/2πZ. This is a novel evolutionary, nonlinear and nonlocal equation with (k + 1)-order nonlinearities, which is not a quasilinear equation when a ̸= 0. It is interesting that for k = 2 the k-abcequation includes the two known integrable Camassa–Holm type equations with cubic nonlinearities, namely the Fokas–Olver–Rosenau–Qiao and the Novikov equations. Also, when a = 0 this equation makes sense for all positive integers k > 1. In particular, for k = 1 it contains the Camassa–Holm and the Degasperis–Procesi equations, which were respectively derived in [6,14] and are the two members of the k-abc-equation with quadratic nonlinearities that are known to be integrable. Furthermore, it is remarkable that the k-abc-equation possesses peakon traveling wave solutions for all values of the four parameters. These, including multipeakons, were recently studied in [32]. In the nonperiodic case, the peakon solutions are of the form (e.g. see Fig. 1.1) k u(x, t) = γe−|x−(1−a)γ t| ,

γ ∈ R,

and in the periodic case they are given by the formula      u(x, t) = γ cosh x − 1 + (1 − a) sinh2 (π) coshk−2 (π)γ k t p − π ,

(1.2)

γ ∈ R,

(1.3)

x . where [x]p = x − 2π 2π , provided that 6a + b + 2c = 3k. Furthermore, under appropriate conditions on the four parameters the k-abc-equation also admits multipeakon solutions. The k-abc-equation unifies several important equations of Camassa–Holm type. For k = 2 and c = (6 − 6a − b)/2, it corresponds to the ab-family of equations (ab-equation)     2a + b − 2 3 b 3 6 − 6a − b 2 u + uux + D−2 ux = 0, ut + u2 ux − au3x + D−2 ∂x (1.4) 3 2 2 which was introduced in [32] and which was shown there to possess non-periodic multipeakon traveling wave solutions u(x, t) =

n 

pj (t)e−|x−qj (t)| ,

(x, t) ∈ R × R, n ∈ N,

(1.5)

j=1

provided that the positions qj and momenta pj satisfy an appropriate system of nonlinear differential equations. Moreover, for a = 1/3 and b = 2 the ab-equation becomes the Fokas–Olver–Rosenau–Qiao (FORQ) equation     2 3 2 3 2 3 −2 −2 1 1 (1.6) ∂t u + u ∂x u − 3 (∂x u) + D ∂x u + u (∂x u) + D (∂x u) = 0, 3 3 which was derived in different ways by Fokas [18], Olver and Rosenau [49] and Qiao [50], and also appeared in a work by Fuchssteiner [19]. The choice a = 0 and b = 3 in Eq. (1.4) gives the Novikov equation (NE)     3 2 2 −2 3 −2 1 3 ut + u ux + D ∂x u + uux + D u = 0, (1.7) 2 2 x which was derived by V. Novikov [47] in his effort to classify all integrable CH-type equations with quadratic and cubic nonlinearities. The Lax pair for the FORQ equation was derived in [50] and reads         ψ1 ψ1 ψ1 ψ1 = U (m, λ) , = V (m, u, λ) , (1.8a) ψ2 ψ2 ψ2 ψ2 x

t

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

Fig. 1.1. The non-periodic peakon (1.2) for k = 2, a =

1 3

and γ =

√

163

2 at time t = 3.

where m = u − uxx and   1 1 − λm   2 U (m, λ) =  2 , λ ∈ R, 1 1 − λm 2 2    2  1 2 1 1 1 2 2 + u − u − (u − u ) − λm u − u x x x   2 2 λ 2 V (m, u, λ) =  λ  2    . 1 1 1 1 (u + ux ) + λm u − u2x − 2− u2 − u2x λ 2 λ 2 The Lax pair for NE was derived by Hone and Wang in [38] and is given by         ψ1 ψ1 ψ1 ψ1         ψ2  = U (m, λ) ψ2  , ψ2  = V (m, u, λ) ψ2  , ψ3 ψ3 ψ3 ψ3 x

(1.8b)

(1.8c)

(1.9a)

t

where, as before, m = u − uxx and  0 λm  U (m, λ) = 0 0 1 0  1

 1  λm , 0

 ux (1.9b) − λmu2 u2x λ    u 2 ux V (m, u, λ) =  − 2 − λmu2  −  , λ ∈ R. λ 3λ λ   u 1 2 −u + uux λ 3λ2 Finally, for a = 0 and c = (3k−b)/2 the k-abc-equation turns into the following generalized Camassa–Holm equation (g-kbCH)     3k − b k−1 2 (k − 1)(b − k) k−2 3 b uk+1 + u ux + D−2 u ux = 0, (1.10) ut + uk ux + D−2 ∂x k+1 2 2 3λ2

− uux

whose well-posedness in H s with s > 23 was proved in [25], while its multipeakon traveling waves were derived in [21]. We note that the g-kbCH equation makes sense for all positive integers k with k > 1 and all real numbers b. In fact, for k = 1 it gives the well-known b-family of equations (b-equation) with quadratic nonlinearities, which was introduced by Holm and Staley [35,36] in the local form ut + 

evolution

umx 

convection

+ bux m = 0,   

m = u − uxx ,

(1.11)

stretching

expressing a balance between evolution, convection and stretching. Holm and Staley made the important observation that the b-equation has peakon (and multipeakon) traveling wave solutions of the form u(x, t) = γe−|x−γt| for all values of b.

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Note that the b-equation (1.11) contains two integrable members, namely the Camassa–Holm (CH) and the Degasperis–Procesi (DP) equations, which correspond to b = 2 and b = 3 respectively. In fact, it has been shown by Mikhailov and Novikov [46] that no other integrable equations are included in the b-family (1.11). Moreover, V. Novikov [48] recently proved that the only other integrable member of the g-kbCH family (1.10) apart from those already included in the b-equation is the NE (1.7). On the other hand, the existence of additional integrable members of the k-abc-equation (1.1) apart from the aforementioned three equations and the FORQ equation remains an open question. Regarding conservation laws, we note that solutions to the k-abc-equation conserve the Sobolev H 1 norm under certain conditions on the four parameters. Indeed, integration by parts yields     1 d 2 k−3 5 ∥u(t)∥H 1 = 2k − 2c − 1 − k2 (9a + b + 2c − 3k) uk ux uxx − 3a (k − 1)(k − 2)u u x dx, 4 2 dt thus, the H 1 norm is conserved if 9a + b + 4c = 9 when k = 2, and if a = 0 and 2c + k2 (b + 2c − 3k) + 1 = 2k when k > 3. Observe that in the case of the g-kbCH equation (1.10) we have a = 0 and c = (3k − b)/2, so the conservation condition simplifies to b = k + 1, which is the condition stated in [25]. Next, we state the main results of this work precisely. Since the case a = 0 yields (up to coefficients) the g-kbCH equation, whose well-posedness in Sobolev spaces H s for all s > 32 was proved in [25], here we assume that a ̸= 0. Theorem 1 (Hadamard Well-Posedness). Let a, b, c ∈ R with a ̸= 0 and k ∈ N with k > 2. Then, the Cauchy problem for the k-abc-equation (1.1) with initial condition u(x, 0) = u0 (x) ∈ H s , x ∈ R or T, s > 25 , has a unique solution u ∈ C([0, T ]; H s ) which admits the estimate  1 k −1 , (1.12) ∥u(t)∥H s 6 21+ k ∥u0 ∥H s , 0 6 t 6 T 6 2k+1 k cs ∥u0 ∥H s where the constant cs > 0 depends only on s. Furthermore, the data-to-solution map u0 → u(t) is continuous. The next result states that the continuity of the data-to-solution map given in Theorem 1 is optimal. Theorem 2 (Non-Uniform Continuity). The data-to-solution map of the Cauchy problem for the k-abcequation is not uniformly continuous from any bounded subset of H s into C([0, T ]; H s ), both on the line and on the circle. According to the following theorem, however, the data-to-solution map is better than continuous if the topology of H s is replaced with that of H r for r < s. Theorem 3 (H¨ older Continuity). If s > 52 and 0 6 r < s, then the data-to-solution map of the Cauchy problem for the k-abc-equation on the line and on the circle is H¨ older continuous in H s under the H r norm. s More precisely, for initial data u0 , u ˜0 ∈ B(0, ϱ) = {f ∈ H : ∥f ∥H s 6 ϱ} ⊂ H s the corresponding solutions u, u ˜ of the k-abc-equation satisfy the inequality α

∥u − u ˜∥C([0,T ];H r ) 6 cs,r ∥u0 − u ˜ 0 ∥H r ,

(1.13)

where the H¨ older exponent α is given by  1, (s, r) ∈ A1 ,    2s − 3   , (s, r) ∈ A2 ,  s−r α= s−r   , (s, r) ∈ A3 ,     2 s − r, (s, r) ∈ A4 ,

(1.14)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

165

Fig. 1.2. The regions of H¨ older continuity in the sr-plane.

and the regions A1 , A2 , A3 and A4 of the sr-plane, which are shown in Fig. 1.2, are defined by 

   3 3 ,3 − s 6 r 6 s − 2 ∪ , s − 2 6 r 6 , 2 2   5 A4 = s > , s − 1 6 r < s . 2 A1 =

06r6

(1.15)

The constant cs,r > 0 depends only on s, r, ϱ. The present work brings in two important novelties in the analysis of Camassa–Holm type equations. It provides for the first time a comprehensive treatment of a large family of Camassa–Holm type equations whose local part is not of a purely Burgers nature (e.g. uk ux ) but also includes non-quasilinear terms like uk−2 u3x . Indeed, with the exception of the FORQ equation [31,30], all existing works on the well-posedness and continuity properties of Camassa–Holm type equations such as the CH itself [28,29] and the DP and Novikov equations [23,24,27], are concerned with models whose local part is entirely quasilinear. The lack of quasilinearity encountered in the present work is handled by converting the k-abc-equation into a 2 × 2 system. Although such an approach was also employed in [31,30] for the FORQ equation, here we analyze a much more general model which is of kth, as opposed to cubic, nonlinear order. This fact introduces additional complications which arise, perhaps surprisingly, at the level of both the local and the nonlocal terms of the k-abc-equation (see the nonlocal form (2.1)) and require clever use of delicate estimates such as the “negative multiplier” estimate of Lemma 5. Also, the k-abc-equation unifies for the first time all known Camassa–Holm type equations, including those that have been shown to be integrable (i.e. CH, DP, NE, FORQ) and those that possess multipeakon traveling wave solutions (e.g. b-equation, g-kbCH, abequation). In this respect, it is the product of the culmination of nearly two decades of research by several mathematicians, and hence it provides the state-of-the-art techniques for studying the well-posedness and continuity properties of Camassa–Holm type equations.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

166

We conclude this introduction by noting that there is an extensive literature about well-posedness, continuity properties of the solution map, traveling wave solutions, unique continuation, and other analytic, geometric and physical properties of Camassa–Holm type equations and related nonlinear evolution equations. Additional results on these topics can be found in the following works and the references therein [1–3, 5,7,8,11–13,16,17,20,23,28,29,34,39,44,45,51,4,9,41,10,22,26,33,37,42,43,52,54,55]. The paper is organized as follows. In Section 2, we establish Theorem 1 on the Hadamard well-posedness in H s , s > 52 . In Sections 3 and 4, we give the proof of Theorem 2 on the non-uniform continuity of the data-to-solution map in the periodic and in the non-periodic case, respectively. Finally, in Section 5 we prove Theorem 3 on the H¨ older continuity of the data-to-solution map in H r , r < s. 2. Hadamard well-posedness In this section we give the proof of Theorem 1 on the Hadamard well-posedness of the k-abc-equation (1.1) with initial data u(x, 0) = u0 (x) in Sobolev spaces H s . Since the proof is similar for both the circle T and the line R, we provide the details for T and then conclude the section with the modifications required in the case of R. The most convenient form of the k-abc-equation for showing well-posedness is the nonlocal form ∂t u = auk−2 u3x − uk ux − Φ(u),

(2.1)

where the term Φ(u) contains the nonlocal terms of the k-abc equation and is equal to   b Φ(u) = D−2 ∂x k+1 uk+1 + c uk−1 u2x − a (k − 2) uk−3 u4x    + D−2 k (k + 2) − 8a − b − c (k + 1) uk−2 u3x − 3a (k − 2) uk−3 u3x uxx .

(2.2)

 −1 We recall that D−2 = 1 − ∂x2 and, more generally, for any s ∈ R the operator Ds is defined by  s s f (ξ) = 1 + ξ 2 2 fˆ(ξ),  D where fˆ denotes the Fourier transform of a function f over T. The Sobolev space H s (T) is then defined through Ds by    2  21 s  . H s (T) = f ∈ D′ (T) : ∥f ∥H s = ∥Ds f ∥L2 ≃ 1 + ξ 2 fˆ(ξ) <∞ ,

s ∈ R.

ξ∈Z

For simplicity of notation, we hereafter denote H s (T) by H s unless stated otherwise. Hadamard wellposedness consists of three parts: existence, uniqueness, and continuous dependence of the solution on the initial data. We begin with existence and then proceed to the other two components. 2.1. Existence We wish to treat the k-abc-equation (2.1) as an ODE in H s and thus establish existence of solution by invoking Cauchy’s existence theorem for ODEs in Banach spaces [15, 10.4.5, p. 283]. In this regard, we note that the two local terms on the right-hand side of Eq. (2.1) require mollification due to the presence of ux . Furthermore, it actually turns out that the first of these local terms, which involves u3x , cannot be handled even after mollification. Therefore, in order to implement our idea we convert the k-abc-equation (2.1) into a first-order system of equations. In particular, differentiating our equation once with respect to x and then setting w = ux gives rise to the system

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

∂t u = auk−2 w3 − uk w − F1 (u, w), k−2

∂t w = 3au

k−1

2

w wx + (c − k) u

167

(2.3a) 2

k

w − u wx +

b k+1

k+1

u

− F2 (u, w),

(2.3b)

supplemented with the initial conditions u(x, 0) = u0 (x),

w(x, 0) = ∂x u0 (x),

where the nonlocal terms F1 and F2 are defined by    Fj (u, w) = D−2 ∂xj−1 k (k + 2) − 8a − b − c (k + 1) uk−2 w3 − 3a (k − 2) uk−3 w3 wx   b uk+1 + cuk−1 w2 − a(k − 2)uk−3 w4 , j = 1, 2. + D−2 ∂x2−j k+1

(2.3c)

(2.4)

We now mollify system (2.3) by applying a Friedrichs mollifier Jε which is constructed as follows. Fix a Schwartz function j ∈ S(R) and define the family of periodic functions jε by jε (x) =

1  iξx  e j(εξ), 2π

ε ∈ (0, 1],

ξ∈Z

so that  jε (ξ) =  j(εξ). The operator Jε is then defined through its action on a function f as the convolution Jε f = jε ∗ f, ε ∈ (0, 1]. It is straightforward to establish the estimate ∥∂x Jε f ∥H s .

1 ∥f ∥H s , ε

(2.5)

(the symbol . denotes inequality up to a constant) which is the one allowing us to apply Cauchy’s existence theorem for systems of ODEs in H s−1 × H s−1 . Indeed, mollification by means of Jε of the local terms of system (2.3) that involve spatial derivatives gives rise to the system ∂t uε = auk−2 wε3 − ukε wε − F1 (uε , wε ), ε     k−2 2 k ∂t wε = 3aJε (Jε uε ) (Jε wε ) (Jε wε )x − Jε (Jε uε ) (Jε wε )x b uk+1 − F2 (uε , wε ), + (c − k) uk−1 wε2 + k+1 ε ε

(2.6b)

uε (x, 0) = u0 (x),

(2.6c)

wε (x, 0) = ∂x u0 (x),

(2.6a)

which is a system of ODEs in H s−1 × H s−1 thanks to estimate (2.5). Therefore, according to Cauchy’s existence theorem for ODEs in Banach spaces, there exists a unique solution (uε , wε ) of this system in the space C([0, Tε ]; H s−1 × H s−1 ). The idea is to use this result in the limit ε → 0 in order to infer existence of solution for the original, non-mollified system (2.3). To achieve this, we first need to estimate (uε , wε ) in terms of the associated initial data (u0 , ∂x u0 ) and for an appropriate lifespan independent of the parameter ε. Energy estimates for uε and wε . We will make extensive use of the inequalities ∥Jε f ∥H s 6 ∥f ∥H s ,

∥∂x f ∥H s−1 6 ∥f ∥H s ,

 −2  D f  s 6 ∥f ∥ s−2 , H H

(2.7)

as well as of the algebra property in H s , which reads ∥f g∥H s 6 cs ∥f ∥H s ∥g∥H s ,

s > 12 , cs > 0.

(2.8)

Applying the operator Ds−1 on the left of Eq. (2.6b), multiplying by Ds−1 wε on the right, commuting the operators Jε and Ds−1 twice and integrating over T, we obtain the “energy” equation

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

168

1 d 2 ∥wε ∥H s−1 = 3a 2 dt



  k−2 2 (Jε wε ) (Jε wε )x · Ds−1 Jε wε dx Ds−1 (Jε uε )

(2.9a)

T



k

Ds−1 (Jε uε ) (Jε wε )x · Ds−1 Jε wε dx

−

(2.9b)

T



 Ds−1 (c − k) uεk−1 wε2 +

+ T −

b k+1

 · Ds−1 wε dx uk+1 ε

Ds−1 F2 (uε , wε ) · Ds−1 wε dx.

(2.9c) (2.9d)

T

The terms (2.9a) and (2.9b) can be estimated in the same way. Hence, we only give the details for the latter term. Noting that  k k k Ds−1 (Jε uε ) (Jε wε )x = Ds−1 , (Jε uε ) (Jε wε )x − (Jε uε ) Ds−1 (Jε wε )x and using the Cauchy–Schwarz inequality, we find      s−1 2  1    k k |(2.9b)| 6  Ds−1 , (Jε uε ) (Jε wε )x  ∥wε ∥H s−1 + J (J ε uε ) · ∂x D ε wε  dx. 2 T L2

(2.10)

The first term on the right-hand side of (2.10) can be estimated by employing the following lemma. Lemma 1 (Kato–Ponce [40]). If s > 0 then there is cs > 0 such that  s        D , f g  2 6 cs ∥Ds f ∥ 2 ∥g∥ ∞ + ∥∂x f ∥ ∞ Ds−1 g  2 . L L L L L Lemma 1 in combination with the Sobolev embedding and the algebra property yields the estimate     s−1 k k (2.11)  D , (Jε uε ) (Jε wε )x  2 . ∥uε ∥H s−1 ∥wε ∥H s−1 , s > 52 . L

Moreover, for the second term of (2.10) we have      2   2    k  k 2 k ∂x (Jε uε ) · Ds−1 Jε wε  dx 6 ∂x (Jε uε ) L∞ ∥Jε wε ∥H s−1 , (Jε uε ) · ∂x Ds−1 Jε wε  dx = T

T

thus, by the Sobolev embedding and the algebra property we find    k  2   k 2 (Jε uε ) · ∂x Ds−1 Jε wε  dx 6 uε H s−1 ∥wε ∥H s−1 ,

s > 25 .

(2.12)

T

Combining estimates (2.11) and (2.12) with (2.10) yields the estimate  k 2 |(2.9b)| . uε H s−1 ∥wε ∥H s−1 ,

s > 25 .

(2.13)

 k−2 4 |(2.9a)| . uε H s−1 ∥wε ∥H s−1 ,

s > 25 .

(2.14)

Similarly, we find

The estimation of the term (2.9c) is straightforward, as this term does not contain any derivatives. Indeed, by using just the Cauchy–Schwarz inequality and the algebra property in H s−1 we have k−1

3

k+1

|(2.9c)| . ∥uε ∥H s−1 ∥wε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 ,

s > 23 .

(2.15)

Finally, concerning the estimation of the term (2.9d) we note that by the Cauchy–Schwarz inequality     |(2.9d)| 6 Ds−1 F2 (uε , wε )L2 Ds−1 wε L2 = ∥F2 (uε , wε )∥H s−1 ∥wε ∥H s−1 . (2.16) Let F2 = F2,1 + F2,2 with

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

169

   F2,1 (uε , wε ) = D−2 ∂x k (k + 2) − 8a − b − c (k + 1) uk−2 w3 − 3a (k − 2) uk−3 wε3 ∂x wε , ε ε   k−1 2 k−3 4 b uk+1 + c u w − a (k − 2) u w F2,2 (uε , wε ) = D−2 k+1 ε ε ε ε ε . Then, the last two inequalities in (2.7), the triangle inequality and the algebra property imply k−2

k−3

3

4

s > 25 ,

∥F2,1 (uε , wε )∥H s−1 . ∥uε ∥H s−1 ∥wε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 , k+1

k−1

k−3

2

4

∥F2,2 (uε , wε )∥H s−1 . ∥uε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 ,

s > 23 .

The above two estimates in combination with inequality (2.16) imply k+1

k−1

k−2

3

4

|(2.9d)| . ∥uε ∥H s−1 ∥wε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 + ∥uε ∥H s−1 ∥wε ∥H s−1 k−3

5

+ ∥uε ∥H s−1 ∥wε ∥H s−1 t ∥H s−1 ∥wε ∥H s−1 ,

s > 25 .

(2.17)

Inserting estimates (2.13)–(2.15) and (2.17) in inequality (2.9) yields the differential inequality  k+1 d ∥wε ∥H s−1 6 cs ∥uε ∥H s−1 + ∥wε ∥H s−1 , dt

s > 52 , cs > 0.

(2.18)

The corresponding differential inequality for ∥uε ∥H s−1 can be obtained in an analogous way and reads k+1  d , s > 32 , cs > 0. ∥uε ∥H s−1 6 cs ∥uε ∥H s−1 + ∥wε ∥H s−1 (2.19) dt . . Size estimate for (uε , wε ). Let Uε = (uε , wε ) with norm ∥Uε ∥H s = ∥uε ∥H s + ∥wε ∥H s . Inequalities (2.18) and (2.19) then imply the differential inequality d k+1 ∥Uε (t)∥H s−1 6 cs ∥Uε (t)∥H s−1 , dt

s > 25 , cs > 0,

which in turn yields  − 1 k ∥Uε (t)∥H s−1 6 1 − kcs ∥Uε (0)∥H s−1 t k ∥Uε (0)∥H s−1 ,

s > 25 , cs > 0.

(2.20)

Hence, for lifespan Tε equal to Tε =

1 k

2kcs ∥Uε (0)∥H s−1

,

cs > 0,

the solution Uε of the initial value problem (2.6) satisfies the estimate 1

0 < t 6 Tε , s > 25 .

∥Uε (t)∥H s−1 6 2 k ∥Uε (0)∥H s−1 ,

(2.21)

However, according to the initial conditions of (2.6) and the second inequality in (2.7) we have ∥Uε (0)∥H s−1 = ∥u0 ∥H s−1 + ∥∂x u0 ∥H s−1 6 2 ∥u0 ∥H s , hence, we conclude that for the common lifespan T given by T =

1 k

2k+1 kcs ∥u0 ∥H s

,

cs > 0,

(2.22)

the solution Uε of the initial value problem (2.6) satisfies the size estimate 1

∥Uε (t)∥H s−1 6 21+ k ∥u0 ∥H s ,

0 < t 6 T, s > 52 .

In addition, estimating Eqs. (2.6a) and (2.6b) in H s−2 we have     ∥∂t uε ∥H s−2 6 auk−2 wε3 H s−2 + ukε wε H s−2 + ∥F1 (uε , wε )∥H s−2 ε

(2.23)

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and       k−2 2 k ∥∂t wε ∥H s−2 6 3|a| Jε (Jε uε ) (Jε wε ) (Jε wε )x H s−2 + Jε (Jε uε ) (Jε wε )x H s−2     |b|  k+1  + ∥F2 (uε , wε )∥H s−2 . + |c − k|uk−1 wε2 H s−2 + k+1 uε ε H s−2 Hence, estimating the terms F1 and F2 as before and using the size estimate (2.23), we obtain the derivative size estimate k+1

∥∂t Uε (t)∥H s−2 . ∥u0 ∥H s ,

0 < t 6 T, s > 25 .

(2.24)

Starting from the solution Uε ∈ C([0, T ]; H s−1 × H s−1 ) of the mollified system (2.6) and taking the limit . ε → 0, we will establish the existence of a solution U = (u, w) ∈ C([0, T ]; H s−1 ×H s−1 ) of the original system (2.3). For simplicity of notation, we hereafter let I = [0, T ] and denote the space C([0, T ]; H s−1 × H s−1 ) by C(I; H s−1 ). Size estimate for U . Recall that {Uε }ε∈(0,1] ⊂ C(I; H s−1 ) ⊂ L∞ (I; H s−1 ). Moreover, the size estimate 1 (2.23) implies that the family {Uε }ε∈(0,1] is a subset of the closed ball B(0, 21+ k ∥u0 ∥H s ) in L∞ (I; H s−1 ). Let the infinite sequence {Un }n∈N consist of the members of the family {Uε }ε∈(0,1] with ε = 1/n, n ∈ N. 1 Then, {Un }n∈N belongs to the closed ball B(0, 21+ k ∥u0 ∥H s ), which is compact in L1 (I; H s−1 ) in the weak* topology by Alaoglu’s theorem. Thus, by the Bolzano–Weierstrass theorem there is a subsequence of {Un }n∈N   1 that converges to an element U of B 0, 21+ k ∥u0 ∥H s in the weak* sense. Denoting this subsequence by   1 {Unj }j∈N we thus have Unj −→ U as j → ∞ for some U that belongs to B 0, 21+ k ∥u0 ∥H s and, as such, satisfies the size estimate 1

∥U (t)∥H s−1 6 21+ k ∥u0 ∥H s ,

0 < t 6 T, s > 25 .

(2.25)

Convergence in C(I; H s−1−σ ), σ ∈ (0, 1). We will now show that U , which was obtained above as the weak* limit of the subsequence {Unj }j∈N , belongs to C(I; H s−1 ) and satisfies system (2.3). For convenience of notation, let the subsequence {Unj }j∈N be denoted simply by {Un }n∈N . Recall that {Un }n∈N ⊂ C(I; H s−1 ) and hence {Un }n∈N ⊂ C(I; H s−1−σ ) for any σ ∈ (0, 1). In addition, for any t1 , t2 ∈ I we have   σ   1 ∥Un (t1 ) − Un (t2 )∥H s−1−σ 2 s−1 1 n (ξ, t1 ) − U n (ξ, t2 )2 2 , U (1 + ξ ) = sup sup 2 2 (1+ξ )|t1 −t2 | |t1 − t2 |σ t1 ̸=t2 t1 ̸=t2 ξ∈Z

σ

so using the fact that ν < 1 + ν for any ν > 0 and σ ∈ (0, 1) together with the triangle inequality and the size estimate (2.23), we find sup t1 ̸=t2

1 ∥Un (t1 ) − Un (t2 )∥H s−2 ∥Un (t1 ) − Un (t2 )∥H s−1−σ . 22+ k ∥u0 ∥H s + sup . |t1 − t2 |σ |t1 − t2 | t1 ̸=t2

(2.26)

Employing the Mean Value Theorem for U (t) in the interval [t1 , t2 ] and the derivative size estimate (2.24), we conclude that sup t1 ̸=t2

∥Un (t1 ) − Un (t2 )∥H s−1−σ k+1 . ∥u0 ∥H s + ∥u0 ∥H s , |t1 − t2 |σ

∀t1 , t2 ∈ I, σ ∈ (0, 1),

(2.27)

which implies that the sequence {Un }n∈N is equicontinuous. Moreover, note that {Un }n∈N is supported in I, which is fixed and compact, and is bounded in H s−1 since it belongs to C(I; H s−1 ). Thus, by Rellich’s theorem for each t ∈ I the set {Un (t)}n∈N is precompact in H s−1−σ . Equicontinuity and precompactness for each t ∈ I allow us to invoke Ascoli’s theorem to conclude that {Un }n∈N is precompact in C(I; H s−1−σ ). Equivalently, there exists a subsequence {Unj }j∈N of {Un }n∈N which converges to some element U of C(I; H s−1−σ ). Convergence in C(I; C 1 ). We have shown that for ε = 1/n the solutions Uε converge to U in C(I; H s−1−σ ) as ε → 0, hence limε→0 supt∈I ∥Uε − U ∥H s−1−σ = 0. Thus, for σ ∈ (0, 1) such that σ < s − 25 , the Sobolev embedding implies that limε→0 ∥Uε − U ∥C(I;C 1 ) = 0.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

171

Proving that U satisfies system (2.6). Next, we show that U , which was obtained above as the limit of Uε in C(I; C 1 ), is a solution of system (2.6). We need the following standard analysis result (see [15, 8.6.3 & 8.6.4, p. 163]): Let {fn } be a sequence of differentiable mappings from an open interval I ⊂ R into a Banach space F . Suppose that (i) there is a t0 ∈ I such that {fn (t0 )} converges as an element of F , and (ii) for every t ∈ I there is a neighborhood B(t) ⊂ I centered at t such that in B(t) the sequence {fn′ (t)} converges uniformly. Then, for each t ∈ I the sequence {fn (t)} converges uniformly in B(t) and for limn→∞ fn (t) = f (t) and limn→∞ fn′ (t) = g(t) we have f ′ (t) = g(t). We make use of the above theorem for fn = Uε and F = C(I; C). The convergence in C(I; C 1 ) of Uε to U established earlier implies that the first hypothesis of the theorem is satisfied. In fact, note that it also implies the first conclusion of the theorem but not the second one, which is needed for showing that U satisfies system (2.6). Thus, we now verify that the second hypothesis of the theorem is also satisfied, namely that ∂t Uε converges uniformly in C(I; C). Recall that ∂t Uε = (∂t uε , ∂t wε ), where uε and wε satisfy the mollified system (2.6). Due to the uniform convergence of Uε to U , the non-mollified terms of (2.6) converge uniformly in C(I; C 1 ) to the corresponding converge to uk−2 w3 , uk w, uk−1 w2 and uk+1 wε2 and uk+1 wε3 , ukε wε , uk−1 terms that involve u and w, i.e. uk−2 ε ε ε −2 respectively. Also, due to the continuity of the operators D , D−2 ∂x it follows that the nonlocal terms Fj (uε , wε ) converge to Fj (u, w) uniformly in C(I; C 1 ). Regarding the mollified terms of system (2.6), using the triangle inequality we have ∥Jε wε − w∥C(I;C) 6 ∥(Jε − I) wε ∥C(I;C) + ∥wε − w∥C(I;C) .

(2.28)

By the Sobolev embedding for 21 < r < s−1 (recall that s > 52 ) and the fact that the map Jε −I : H s−1 → H r satisfies the operator norm estimate   ∥Jε − I∥L(H s−1 ;H r ) = o εs−1−r , (2.29) we obtain     ∥(Jε − I) wε ∥C(I;C) 6 o εs−1−r ∥wε ∥C(I;H s−1 ) 6 o εs−1−r ∥u0 ∥H s . Using this inequality in (2.28) and recalling that wε converges uniformly to w in C(I; C 1 ), we find   ∥Jε wε − w∥C(I;C) 6 o εs−1−r ∥u0 ∥H s + ∥wε − w∥C(I;C) −−−−→ 0. ε→0

(2.30)

Similarly,   ∥Jε uε − u∥C(I;C) 6 o εs−1−r ∥u0 ∥H s + ∥uε − u∥C(I;C) −−−−→ 0. ε→0

Finally, using the triangle inequality, the Sobolev embedding for r > we have

3 2

(2.31)

and estimate (2.29) for r < s − 1,

∥(Jε wε )x − wx ∥C(I;C) 6 ∥(Jε − I) wε ∥C(I;H r ) + ∥wε − w∥C(I;C 1 )   6 o εs−1−r ∥u0 ∥H s + ∥wε − w∥C(I;C 1 ) −−−−→ 0. ε→0

(2.32)

   k−2 2 k Estimates (2.30)–(2.32) imply that the mollified terms Jε (Jε uε ) (Jε wε ) (Jε wε )x and Jε (Jε uε )  (Jε wε )x converge uniformly in C(I; C) to uk−2 w2 wx and uk wx respectively. Overall, ∂t Uε converges uniformly in C(I; C) to the vector   b auk−2 w3 − uk w − F1 (u, w), 3auk−2 w2 wx − uk wx + (c − k) uk−1 w2 + k+1 uk+1 − F2 (u, w) . (2.33) The two hypotheses of the convergence theorem have been satisfied, thus since limε→0 Uε (t) = U (t) and limε→0 ∂t Uε (t) = (2.33) we deduce that ∂t U = (2.33), i.e. that U satisfies the non-mollified system (2.3).

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In addition, due to the uniform convergence of ∂t Uε to ∂t U in C(I; C), the derivative size estimate (2.24) yields an analogous estimate for U , i.e. k+1

∥∂t U (t)∥H s−2 . ∥u0 ∥H s ,

0 < t 6 T, s > 25 .

(2.34)

Then, the Mean Value Theorem inequality ∥U (t1 ) − U (t2 )∥H s−2 6 supt∈I ∥∂t U ∥H s−2 |t1 −t2 | for all t1 , t2 ∈ I implies k+1

∥U (t1 ) − U (t2 )∥H s−2 6 ∥u0 ∥H s |t1 − t2 | i.e. that U ∈ Lip(I; H

s−2

∀t1 , t2 ∈ I, s > 25 ,

(2.35)

).

Improving the regularity of U up to C(I; H s−1 ). We have shown that there exists a solution U of system (2.3) which belongs to L∞ (I; H s−1 ) ∩ Lip(I; H s−2 ) ∩ C(I; C 1 ). We will now prove that U ∈ C(I; H s−1 ). More precisely, we will show that if {tn }n∈N ⊂ I converges to t ∈ I then limn→∞ ∥U (tn ) − U (t)∥H s−1 = 0. By the definition of the H s−1 norm, we have 2

2

2

∥U (tn ) − U (t)∥H s−1 = ∥U (tn )∥H s−1 − 2 ⟨U (tn ), U (t)⟩H s−1 + ∥U (t)∥H s−1 . Thus, it suffices to show that   2 2 lim ∥U (tn )∥H s−1 − 2 ⟨U (tn ), U (t)⟩H s−1 + ∥U (t)∥H s−1 = 0. n→∞

(2.36)

Lemma 2. The solution U ∈ L∞ (I; H s−1 )∩Lip(I; H s−2 ) is continuous on I with respect to the weak topology in H s−1 , i.e. limn→∞ ⟨U (tn ) − U (t), ϕ⟩H s−1 = 0 for all ϕ ∈ H s−1 . Proof. Let ϕ ∈ H s−1 . Given ε > 0, choose ψ ∈ S(T) such that ∥ϕ − ψ∥H s−1 <

1 23+ k

ε . ∥u0 ∥H s

(2.37)

Applying the triangle inequality and the Cauchy–Schwarz inequality, we find |⟨U (tn ) − U (t), ϕ⟩H s−1 | 6 ∥U (tn ) − U (t)∥H s−1 ∥ϕ − ψ∥H s−1 + ∥U (tn ) − U (t)∥H s−2 ∥ψ∥H s . But from the size estimate (2.25) we have 1

∥U (tn ) − U (t)∥H s−1 6 ∥U (tn )∥H s−1 + ∥U (t)∥H s−1 6 2 · 21+ k ∥u0 ∥H s . Hence, using the Lipschitz estimate (2.35) together with inequality (2.37), we find ε k+1 ⟨U (tn ) − U (t), ϕ⟩H s−1 6 + ∥u0 ∥H s |tn − t| ∥ψ∥H s . 2  k+1 Since ∥ψ∥H s is bounded, there is an N such that |tn − t| < ε (2 ∥u0 ∥H s ∥ψ∥H s ) for all n > N . Thus, we conclude that |⟨U (tn ) − U (t), ϕ⟩H s−1 | < ε for all n > N , which completes the proof.  Lemma 2 with ϕ = U reduces (2.36) to limn→∞ ∥U (tn )∥H s−1 = ∥U (t)∥H s−1 . Equivalently, we need to show that the map t → ∥U (t)∥H s−1 is continuous. Let Yε (t) = ∥Jε U (t)∥H s−1 and Y (t) = ∥U (t)∥H s−1 . Using 2 2 the size estimate (2.25) we find that Yε (t) 6 22+ k ∥u0 ∥H s . Thus, by the dominated convergence theorem we deduce that Yε converges to Y pointwise in t for all t ∈ I. Consequently, the following lemma for Yε implies that Y is continuous on I. . Lemma 3. The function Yε (t) = ∥Jε U (t)∥H s−1 is Lipschitz on the interval I. Proof. Applying the Friedrichs mollifier to the first equation of system (2.3) yields the equation ∂t Jε u = aJε (uk−2 w3 ) − Jε (uk w) − Jε F1 (u, w), which in turn implies the energy inequality

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

173

      Ds−1 Jε (uk−2 w3 ) · Ds−1 Jε u dx +  Ds−1 Jε (uk w) · Ds−1 Jε u dx T T     s−1 s−1 +  D Jε F1 (u, w) · D Jε u dx.

 1 d  2 ∥Jε u∥H s−1 .  2 dt



(2.38)

T

Using (2.7) and the algebra property, we obtain the following estimates for the three terms on the right-hand side of inequality (2.38):         k+1 k−1 3 s−1 k s−1 s−1 k−2 3 s−1 . ∥u∥ ∥w∥ , D J (u w ) · D J u dx  D Jε (u w) · D Jε u dx . ∥u∥H s−1 ∥w∥H s−1 ,   ε ε H s−1 H s−1 T T     k−2 4 k−1 3 k 2 k+2 s−1 s−1  D Jε F1 (u, w) · D Jε u dx . ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 . T

Hence, using also the size estimate (2.25) we find  k+1 d k+1 ∥Jε u(t)∥H s−1 . ∥u(t)∥H s−1 + ∥w(t)∥H s−1 . ∥u0 ∥H s . dt Similarly to inequality (2.38), we obtain the following energy inequality for Jε w:   1 d   2 ∥Jε w∥H s−1 .  Ds−1 Jε (uk−2 w2 wx ) · Ds−1 Jε w dx 2 dt T     +  Ds−1 Jε (uk wx ) · Ds−1 Jε w dx T         s−1 k−1 2 s−1 +  D Jε (u w ) · D Jε w dx +  Ds−1 Jε (uk+1 ) · Ds−1 Jε w dx T T     s−1 s−1 +  D Jε F2 (u, w) · D Jε w dx.

(2.39)

(2.40a) (2.40b) (2.40c) (2.40d)

T

The terms (2.40a) and (2.40b) involve the derivative wx and hence they require use of the Kato–Ponce   commutator estimate of Lemma 1. In particular, using the fact that Ds−1 , Jε = 0 and also that ⟨Jε f, g⟩L2 = ⟨f, Jε g⟩L2 , we write (2.40a) in the form        (2.40a) =  Ds−1 , uk−2 w2 wx · Jε Ds−1 Jε w dx + uk−2 w2 Ds−1 wx · Jε Ds−1 Jε w dx. T

T

Employing the Cauchy–Schwarz inequality in the first integral and commuting the mollifier in the second integral then yields      (2.40a) 6  Ds−1 , uk−2 w2 wx L2 Ds−1 Jε wL2 (2.41a)       +  Jε uk−2 w2 Ds−1 wx · Ds−1 Jε w dx. (2.41b) T

Employing Lemma 1, inequalities (2.7), the algebra property and the Sobolev embedding, we obtain k−2

4

(2.41a) . ∥u∥H s−1 ∥w∥H s−1 . Moreover, writing      s−1   k−2 2 s−1 (2.41b) =  Jε , u w D wx · D Jε w dx + uk−2 w2 Jε Ds−1 wx · Ds−1 Jε w dx T T     s−1  2  1  k−2 2 s−1 =  Jε , u w D wx · D Jε w dx + uk−2 w2 · ∂x Ds−1 Jε w dx, 2 T T integrating by parts and using the Cauchy–Schwarz inequality, we have

(2.42)

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   1   2        uk−2 w2 x · Ds−1 Jε w dx (2.41b) 6  Jε , uk−2 w2 Ds−1 wx · Ds−1 Jε w dx +  2 T T       2  1   k−2 2   s−1 6  Jε , uk−2 w2 Ds−1 wx L2 Ds−1 Jε wL2 +  u w x · D Jε w dx. 2 T

(2.43)

We now use the following lemma, whose proof can be found in [28]. Lemma 4. Let f be such that ∥∂x f ∥L∞ < ∞. Then, there is a c > 0 such that for any g ∈ L2 we have ∥[Jε , f ] ∂x g∥L2 6 c ∥∂x f ∥L∞ ∥g∥L2 . Employing Lemma 4 for f = uk−2 w2 and g = Ds−1 wx , we find 2            (2.41b) 6  uk−2 w2 x L∞ Ds−1 wx L2 Ds−1 Jε wL2 +  uk−2 w2 x L∞ Ds−1 Jε wH s−1 k−2

4

. ∥u∥H s−1 ∥w∥H s−1 .

(2.44)

Estimates (2.42) and (2.44) imply the estimate k−2

4

(2.45)

k

2

(2.46)

(2.40a) . ∥u∥H s−1 ∥w∥H s−1 . Similarly, the term (2.40b) admits the estimate (2.40b) . ∥u∥H s−1 ∥w∥H s−1 .

The term (2.40c), which consists of two parts, does not involve wx . Hence, a simple application of the algebra property yields the estimate k−1

k+1

3

(2.40c) . ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 .

(2.47)

For the nonlocal term (2.40d) we use the Cauchy–Schwarz inequality and the algebra property to obtain   k−3 4 k−2 3 k−1 2 k+1 (2.40d) . ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 + ∥u∥H s−1 ∥w∥H s−1 . (2.48) Finally, combining estimates (2.45)–(2.48) and the size estimate (2.25) we find  k+1 d k+1 ∥Jε w∥H s−1 . ∥w∥H s−1 + ∥u∥H s−1 . ∥u0 ∥H s . dt Estimates (2.39) and (2.49) imply

d dt Yε (t)

k+1

. ∥u0 ∥H s for all t ∈ I. Therefore, we have k+1

|Yε (t1 ) − Yε (t2 )| . ∥u0 ∥H s |t1 − t2 | The proof of Lemma 3 is complete.

(2.49)

∀t1 , t2 ∈ I.



2.2. Uniqueness of solution of system (2.3) We have shown that system (2.3) has a solution U = (u, w) ∈ C([0, T ]; H s−1 ) which satisfies the estimates (2.25) and (2.34) and has lifespan T given by Eq. (2.22). We will now prove that U is the unique solution . of system (2.3) in the space C([0, T ]; H s−1 ). Let W = (v, z) ∈ C([0, T ]; H s−1 ) be another solution of system (2.3). Then, the differences ϕ = u − v and ψ = w − z must satisfy the system     ∂t ϕ = auk−2 g2 − uk ψ + af3 z 3 − f1 z ϕ − F1 (u, w) + F1 (v, z), (2.50a)  k−2    ∂t ψ = au g2 − uk ψx + auk−2 ∂x g2 + (c − k) uk−1 g1 ψ   b + af3 (z 3 )x + (c − k) f2 z 2 − f1 zx + k+1 f0 ϕ − F2 (u, w) + F2 (v, z), (2.50b) ϕ(x, 0) = ψ(x, 0) = 0,

(2.50c)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

175

where the nonlocal terms F1 , F2 are defined by Eq. (2.4) and the quantities fj , gj are defined by fj =

k−j 

uk−j−m v m ,

gj =

m=0

j 

wj−m z m ,

j ∈ N.

(2.51)

m=0

We will show that the unique solution of system (2.50) in the space C(I; H σ ), 12 < σ < s − 2, is the trivial solution ϕ = ψ = 0. Note that choosing to work in a larger space than C(I; H s−1 ) simplifies our proof significantly. Applying Dσ on the left of Eqs. (2.50a) and (2.50b) and multiplying on the right by Dσ ϕ and Dσ ψ respectively, we find    k−2    1 d 2 σ k σ ∥ϕ∥H σ = D au g2 − u ψ · D ϕ dx − Dσ af3 z 3 − f1 z ϕ · Dσ ϕ dx 2 dt T T    σ σ − D F1 (u, w) − F1 (v, z) · D ϕ dx (2.52) T

and 1 d 2 ∥ψ∥H σ = 2 dt



  Dσ auk−2 g2 − uk ψx · Dσ ψ dx T    + Dσ auk−2 ∂x g2 + (c − k) uk−1 g1 ψ · Dσ ψ dx T   b + Dσ af3 (z 3 )x + (c − k) f2 z 2 − f1 zx + k+1 f0 ϕ · Dσ ψ dx T   − Dσ F2 (u, w) − F2 (v, z) · Dσ ψ dx.

(2.53a) (2.53b) (2.53c) (2.53d)

T

We first consider Eq. (2.52). The first two terms on the right-hand side of this equation can be handled easily via the algebra property and the size estimate (2.25) and admit the estimates       k (2.54a)  Dσ auk−2 g2 − uk ψ · Dσ ϕ dx . ∥u0 ∥H s ∥ϕ∥H σ ∥ψ∥H σ , T        k 2 (2.54b)  Dσ af3 z 3 − f1 z ϕ · Dσ ϕ dx . ∥u0 ∥H s ∥ϕ∥H σ . T

Concerning the third term of (2.52), we first express the difference of the nonlocal terms as    F1 (u, w) − F1 (v, z) = D−2 k(k + 2) − 8a − b − c(k + 1) z 3 f3 − 43 a(k − 2)f4 (z 4 )x ϕ    + k(k + 2) − 8a − b − c(k + 1) uk−2 g2 ψ − 43 a(k − 2)uk−3 (g3 ψ)x      b + D−2 ∂x k+1 f0 + cf2 z 2 − a(k − 2)f4 z 4 ϕ + c uk−1 g1 − a(k − 2)uk−3 g3 ψ .

(2.55)

All the terms of the above expression can be estimated via the algebra property apart from the term that involves ψx . The reason for this exception is that the derivative would require application of the algebra property in H σ−1 . This would in turn require σ > 23 , which conflicts with the restriction σ < s − 2 (recall that s > 52 ). Thus, the term involving ψx will be handled via the following “negative multiplier” estimate instead of the algebra property. Lemma 5. If σ >

1 2

then ∥f g∥H σ−1 6 cσ ∥f ∥H σ ∥g∥H σ−1 .

The proof of Lemma 5 for 12 < σ < 1 can be found in [29], while the case σ > 1 is proved in [25]. We then find  −2 k−3    k−3 D u (g3 ψ)x H σ . uk−3 H σ ∥(g3 ψ)x ∥H σ−1 . ∥u∥H s−1 ∥g3 ∥H s−1 ∥ψ∥H σ ,

176

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

which yields       k  2   Dσ F (u, w) − F (v, z) · Dσ ϕ dx . ∥u0 ∥H s ∥ϕ∥H σ ∥ψ∥H σ + ∥ϕ∥H σ .

(2.56)

T

Estimates (2.54) and (2.56) together with Eq. (2.52) imply the inequality  d k  ∥ϕ∥H σ . ∥u0 ∥H s ∥ψ∥H σ + ∥ϕ∥H σ . dt   Next, we consider Eq. (2.53). By commuting ∂x with auk−2 g2 − uk , we have      σ   k−2 (2.53a) = D ∂x , auk−2 g2 − uk ψ · Dσ ψ dx + au g2 − uk Dσ ∂x ψ · Dσ ψ dx T T   k−2  σ k σ − D ∂x au g2 − u ψ · D ψ dx.

(2.57)

(2.58)

T

The first term of Eq. (2.58) can be estimated using the following lemma, which is proved in [53]. Lemma 6 (Calderon–Coifman–Meyer). If ρ >

3 2

and 0 6 σ + 1 6 ρ, then ∥[Dσ ∂x , f ]g∥L2 . ∥f ∥H ρ ∥g∥H σ .

Lemma 6 for ρ = s − 1, which is an admissible choice since s > 25 and σ < s − 2, together with the algebra property and the size estimate (2.25) imply        k 2  Dσ ∂x , auk−2 g2 − uk ψ · Dσ ψ dx . ∥u0 ∥H s ∥ψ∥H σ . T

The same estimate can be obtained for the other two terms of (2.58) via integration by parts, the algebra property, the Sobolev embedding and the size estimate (2.25). Overall, we have 2

2

|(2.53a)| . ∥u0 ∥H s ∥ψ∥H σ .

(2.59)

For the two remaining local terms of (2.53), we find   2 k 2 |(2.53b)| . ∥uk−2 ∂x g2 ∥H s−2 + ∥uk−1 g1 ∥H s−2 ∥ψ∥H σ . ∥u0 ∥H s ∥ψ∥H σ

(2.60)

k

(2.61)

and, similarly, |(2.53c)| . ∥u0 ∥H s ∥ϕ∥H σ ∥ψ∥H σ .

Finally, handling the difference F2 (u, w)−F2 (v, z) of the nonlocal terms in the same way as F1 (u, w)−F1 (v, z), we obtain the estimate k  2  |(2.53d)| . ∥u0 ∥H s ∥ϕ∥H σ ∥ψ∥H σ + ∥ψ∥H σ . (2.62) Combining estimates (2.59)–(2.62), we find d k ∥ψ∥H σ . ∥u0 ∥H s (∥ϕ∥H σ + ∥ψ∥H σ ) . dt

(2.63)

Adding inequalities (2.57) and (2.63) and solving the resulting differential inequality, we obtain   k ∥ϕ∥H σ + ∥ψ∥H σ . ∥ϕ(0)∥H σ + ∥ψ(0)∥H σ e∥u0 ∥H s t .

(2.64)

Since ϕ(0) = ψ(0) = 0, we conclude that ϕ = ψ = 0 and therefore, the solution U to system (2.3) is unique in C([0, T ]; H σ ). Since σ < s − 2, we have C([0, T ]; H s−1 ) ⊂ C([0, T ]; H σ ) and hence, we deduce uniqueness in C([0, T ]; H s−1 ).

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

177

2.3. Return to the k-abc-equation Differentiating Eq. (2.3a) with respect to x and rearranging the resulting expression turns system (2.3) into the following system: b ∂t ux = a(k − 2)uk−3 w3 (ux − w) + 3auk−2 w2 wx + uk−1 (cw − kux ) w − uk wx + k+1 uk+1    − D−2 ∂x k (k + 2) − 8a − b − c (k + 1) uk−2 w3 − 3a (k − 2) uk−3 w3 wx   b − D−2 k+1 uk+1 + cuk−1 w2 − a (k − 2) uk−3 w4 , b ∂t w = 3auk−2 w2 wx + (c − k) uk−1 w2 − uk wx + k+1 uk+1    − D−2 ∂x k(k + 2) − 8a − b − c (k + 1) uk−2 w3 − 3a(k − 2)uk−3 w3 wx   b uk+1 + cuk−1 w2 − a (k − 2) uk−3 w4 , − D−2 k+1

ux (x, 0) = w(x, 0) = ∂x u0 (x).

(2.65a)

(2.65b) (2.65c)

Subtracting Eq. (2.65b) from Eq. (2.65a), letting y = ux − w, and integrating, we find  t k−3 − u w[a(k−2)w2 −ku2 ]dt′ y(x, t) = y(x, 0) e 0 . Noting that the initial conditions (2.65c) imply y(x, 0) = 0, we deduce that y = 0 ⇔ w = ux . Hence, if U = (u, w) solves system (2.3) then w = ux where u solves the k-abc-equation (2.1). On the other hand, if u solves Eq. (2.1) then by construction (u, ux ) solves system (2.3). Therefore, a solution of system (2.3) always gives rise to a solution of Eq. (2.1) and vice-versa. Since w ∈ C([0, T ]; H s−1 ), we deduce that u ∈ C([0, T ]; H s ). Moreover, setting w = ux in inequality (2.18) implies the size estimate and lifespan for u as stated in Theorem 1. Finally, the uniqueness of solution of the k-abc-equation (2.1) in C([0, T ]; H s ) follows immediately from the uniqueness of solution of system (2.3) in C([0, T ]; H s−1 × H s−1 ). 2.4. Continuity of the data-to-solution map We will now complete the proof of Theorem 1 on the Hadamard well-posedness of the k-abc-equation on the circle by establishing continuity of the data-to-solution map u0 ∈ H s → u ∈ C([0, T ]; H s ). Let u and un denote the solutions to the k-abc-equation with initial data u0 and u0,n respectively, i.e. ∂t u = auk−2 (∂x u)3 − uk ∂x u − Φ(u), u(x, 0) = u0 (x),

3 k ∂t un = auk−2 n (∂x un ) − un ∂x un − Φ(un ), un (x, 0) = u0,n (x),

(2.66)

with the nonlocal quantity Φ defined by Eq. (2.2). We will show that if u0,n −→ u0 in H s then un −→ u in C([0, T ]; H s ). As we have already seen in the proof of existence, the local terms of the k-abc-equation need special care. In particular, recall that even after mollification the term uk ∂x u required the use of the Kato–Ponce Lemma 1, while the term uk−2 (∂x u)3 caused the transition from the k-abc-equation (2.1) to system (2.3). The same techniques are necessary for proving continuity of the data-to-solution map. It turns out, however, that the mollification of the local terms involving derivatives can now be avoided. Instead, the distance between un and u can be estimated indirectly, with the help of the functions uεn and uε which satisfy the following initial value problems with mollified initial conditions: ∂t uε = a(uε )k−2 (∂x uε )3 − (uε )k ∂x uε − Φ(uε ), . uε (x, 0) = Jε u0 (x) = jε ∗ u0 (x),

∂t uεn = a(uεn )k−2 (∂x uεn )3 − (uεn )k ∂x uεn − Φ(uεn ), (2.67) . uεn (x, 0) = Jε u0,n (x) = jε ∗ u0,n (x).

Indeed, by the triangle inequality ∥un − u∥H s−1 6 ∥un − uεn ∥H s−1 + ∥uεn − uε ∥H s−1 + ∥u − uε ∥H s−1 ,

(2.68)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

178

hence, we can estimate the difference un − u indirectly by estimating the three differences on the right-hand side of inequality (2.68). The estimation of those three terms requires transition from equations to systems, as in the case of existence. In particular, differentiating the first of Eqs. (2.66) once with respect to x and letting w = ∂x u, we obtain the system ∂t u = auk−2 w3 − uk w − F1 (u, w), k−2

∂t w = 3au

(2.69a)

k

2

k−1

w ∂x w − u ∂x w + (c − k) u

2

w +

b k+1

u

k+1

− F2 (u, w),

(2.69b)

u(x, 0) = u0 (x), w(x, 0) = ∂x u0 (x),

(2.69c)

where the nonlocal quantities F1 and F2 are defined by Eq. (2.4). Moreover, differentiating the second of Eqs. (2.66) once with respect to x and letting wn = ∂x un , we arrive at the system 3 k ∂t un = auk−2 n wn − un wn − F1 (un , wn )

(2.70a)

2 k k−1 2 ∂t wn = 3auk−2 n wn ∂x wn − un ∂x wn + (c − k) un wn +

b k+1

uk+1 − F2 (un , wn ) n

(2.70b)

un (x, 0) = u0,n (x), wn (x, 0) = ∂x u0,n (x).

(2.70c)

Similarly, differentiating the first of Eqs. (2.67) and setting wε = ∂x uε gives rise to the system ∂t uε = a (uε )k−2 (wε )3 − (uε )k wε − F1 (uε , wε ), ε

ε k−2

∂t w = 3a(u )

ε 2

ε

ε k

(2.71a)

ε

ε k−1

(w ) ∂x w − (u ) ∂x w + (c − k) (u )

ε

ε 2

(w ) +

b k+1

ε k+1

(u )

ε

ε

− F2 (u , w ), (2.71b)

ε

u (x, 0) = Jε u0 (x), w (x, 0) = ∂x Jε u0 (x),

(2.71c)

while differentiating the second of Eqs. (2.67) and setting wnε = ∂x uεn yields the system ∂t uεn = a (uεn )k−2 (wnε )3 − (uεn )k wnε − F (uεn , wnε ), ∂t wnε

=

3a(uεn )k−2 (wnε )2 ∂x wnε −(uεn )k ∂x wnε

uεn (x, 0) = Jε u0,n (x),

+ (c −

(2.72a) k) (uεn )k−1 (wnε )2

+

b k+1

(uεn )k+1 −F2 (uεn , wnε ),

wnε (x, 0) = ∂x Jε u0,n (x).

(2.72b) (2.72c)

We therefore supplement inequality (2.68) with the analogous inequality for w, namely ∥wn − w∥H s−1 6 ∥wn − wnε ∥H s−1 + ∥wnε − wε ∥H s−1 + ∥w − wε ∥H s−1 ,

(2.73)

and proceed to the estimation of the relevant differences on the right-hand sides of these two inequalities. Estimation of uεn − uε and wnε − wε . For ϕ = uεn − uε and ψ = wnε − wε , we subtract system (2.71) from system (2.72) to obtain the system     ∂t ϕ = a (uεn )k−2 g2 − (uεn )k ψ + af3 (wε )3 − f1 wε ϕ − F1 (uεn , wnε ) + F1 (uε , wε ), (2.74a)     ∂t ψ = a (uεn )k−2 g2 − (uεn )k ∂x ψ + a (uεn )k−2 ∂x g2 + (c − k) (uεn )k−1 g1 ψ   b f0 ϕ − F2 (uεn , wnε ) + F2 (uε , wε ), (2.74b) + af3 ∂x (wε )3 + (c − k) f2 (wε )2 − f1 ∂x wε + k+1   ϕ(x, 0) = Jε u0,n (x) − Jε u0 (x), ψ(x, 0) = ∂x Jε u0,n (x) − Jε u0 (x) , (2.74c) where the quantities fj and gj are defined as in (2.51) but with u, v, w, z replaced by uεn , uε , wnε , wε respectively. Note that the size estimate (2.25) implies  k−j ∥fj ∥H s−1 . ∥uε0,n ∥H s + ∥uε0 ∥H s ,

 j ∥gj ∥H s−1 . ∥uε0,n ∥H s + ∥uε0 ∥H s ,

j ∈ N.

(2.75)

The estimation of ψ is more involved technically than the one of ϕ. Hence, we derive in every detail the estimate for ψ and then deduce the corresponding estimate for ϕ. Applying the operator Ds−1 on the left of Eq. (2.74b), multiplying by Ds−1 ψ on the right and integrating over T yields

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

1 d 2 ∥ψ∥H s−1 = 2 dt

179



  Ds−1 a (uεn )k−2 g2 − (uεn )k ∂x ψ · Ds−1 ψ dx T    + Ds−1 a (uεn )k−2 ∂x g2 + (c − k) (uεn )k−1 g1 ψ · Ds−1 ψ dx T   b f0 ϕ · Ds−1 ψ dx + Ds−1 af3 ∂x (wε )3 + (c − k) f2 (wε )2 − f1 ∂x wε + k+1 T   − Ds−1 F2 (uεn , wnε ) − F2 (uε , wε ) · Ds−1 ψ dx.

(2.76a) (2.76b) (2.76c) (2.76d)

T

We begin with term (2.76a). Commuting Ds−1 and a (uεn )k−2 g2 − (uεn )k and then applying the Cauchy–Schwarz inequality, we find    s−1     1    ε k−2 ε k   a (uεn )k−2 g2 − (uεn )k ∂x (Ds−1 ψ)2 dx. |(2.76a)| 6 D , a (un ) g2 − (un ) ∂x ψ L2 ∥ψ∥H s−1 +  2 T Using the Kato–Ponce Lemma 1 for the first term, integrating by parts in the second term, and employing the algebra property, the Sobolev embedding and estimates (2.25) (2.75), we then obtain   2 ε k 2 s−1 + ∥u ∥ s−1 ∥ψ∥ s−1 . ∥ψ∥ s−1 . |(2.76a)| . ∥uεn ∥k−2 (2.77) n H H H H s−1 ∥g2 ∥H Regarding term (2.76b), the Cauchy–Schwarz inequality implies   k−2 k−1 2 |(2.76b)| . ∥uεn ∥H s−1 ∥∂x g2 ∥H s−1 + ∥uεn ∥H s−1 ∥g1 ∥H s−1 ∥ψ∥H s−1 .

(2.78)

The only term on the right-hand side that requires special attention is ∥∂x g2 ∥H s−1 . We have ∥∂x g2 ∥H s−1 = ∥2wnε ∂x (wnε ) + ∂x (wnε )wε + wnε ∂x (wε ) + 2wε ∂x (wε )∥H s−1 . ∥wnε ∥H s−1 ∥wnε ∥H s + ∥wnε ∥H s ∥wε ∥H s−1 + ∥wnε ∥H s−1 ∥wε ∥H s + ∥wε ∥H s−1 ∥wε ∥H s . (2.79) In order to estimate the terms wε and wnε , we note that systems (2.71) and (2.72) are equivalent to system (2.3) after setting the initial data (u0 , ∂x u0 ) of this system equal to those specified by Eqs. (2.71c) and (2.72c) respectively. Hence, the solutions of systems (2.71) and (2.72) both admit the well-posedness size estimate (2.25), i.e. ∥uε ∥H s−1 + ∥wε ∥H s−1 . ∥Jε u0 ∥H s−1 + ∥∂x Jε u0 ∥H s−1 , ∥uεn ∥H s−1

+

∥wnε ∥H s−1

. ∥Jε u0,n ∥H s−1 + ∥∂x Jε u0,n ∥H s−1 ,

s > 25 ,

(2.80a)

5 2.

(2.80b)

s>

Estimates (2.80) will be used in two different ways for the H s−1 and H s norms on the right-hand side of inequality (2.79). Concerning the H s−1 norms, in combination with the first property in (2.7) estimates (2.80) yield ∥wε ∥H s−1 . ∥u0 ∥H s ,

∥wnε ∥H s−1 . ∥Ju0,n ∥H s ,

s > 52 .

(2.81)

Concerning the H s norms of (2.79), we shift s − 1 to s so that estimates (2.80) read ∥uε ∥H s + ∥wε ∥H s . ∥Jε u0 ∥H s + ∥∂x Jε u0 ∥H s ,

s > 32 ,

∥uεn ∥H s + ∥wnε ∥H s . ∥Jε u0,n ∥H s + ∥∂x Jε u0,n ∥H s ,

s > 32 .

Then, using inequality (2.5) for the Friedrichs mollifier Jε we arrive at the estimates     ∥wnε ∥H s . 1 + 1ε ∥u0,n ∥H s , s > 32 . ∥wε ∥H s . 1 + 1ε ∥u0 ∥H s ,

(2.82)

Inserting (2.81) and (2.82) in (2.79) then yields the estimate   2 ∥∂x g2 ∥H s−1 . 1 + 1ε ∥u0,n ∥H s + ∥u0 ∥H s ,

s > 25 .

(2.83)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

180

Therefore, (2.78) we obtain |(2.76b)| .

1 ε

2

∥ψ∥H s−1 ,

s > 25 .

(2.84)

Similarly, we find |(2.76c)| .

1 ε

∥ϕ∥H s−1 ∥ψ∥H s−1 ,

s > 25 .

(2.85)

Concerning the nonlocal term (2.76d), expressing the difference F2 (uεn , wnε ) − F2 (uε , wε ) analogously to (2.55) with u = uεn , w = wnε , v = uε , z = wε , and performing straightforward computations that involve use of the Cauchy–Schwarz inequality, the algebra property and estimates (2.25) and (2.75), we obtain 2

|(2.76d)| . ∥ψ∥H s−1 + ∥ψ∥H s−1 ∥ϕ∥H s−1 .

(2.86)

Combining estimates (2.77), (2.84), (2.85) and (2.86) with Eq. (2.76) then yields the inequality   d ∥ψ∥H s−1 . 1ε ∥ψ∥H s−1 + ∥ϕ∥H s−1 . dt

(2.87)

The corresponding energy estimate for ϕ is actually easier to derive and reads d ∥ϕ∥H s−1 . ∥ϕ∥H s−1 + ∥ψ∥H s−1 . dt

(2.88)

Adding inequalities (2.87) and (2.88), we have  cs   d ∥ϕ∥H s−1 + ∥ψ∥H s−1 6 ∥ϕ∥H s−1 + ∥ψ∥H s−1 . dt ε Solving this differential inequality for all 0 6 t 6 T with T defined by Eq. (2.22), and restoring the differences uεn − uε and wnε − wε in place of ϕ and ψ, we obtain  cs T  ∥uεn − uε ∥H s−1 + ∥wnε − wε ∥H s−1 6 ∥uεn (0) − uε (0)∥H s−1 + ∥wnε (0) − wε (0)∥H s−1 e ε .

(2.89)

Remark 2.1. Eventually, we will consider estimate (2.89) in the limit ε → 0, in which case the exponential on the right-hand side grows to infinity. However, exploiting the fact that uεn (0) −→ uε (0) and wnε (0) −→ wε (0) as n → ∞, for any ε > 0 we can choose N = N (ε) such that for all n > N the right-hand side of (2.89) is bounded. Estimating the remaining differences. The estimation of the differences (u − uε , w − wε ) and (un − uεn , wn − wnε ) is more challenging, due to the fact that the convergence exploited in Remark 2.1 in order to keep the right-hand side of estimate (2.89) bounded as ε → 0 is not available now. Note that these two differences are very similar to each other from the point of view of estimates, since both of them involve a solution corresponding to mollified data and a solution corresponding to non-mollified data. In fact, their treatment turns out to be identical, thus we provide the details only for (u − uε , w − wε ). Subtracting system (2.71) from system (2.69) and letting ϕ = u − uε and ψ = w − wε , we obtain     ∂t ϕ = auk−2 g2 − uk ψ + af3 (wε )3 − f1 wε ϕ − F1 (u, w) + F1 (uε , wε ), (2.90a)  k−2    ∂t ψ = au g2 − uk ∂x ψ + auk−2 ∂x g2 + (c − k) uk−1 g1 ψ   b + af3 ∂x (wε )3 + (c − k) f2 (wε )2 − f1 ∂x wε + k+1 (2.90b) f0 ϕ − F2 (u, w) + F2 (uε , wε ),   ϕ(x, 0) = u0 (x) − Jε u0 (x), ψ(x, 0) = ∂x u0 (x) − Jε u0 (x) , (2.90c) where the quantities fj and gj are defined as in (2.51) but with v, z replaced by uε , wε respectively.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

181

Following the approach leading to estimates (2.87) and (2.88), it is straightforward to obtain the following energy estimate for ϕ: d ∥ϕ∥H s−1 . ∥ϕ∥H s−1 + ∥ψ∥H s−1 . dt

(2.91)

For ψ, however, as explained below Remark 2.1 we need an inequality sharper than (2.87). Applying the operator Ds−1 on the left of Eq. (2.90b), multiplying by Ds−1 ψ on the right, integrating over T, and noting that ∂x g2 = (2w + wε ) ∂x ψ + 3g1 ∂x wε , we find      1 d 2 ∥ψ∥H s−1 = Ds−1 auk−2 2g2 − 3w2 − uk ∂x ψ · Ds−1 ψ dx (2.92a) 2 dt T  + Ds−1 3auk−2 g1 ψ ∂x wε · Ds−1 ψ dx (2.92b) T    + Ds−1 3af3 (wε )2 − f1 ϕ ∂x wε · Ds−1 ψ dx (2.92c) T + Ds−1 (c − k) uk−1 g1 ψ · Ds−1 ψ dx (2.92d) T    b f0 ϕ · Ds−1 ψ dx (2.92e) + Ds−1 (c − k) f2 (wε )2 + k+1 T   − Ds−1 F2 (u, w) − F2 (uε , wε ) · Ds−1 ψ dx. (2.92f) T

Employing the Cauchy–Schwarz inequality, the Kato–Ponce Lemma 1, the algebra property, the Sobolev embedding, and estimates (2.25) and (2.75), it is straightforward to establish the estimates |(2.92a)| . ∥ψ∥2H s−1 ,

|(2.92d)| . ∥ψ∥2H s−1 ,

|(2.92e)| . ∥ϕ∥H s−1 ∥ψ∥H s−1 ,

|(2.92f)| . ∥ψ∥2H s−1 . (2.93)

Furthermore, the terms (2.92b) and (2.92c) can be handled in the same way, thus we only give the details for the former one. Commuting Ds−1 with 3auk−2 g1 ψ and using the triangle inequality, we have       |(2.92b)| 6  Ds−1 , 3auk−2 g1 ψ ∂x wε L2 ∥ψ∥H s−1 + 3auk−2 g1 ψ Ds−1 ∂x wε · Ds−1 ψ dx. T

Using the Kato–Ponce Lemma 1 and estimates (2.25) and (2.75), we find  s−1 k−2   k  D ,u g1 ψ ∂x wε L2 . ∥u0 ∥H s ∥ψ∥H s−1 . ∥ψ∥H s−1 .

(2.94)

Also, by the Sobolev embedding, inequality (2.5) and estimates (2.25), (2.75) and (2.82) it follows that   k−2  ε u s−2 ∥ψ∥H s−2 ∥w ∥H s ∥ψ∥ s−1 g1 ψ Ds−1 ∂x wε · Ds−1 ψ  dx . ∥u∥k−2 H H s−2 ∥g1 ∥H T   k−2 . ∥u0 ∥H s (∥uε0 ∥H s + ∥u0 ∥H s ) ∥ψ∥H s−2 1 + 1ε ∥u0 ∥H s ∥ψ∥H s−1 .

1 ε

∥ψ∥H s−2 ∥ψ∥H s−1 .

(2.95)

Overall, we obtain the estimate 2

|(2.92b)| . ∥ψ∥H s−1 +

1 ε

∥ψ∥H s−2 ∥ψ∥H s−1 .

(2.96)

Similarly, we find |(2.92c)| . ∥ϕ∥H s−1 ∥ψ∥H s−1 +

1 ε

∥ϕ∥H s−2 ∥ψ∥H s−1 .

(2.97)

Combining estimates (2.93), (2.96) and (2.97) with Eq. (2.92) yields the inequality     d ∥ψ∥H s−1 . ∥ϕ∥H s−1 + ∥ψ∥H s−1 + 1ε ∥ϕ∥H s−2 + ∥ψ∥H s−2 . dt

(2.98)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

182

Adding (2.91) and (2.98), we obtain      d ∥ϕ∥H s−1 + ∥ψ∥H s−1 . ∥ϕ∥H s−1 + ∥ψ∥H s−1 + 1ε ∥ϕ∥H s−2 + ∥ψ∥H s−2 . dt

(2.99)

We handle the H s−2 norms involved in the above inequality via the following interpolation lemma, whose proof can be found in [30]. r2 −r

r−r1

r2 −r1 r2 −r1 ∥f ∥H . Lemma 7 (Interpolation). Suppose r1 < r < r2 and f ∈ H r . Then, ∥f ∥H r 6 ∥f ∥H r1 r2

1 2

< σ < s − 2, using Lemma 7 with r = s − 2, r1 = σ and r2 = s − 1, we have   1   s−2−σ   s−2−σ  ∥ϕ∥H s−2 + ∥ψ∥H s−2 . ∥ϕ∥H σ + ∥ψ∥H σ s−1−σ ∥u∥H s−1 + ∥uε ∥H s−1 s−1−σ + ∥w∥H s−1 + ∥wε ∥H s−1 s−1−σ For



. ∥ϕ∥H σ + ∥ψ∥H σ

1  s−1−σ

s−2−σ s−1−σ ∥u0 ∥H . s

(2.100)

Moreover, using the approach of energy estimates employed in the proof of uniqueness we can obtain inequality (2.64) for the current definitions of ϕ and ψ. Recalling that the definition (2.22) implies k ∥u0 ∥H s t 6 2k+11 kcs for all 0 6 t 6 T , we further deduce the inequality ∥ϕ∥H σ + ∥ψ∥H σ . ∥ϕ(0)∥H σ + ∥ψ(0)∥H σ ,

1 2

< σ < s − 2.

(2.101)

In addition, using estimate (2.29) we have   ∥ϕ(0)∥H σ = ∥(I − Jε )u0 ∥H σ 6 ∥I − Jε ∥L(H s−1 ;H σ ) ∥u0 ∥H s−1 6 o εs−1−σ ∥u0 ∥H s ,   ∥ψ(0)∥H σ = ∥(I − Jε )u0 ∥H σ+1 6 ∥I − Jε ∥L(H s ;H σ+1 ) ∥u0 ∥H s 6 o εs−1−σ ∥u0 ∥H s . Hence, inequality (2.101) implies   ∥ϕ(t)∥H σ + ∥ψ(t)∥H σ . o εs−1−σ ∥u0 ∥H s ,

1 2

< σ < s − 2,

and inequality (2.100) in turn yields ∥ϕ∥H s−2 + ∥ψ∥H s−2 . o(ε) ∥u0 ∥H s . Combining inequalities (2.99) and (2.102) and setting δ =

o(ε) ε

(2.102)

∥u0 ∥H s so that limε→0 δ = 0, we obtain

   d ∥ϕ∥H s−1 + ∥ψ∥H s−1 . ∥ϕ∥H s−1 + ∥ψ∥H s−1 + δ. dt Solving the above differential inequality and substituting ϕ = u − uε and ψ = w − wε , we conclude that   ∥u − uε ∥H s−1 + ∥w − wε ∥H s−1 . ∥u(0) − uε (0)∥H s−1 + ∥w(0) − wε (0)∥H s−1 et − δ. (2.103) Analyzing the system for the differences uεn − un and wnε − wn in the same way as system (2.90) yields   ∥uεn − un ∥H s−1 + ∥wnε − wn ∥H s−1 . ∥uεn (0) − un (0)∥H s−1 + ∥wnε (0) − wn (0)∥H s−1 et − δ. (2.104) Furthermore, recalling the prescribed initial conditions of systems (2.69)–(2.72) we observe that the norms on the right-hand side of inequalities (2.103) and (2.104) vanish in the limit ε → 0. In particular, given η > 0 we may choose ε = ε(η) sufficiently small so that ∥uε − u∥H s−1 + ∥wε − w∥H s−1 < η3 ,

∥uεn − un ∥H s−1 + ∥wnε − wn ∥H s−1 < η3 .

For this choice of ε, we choose N = N (ε) sufficiently large so that for all n > N we have ∥uε (0) − uεn (0)∥H s−1 + ∥wε (0) − wnε (0)∥H s−1 <

η 3

e−

cs T ε

.

(2.105)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

183

Then, estimate (2.89) implies ∥uε − uεn ∥H s−1 + ∥wε − wnε ∥H s−1 <

η 3

∀n > N.

(2.106)

Combining inequalities (2.68), (2.105) and (2.106), we conclude that ∥un − u∥H s−1 + ∥wn − w∥H s−1 < η

∀n > N,

which establishes continuity of the data-to-solution map of system (2.3) and hence of the k-abc-equation. Hadamard well-posedness on the line. The non-periodic Cauchy problem for the k-abc-equation with initial data u0 ∈ H s (R), s > 25 , can be treated similarly to the Cauchy problem on the circle. One first . establishes well-posedness of the mollified system (2.6), with the Friedrichs mollified Jε f = jε ∗ f now constructed by fixing a Schwartz function j ∈ S(R) such that 0 6  j(ξ) 6 1 for all ξ ∈ R and  j(ξ) = 1 for . |ξ| 6 1, and then setting jε (x) = j(x/ε)/ε. Then, one deduces well-posedness for system (2.3) and hence for the k-abc-equation (2.1). This procedure requires only two minor modifications. First, in order to show the convergence of the family {(uε , wε )}ε∈(0,1] of solutions of the mollified system to (u, w) in the space C(I; H s−1−σ (R)), σ ∈ (0, 1), one now needs to introduce a function ϕ ∈ S(R) such that ϕ(x) > 0 for all x ∈ R and consider the family {(ϕuε , ϕwε )}ε∈(0,1] as opposed to just {(uε , wε )}ε∈(0,1] . The presence of the Schwartz function ϕ ensures that (ϕuε , ϕwε ) has compact support and hence, by Rellich’s theorem the set {(ϕuε (t), ϕwε (t))}ε∈(0,1] ⊂ H s−1 (R) is precompact in H s−1−σ (R). Recall that precompactness is needed for Ascoli’s theorem, which is used to deduce the desired convergence. Then, it is straightforward to show that (ϕuε , ϕwε ) → (ϕu, ϕw) in C(I; C 1 (R)), from which one infers that (uε , wε ) → (u, w) and ∂x (uε , wε ) → ∂x (u, w) pointwise as ε → 0. Second, similarly to the periodic case it is required to prove that (u, w) is actually a solution to the k-abc-equation. The first step in this direction is now to show that ϕ ∂t u −−−−→ aϕ uk−2 w3 − ϕ uk w − ϕ F1 (u, w), ε→0

ϕ ∂t w −−−−→ 3aϕ uk−2 w2 wx + (c − k) ϕ uk−1 w2 − ϕ uk wx + ε→0

b k+1

ϕ uk+1 − ϕ F2 (u, w)

in C(I; H s−σ−2 (R)) by invoking once again Ascoli’s theorem. 3. Non-uniform dependence on the circle It was shown in Section 2 that the k-abc equation formulated either on the line or on the circle with data u0 ∈ H s , s > 52 , is well-posed in the sense of Hadamard (Theorem 1). In particular, it was proved that the data-to-solution map u0 ∈ H s → u(t) ∈ C([0, T ]; H s ) is continuous. We now show that this continuity is non-uniform (Theorem 2). The proof of this result in the periodic case is provided in the present section, while the proof in the case of the line, which is more technical, is given in Section 4. The idea for showing non-uniform continuity is to construct two sequences of solutions, {uω1 ,n }n∈N and {uω2 ,n }n∈N , with the following properties.   (1) Boundedness: uω1 ,n , uω2 ,n ∈ B (0, 1) ⊂ H s for all t ∈ [0, T ], i.e. uωj ,n C([0,T ];H s ) . 1, j = 1, 2. (2) Coming together at t = 0: ∥uω1 ,n (0) − uω2 ,n (0)∥H s −→ 0 as n → ∞. (3) Staying apart for some t∗ > 0: there exists an N such that for all n > N and some fixed t∗ > 0, ∥uω1 ,n (t∗ ) − uω2 ,n (t∗ )∥H s > C(t∗ ) > 0, for some constant C > 0 depending only on t∗ .

184

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We begin our construction by considering the sequences {uω1 ,n (x, 0)}n∈N and {uω2 ,n (x, 0)}n∈N of initial data defined by 1

uω1 ,n (x, 0) = ω1 n− k + n−s cos(nx),

1

uω2 ,n (x, 0) = ω2 n− k + n−s cos(nx),

(3.1)

where the real constants ω1 and ω2 are yet to be determined. It is straightforward to show that the data    (3.1) satisfy Properties (1) and (2). Indeed, noting that cos(nx)(ξ) = π δ(ξ + n) + δ(ξ − n) we have the estimate  √   2  21 s   = 2π 1 + n2 2 . ns . ∥cos(nx)∥H s = (3.2) (1 + ξ 2 )s cos(nx)(ξ) ξ∈Z

Hence, starting from the size estimate (2.25) we obtain 1

1

∥uω,n (t)∥H s . ∥uω,n (0)∥H s = ∥uω,n ∥H s 6 |ω| n− k + n−s ∥cos(nx)∥H s . |ω| n− k + 1 . 1, so Property (1) is satisfied. Furthermore, we have   1  ∥uω1 ,n (·, 0) − uω2 ,n (·, 0)∥H s = (ω1 − ω2 ) n− k 

Hs

1

= 2π |ω1 − ω2 | n− k −−−−−→ 0, n→∞

thus Property (2) is also satisfied. The actual motivation, however, behind the choice (3.1) of initial data emanates from the fact that, apart from associating uω1 ,n (x, 0) and uω2 ,n (x, 0) to the exact solutions uω1 ,n (x, t) and uω2 ,n (x, t) of the k-abc-equation (2.1), we can also associate them to the following family of approximate solutions of Eq. (2.1): 1

uω,n (x, t) = ωn− k + n−s cos(nx − ωt),

ω ∈ R.

(3.3)

It is trivial to verify that uω,n (x, t) satisfy the initial conditions (3.1). Note that uω,n (x, t) consist of a low 1 and a high frequency part, ωn− k and n−s cos(nx − ωt), respectively. Moreover, note that the low frequency part is a constant and hence it is actually an exact solution of the k-abc-equation. Since no explicit formulae are available for the exact solutions uω1 ,n (x, t) and uω2 ,n (x, t) corresponding to the initial data (3.1), Property (3) will be verified instead at the level of the approximate solutions uω1 ,n (x, t) and uω2 ,n (x, t) which correspond to the same initial data. In order to carry out the above plan, we first need to choose ω such that the error associated with the approximate solutions (3.3) vanishes in the limit n → ∞ in the solution space C([0, T ]; H s ). Denoting this error by E ω,n , we deduce from the k-abc-equation that it is given by   E ω,n = ∂t uω,n − a(uω,n )k−2 (∂x uω,n )3 − (uω,n )k ∂x uω,n − Φ(uω,n ) . (3.4) Therefore, we need to identify those values of ω for which ∥E ω,n ∥C([0,T ];H s ) −−−−−→ 0. Substituting (3.3) n→∞

into (3.4), we have   E ω,n = ω − ω k n−s sin(nx − ωt) k−2 j   k − 2  k−2−j 3(1−s) 3 1 +a ωn− k n−s cos(nx − ωt) n sin (nx − ωt) j j=0 k−1 

−

1 2 j=0

k j



1

ωn− k

j  k−j−1 1−2s   n−s cos(nx − ωt) n sin 2(nx − ωt)

+ Φ(uω,n ). Similarly to estimate (3.2), and using also the identity sin3 θ = 14 [3 sin θ − sin(3θ)], we have √   3  s sin (nx) s . ns . ∥sin(nx)∥H s = 2π 1 + n2 2 . ns , H

(3.5a) (3.5b)

(3.5c) (3.5d)

(3.6)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

Using the first of these estimates, we find      s ∥(3.5a)∥H s = ω − ω k  n−s · 2π 2 1 + n2 −−−−−→ 2π 2 ω − ω k .

185

(3.7)

n→∞

Next, using estimates (3.2) and (3.6) and recalling that s > 52 , we obtain ∥(3.5b)∥H s .

k−2 

   j − j ω  n k · n−s(k−2−j) ∥cos(nx − ωt)∥k−2−j · n3(1−s) sin3 (nx − ωt)H s Hs

j=0

.

k−2 

j

n− k · n−s(k−2−j) · ns(k−2−j) · n3(1−s) · ns = n3−2s ·

j=0

and also ∥(3.5c)∥H s . n1−s · ∥Φ(u

ω,n

n−1 −1 n

−1 k

−1

k−1 k −1 −1 n k −1

n

−

−−−−−→ 0, n→∞

−−−−−→ 0. Finally, it will be shown later (see estimate (3.21)) that n→∞

)∥H s −−−−−→ 0.

n→∞   Collecting all of the above estimates, we deduce that ∥E ω,n (t)∥H s −−−−−→ 2π 2 ω −ω k . Therefore, in order n→∞

for ∥E ω,n ∥C([0,T ];H s ) to vanish as n → ∞ we must have ω k = ω, i.e. we must choose ω=0

or ω = e

2iπm k−1

m ∈ N.

,

(3.8)

Next, for ω given by Eq. (3.8) we use the approximate solutions (3.3) in order to obtain a lower bound for the difference ∥uω1 ,n (t) − uω2 ,n (t)∥H s by means of the triangle inequality ∥uω1 ,n (t) − uω2 ,n (t)∥H s > ∥uω1 ,n (t) − uω2 ,n (t)∥H s − ∥uω1 ,n (t) − uω1 ,n (t)∥H s − ∥uω2 ,n (t) − uω2 ,n (t)∥H s . (3.9)

    2 2 Using the identity cos (nx − ω1 t) − cos (nx − ω2 t) = 2 sin nx − ω1 +ω t sin ω1 −ω t , the first term on the 2 2 right-hand side of inequality (3.9) can be bounded from below as follows:       1   ω1 −ω2 2 ∥uω1 ,n (t) − uω2 ,n (t)∥H s > 2n−s sin nx − ω1 +ω t sin t  s − n− k H s 2 2 H    √   ω −ω  s √ 1   −s √  2 2 −k ω1 −ω2 = 2sin 2π 1 + n 2πn − − − − − → 2 2π sin 1 2 2 t . t · − n 2 n→∞

Hence, there is an N such that for all n > N and some t∗ > 0 we have ∥uω1 ,n (t∗ ) − uω2 ,n (t∗ )∥H s & 1.

(3.10)

Thanks to estimate (3.10), Property (3) for uω1 ,n and uω2 ,n can now be established by showing that the remaining terms on the right-hand side of inequality (3.9) vanish in the limit n → ∞, i.e. ∥uω,n (t) − uω,n (t)∥H s −−−−−→ 0, n→∞

ω ∈ {ω1 , ω2 }.

Recall that uω1 ,n and uω2 ,n are exact solutions of the initial value problem ∂t uω,n = a (uω,n )k−2 (∂x uω,n )3 − (uω,n )k ∂x uω,n − Φ(uω,n ), 1

uω,n (x, 0) = uω,n (x, 0) = ωn− k + n−s cos(nx),

ω ∈ {ω1 , ω2 },

(3.11)

while the approximate solutions uω1 ,n and uω2 ,n satisfy the initial value problem ∂t uω,n = a (uω,n )k−2 (∂x uω,n )3 − (uω,n )k ∂x uω,n − Φ(uω,n ) + E ω,n , 1

uω,n (x, 0) = ωn− k + n−s cos(nx),

ω ∈ {ω1 , ω2 }.

(3.12)

Subtracting (3.11) from (3.12) and letting v = uω,n − uω,n then yields the initial value problem   ∂t v = E ω,n + a (uω,n )k−2 g2 ∂x v + (∂x uω,n )3 f3 v − v(x, 0) = 0,

ω ∈ {ω1 , ω2 },

1 ∂x (f0 v) + Φ(uω,n ) − Φ(uω,n ), k+1

(3.13)

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where the quantities fj and gj are defined as in (2.51) but with u, v, w, z replaced by uω,n , uω,n , ∂x uω,n , ∂x uω,n respectively. Our goal is to estimate ∥v∥H s . By the well-posedness size estimate (2.25) for uω,n and estimate (3.2) for ω,n u , we have ∥uω,n (t)∥H s . 1,

∥uω,n (t)∥H s . 1.

(3.14)

Thus, using also the algebra property we deduce 1 2

∥fj ∥H σ . 1,

< σ < s,

∥gj ∥H σ . 1,

3 2

< σ < s − 1.

(3.15)

The idea is therefore to estimate ∥v∥H σ for 32 < σ < s − 1 (this restriction is required due to (3.15)) and then use Sobolev interpolation to extract an estimate for ∥v∥H s . Note that, as σ < s, no mollification of the initial value problem (3.13) is required. Applying Dσ on the left of Eq. (3.13), multiplying on the right by Dσ v and integrating with respect to x over T, we obtain  1 d 2 ∥v∥H σ = Dσ E ω,n · Dσ v dx (3.16a) 2 dt T  + a Dσ (uω,n )k−2 g2 ∂x v · Dσ v dx (3.16b) T + a Dσ (∂x uω,n )3 f3 v · Dσ v dx (3.16c) T  1 − k+1 Dσ ∂x (f0 v) · Dσ v dx (3.16d) T  + Dσ [Φ(uω,n ) − Φ(uω,n )] · Dσ v dx. (3.16e) T

The estimation of term (3.16a), which involves the error E ω,n , relies on the following lemma. Lemma 8 (Estimation of the Error). Let s > Eq. (3.4) admits the estimate

5 2

and

3 2

< σ < s − 1. Then, the error E ω,n defined by

∥E ω,n ∥H σ . nσ−2s+1 ,

n → ∞.

Employing Lemma 8, which is proved at the end of the present section, we find ∥(3.16a)∥H σ 6 ∥E ω,n ∥H σ ∥v∥H σ . nσ−2s+1 ∥v∥H σ ,

n → ∞.

(3.17)

The treatment of term (3.16b) requires the Cauchy–Schwarz inequality, the Kato–Ponce Lemma 1, the Sobolev embedding for σ > 32 , the algebra property in H σ and estimates (3.14) and (3.15). We thereby obtain 2

∥(3.16b)∥H σ . ∥v∥H σ .

(3.18)

Similar arguments but without the need for the Kato–Ponce Lemma yield the estimates 2

∥(3.16c)∥H σ . ∥v∥H σ ,

2

∥(3.16d)∥H σ . ∥v∥H σ ,

(3.19)

where the latter estimate is obtained by employing the Calderon–Coifman–Meyer estimate of Lemma 6 with ρ = s after expressing this term in the form    σ  (3.16d) = D ∂x , f0 v · Dσ v dx + f0 Dσ ∂x v · Dσ v dx. T

T

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

187

Finally, regarding term (3.16e) we write the difference of the nonlocal terms in the form    b f0 + (∂x uω,n )2 cf2 + a (k − 2) f4 v Φ(uω,n ) − Φ(uω,n ) = D−2 ∂x k+1   k−3  + (uω,n ) c(uω,n )2 g1 − a (k − 2) g3 ∂x v    + D−2 k (k + 2) − 8a − b − c (k + 1) (uω,n )k−2 g2 ∂x v − (∂x uω,n )3 f3 v   − 43 a (k − 2) (uω,n )k−3 ∂x (g3 ∂x v) − ∂x (∂x uω,n )4 f4 v . (3.20) Thanks to the restriction 23 < σ < s − 1, all the terms on the right-hand side of Eq. (3.20) can be handled via the algebra property in H σ−1 , with the exception of the term that involves ∂x (g3 ∂x v). This term requires special care, since the presence of the second derivative does not allow us to employ the algebra property due to the restriction σ < s − 1. Instead, we use the negative multiplier estimate of Lemma 5 and estimates (3.14) and (3.15) to obtain  −2 ω,n k−3    D (u ) ∂x (g3 ∂x v)H σ . (uω,n )k−3 H σ−1 ∥∂x (g3 ∂x v)∥H (σ−1)−1 k−3

. ∥uω,n ∥H σ−1 ∥g3 ∥H σ−1 ∥∂x v∥H σ−1 k−3

. ∥uω,n ∥H s ∥g3 ∥H s ∥v∥H σ . ∥v∥H σ . Hence, we deduce the estimate 2

∥(3.16e)∥H σ . ∥v∥H σ .

(3.21)

Overall, combining estimates (3.17)–(3.19) and (3.21) with the energy equation (3.16), we arrive at the differential inequality d ∥v∥H σ . nσ−2s+1 + ∥v∥H σ , dt

n → ∞.

Recalling further that v(x, 0) = 0, we obtain ∥v∥H σ . nσ−2s+1 (et − 1) as n → ∞. Therefore, for 0 6 t 6 T we have shown that ∥v∥H σ . nσ−2s+1 ,

n ≫ 1.

(3.22)

Imposing the additional restriction σ > s − 2 allows us to estimate ∥v∥H s by combining estimate (3.22)   with Sobolev interpolation between H σ and H σ+2 . Indeed, for max 32 , s − 2 < σ < s − 1 we have 1

∥v∥H σ+2 . ∥uω,n ∥H σ+2 + ∥uω,n (0)∥H σ+2 . n− k + n−s (1 + n2 )

σ+2 2

. n−s+σ+2 .

(3.23)

ω ∈ {ω1 , ω2 } ∀t ∈ [0, T ].

(3.24)

Hence, employing Lemma 7 with r = s, r1 = σ and r2 = σ + 2, we find 1

∥uω,n (t) − uω,n (t)∥H s . n 2 (σ+2−s)(1−s) −−−−−→ 0, n→∞

Inequalities (3.9), (3.10) and (3.24) then imply ∥uω1 ,n (t∗ ) − uω2 ,n (t∗ )∥H s & 1 for t∗ > 0 as specified in the context of (3.10). The proof of Theorem 2 in the case of the circle is complete.  Proof of Lemma 8. Due to condition (3.8), the expression (3.5) for the error E ω,n becomes ∥E

ω,n

∥H σ .

k−2 

j

k−2−j

n− k · n−s(k−2−j) ∥cos(nx − ωt)∥H σ

  · n3(1−s) sin3 (nx − ωt)H σ

(3.25a)

j=0

+

k−1 

j

k−j−1

n− k · n−s(k−j−1) ∥cos(nx − ωt)∥H σ

· n1−2s ∥sin [2(nx − ωt)]∥H σ

(3.25b)

j=0

+ ∥Φ(uω,n )∥H σ .

(3.25c)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

188

The first two terms are straightforward to estimate. Indeed, using estimates (3.2) and (3.6) we have ∥(3.25a)∥H σ .

k−2 

1

j

n− k · n−s(k−2−j) nσ(k−2−j) · n3(1−s) nσ = n3(1−s)+σ · n(k−2)(σ−s) ·

j=0

and since s − σ −

1 k

n(s−σ− k )(k−1) − 1 , 1 n(s−σ− k ) − 1

> 0, we obtain ∥(3.25a)∥H σ . n3(1−s)+σ · n(k−2)(σ−s) · n(s−σ)(k−2) = n3(1−s)+σ .

(3.26)

Similarly, we find ∥(3.25b)∥H σ . nσ−2s+1 .

(3.27)

The estimation of the nonlocal term (3.25c) is more involved. By the triangle inequality and properties (2.7), we have         Φ(uω,n ) σ . (uω,n )k+1  σ−1 + (uω,n )k−1 (∂x uω,n )2  σ−1 + (uω,n )k−3 (∂x uω,n )4  σ−1 H H H H     + (uω,n )k−2 (∂x uω,n )3 H σ−2 + (uω,n )k−3 (∂x uω,n )3 ∂x2 uω,n H σ−2 . Careful application of the algebra property then yields k+1 

∥Φ(uω,n )∥H σ .

k+1−j j  n− k n−s sin (nx − ωt)H σ−1

j=0

+

k−1 

k−1−j   j  n− k n−s sin (nx − ωt)H σ−1 · n2(1−s) cos2 (nx − ωt)H σ−1

j=0

+

k−3 

 k−3−j  j  n− k n−s sin (nx − ωt)H σ−1 · n4(1−s) cos4 (nx − ωt)H σ−1

j=0

+

k−2 

k−2−j   j  n− k n−s sin (nx − ωt)H σ−1 · n3(1−s) cos3 (nx − ωt)H σ−1

j=0

+

k−3 

k−3−j   j  n− k n−s sin (nx − ωt)H σ−1 · n4(1−s) cos4 (nx − ωt)H σ .

j=0

Employing the trigonometric identities cos2 (nx − ωt) = 4

cos (nx − ωt) =

1 2 1 8

cos3 (nx − ωt) =

(cos [2(nx − ωt)] + 1) ,

1 4

(cos [3(nx − ωt)] + 3 cos(nx − ωt)) ,

(cos [4(nx − ωt)] + 4 cos [2(nx − ωt)] + 3) ,

the algebra property in H σ−1 and estimates (3.2) and (3.6), we find ∥Φ(uω,n )∥H σ .

k+1 

j

n− k · n(σ−s−1)(k+1−j) +

j=0

+

k−3 

k−1 

j

n− k · n(σ−s−1)(k−1−j) · n2(1−s) · nσ−1

j=0 j

n− k · n(σ−s−1)(k−3−j) · n4(1−s) · nσ−1 +

j=0

+

k−3 

k−2 

j

n− k · n(σ−s−1)(k−2−j) · n3(1−s) · nσ−1

j=0 j

n− k · n(σ−s−1)(k−3−j) · n4(1−s) · nσ .

j=0

Rearranging the right-hand side of the above expression and summing up the relevant geometric series, we obtain the estimate ∥Φ(uω,n )∥H σ . n−

k+1 k

+ n−

k−1 k +σ−2s+1

+ n−

k−3 k +σ−4s+3

+ n−

k−2 k +σ−3s+2

+ n−

k−3 k +σ−4s+4

.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

Comparing the exponents in the powers of n on the right-hand side for ∥Φ(uω,n )∥H σ . n−

k−1 k +σ−2s+1

3 2

189

< σ < s − 1, we conclude that

. nσ−2s+1 .

(3.28)

Combining estimates (3.26)–(3.28) and noting that σ − 2s + 1 > 3(1 − s) + σ for all s > 2 completes the proof of Lemma 8.  4. Non-uniform dependence on the line This section is devoted to the proof of Theorem 2 in the non-periodic case. We employ the method of approximate solutions that was already used in the previous section for the circle. However, the fact that R, as opposed to T, is not compact, forces us to localize the approximate solutions and hence, the proof becomes more technical. Note that throughout this section, we denote H s = H s (R). We introduce the localizer ϕ ∈ C0∞ (R) defined by  1, |x| < 1, ϕ(x) = (4.1) 0, |x| > 2, and we also introduce a localizer ϕ˜ ∈ C0∞ (R) such that ϕ(x) ˜ = 1 ∀x ∈ supp(ϕ). The family of approximate solutions that will be used for the line is then given by uω,n (x, t) = ul (x, t) + uh (x, t),

(4.2)

where the low frequency part ul is the exact solution of the k-abc-equation initial value problem ∂t ul = auk−2 (∂x ul )3 − ukl ∂x ul − Φ(ul ), l x (4.3) 1 ul (x, 0) = ωn− k ϕ˜ δ , n and the high frequency part is equal to x δ uh (x, t) = n− 2 −s ϕ δ cos(nx − ωt). (4.4) n Note that, as in the periodic case, in order to ensure that the error E ω,n associated with the approximate solution uω,n vanishes under the H s norm as n → ∞, the real constant ω must satisfy the equation ω k = ω, which yields condition (3.8). The error E ω,n is given by   E ω,n = ∂t uω,n − a(uω,n )k−2 (∂x uω,n )3 − (uω,n )k ∂x uω,n − Φ(uω,n ) . However, taking into account that ul is an exact solution of the k-abc-equation, E ω,n can be expressed b E ω,n = E1 − a (3E2 + E4 ) + E3 + k+1 E5 + cE6 − a(k − 2)E7   + k (k + 2) − 8a − b − c (k + 1) E8 − 43 a (k − 2) E9 ,

where the nine error components E1 , . . . , E9 are defined as follows: k−1    E1 = ∂t uh + ukl ∂x uh ,

E2 = uk−2 (∂x ul )2 ∂x uh , l

E3 =

j=0 k−3

E4 =

 j=0

k−2 j



k j

ujl (uh )k−j ∂x ul + ∂x uh ,



ujl (uh )k−2−j (∂x ul )3 + 3(∂x ul )2 (∂x uh ) + 3(∂x ul )(∂x uh )2 + (∂x uh )3



(4.5)





A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

190

E5 = D−2 ∂x

+ uk−2 3(∂x ul )(∂x uh )2 + (∂x uh )3 , l





k    k j=0

E6 = D−2 ∂x

 k−2  j=0

E7 = D−2 ∂x

+



j

 k−3  j=0

+ uk−2 l E9 = D−2

+



ujl (uh )k−1−j (∂x ul )2 + 2(∂x ul )(∂x uh ) + (∂x uh )2 + ulk−1 2(∂x ul )(∂x uh ) + (∂x uh )2









 k−4   k−3

−2

ujl (uh )k+1−j ,

uk−3 6(∂x ul )3 (∂x uh ) + 6(∂x ul )2 (∂x uh )2 + 10(∂x ul )(∂x uh )3 + (∂x uh )4 l

j=0

E8 = D

k−1 j



j

ujl (uh )k−3−j

k−2 j 2





4



h

3

2

h 2

h 3

ujl (uh )k−2−j (∂x ul )3 + 3(∂x ul )2 (∂x uh ) + 3(∂x ul )(∂x uh )2 + (∂x uh )3



h 2

h 3

3(∂x ul ) (∂x u ) + 3(∂x ul )(∂x u ) + (∂x u )

 



j

j=0

h 4

 

,



,

uk−3 ∂x 6(∂x ul )3 (∂x uh ) + 6(∂x ul )2 (∂x uh )2 + 10(∂x ul )(∂x uh )3 + (∂x uh )4 l

 k−4   k−3

,



(∂x ul ) + 6(∂x ul ) (∂x u ) + 6(∂x ul ) (∂x u ) + 10(∂x ul )(∂x u ) + (∂x u )

h

 



ujl (uh )k−3−j ∂x (∂x ul )4 + 6(∂x ul )3 (∂x uh ) + 6(∂x ul )2 (∂x uh )2 + 10(∂x ul )(∂x uh )3 + (∂x uh )4



 

.

Before attempting to estimate the above error terms, we state two useful lemmas. Lemma 9. For 0 < δ < k2 and s > 25 , the initial value problem (4.3) has a unique solution ul ∈ C([0, Tδ ]; H s ). Moreover, for any s > 0 this solution satisfies the estimate δ 1 kδ . −k ∥ul (t)∥H s 6 cs,ϕ˜ |ω| n 2 − k , 0 6 t 6 Tδ = c˜s,ϕ˜ |ω| n1− 2 ,  1 k −1 ˜ H s and c˜s,ϕ˜ = 2k+1 kcs ∥ϕ∥ ˜ Hs where cs,ϕ˜ = 21+ k ∥ϕ∥ with cs > 0. Proof. Since ul is an exact solution of the initial value problem (4.3), the proof of the lemma follows directly from the well-posedness of the k-abc-equation on the line. Suppose first that s > 25 . In this case, Theorem 1 establishes existence of a unique solution ul ∈ C([0, T ]; H s ) with the size estimate  1 k −1 ∥ul (t)∥H s 6 21+ k ∥ul (0)∥H s , 0 6 t 6 2k+1 kcs ∥ul (0)∥H s , cs = c(s) > 0. (4.6) 1

δ

Moreover, a straightforward computation shows that ∥ul (0)∥H s 6 |ω| n− k + 2 ∥ϕ∥ ˜ H s for n ≫ 1, thus estimate (4.6) becomes 1 1 δ kδ  k k −1 ∥ul (t)∥H s 6 21+ k ∥ϕ∥ ˜ H s |ω| n− k + 2 , 0 6 t 6 n1− 2 2k+1 kcs ∥ϕ∥ ˜ H s |ω| . (4.7) The case s 6

5 2

is treated by noting that ∥ul (t)∥H s 6 ∥ul (t)∥H 3 and using estimate (4.7) for s = 3.



Lemma 10. Let ψ ∈ S(R), δ > 0 and α ∈ R. Then, for any s > 0 we have     δ lim n− 2 −s ψ nxδ sin(nx − α)H s = √12 ∥ψ∥L2 . n→∞

The same is true if sine is replaced by cosine. Lemma 10 can be established via a straightforward computation and is proved in [24]. We now proceed   to the estimation of the error terms of Eq. (4.5) under the H σ norm with σ ∈ 32 , s − 2 . We only present the details for the terms E1 and E2 , which turn out to be the most challenging ones.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

191

Estimate for E1 . Note that the definitions of ϕ and ϕ˜ imply that ϕ˜k (x) · ϕ(x) = ϕ(x) for all x ∈ R. Thus, since ω k = ω it follows that       δ δ . ∂t uh = ωn− 2 −s ϕ nxδ sin(nx − ωt) = ω k ϕ˜k nxδ n− 2 −s ϕ nxδ sin(nx − ωt)   δ = ukl (x, 0) n1− 2 −s ϕ nxδ sin(nx − ωt), which in turn yields       δ 3 E1 = ukl (x, 0) − ukl (x, t) n1− 2 −s ϕ nxδ sin(nx − ωt) + ukl (x, t) n− 2 δ−s ϕ′ nxδ cos(nx − ωt). Then, using Lemmas 9 and 10 as well as the algebra property in H σ we find          3 δ ∥E1 ∥H σ 6  ukl (0) − ukl (t) n1− 2 −s ϕ nxδ sin(nx − ωt)H σ + ukl (x, t) n− 2 δ−s ϕ′ nxδ cos(nx − ωt)H σ      δ 1 δ (4.8) .  ukl (0) − ukl (t) n1− 2 −s ϕ nxδ sin(nx − ωt)H σ + n( 2 − k )k−δ−s+σ . Moreover, we have   k     ul (0) − ukl (t) n1− δ2 −s ϕ xδ sin(nx − ωt) σ n H .

k−1 

j

k−1−j

∥ul (0)∥H σ ∥ul (t)∥H σ

    δ · ∥ul (0) − ul (t)∥H σ n1− 2 −s ϕ nxδ sin(nx − ωt)H σ

j=0 δ 1 . n( 2 − k )(k−1) · nσ−s+1 ·

 0

t

∥∂τ ul (τ )∥H σ dτ,

(4.9)

where we have used the Fundamental Theorem of Calculus in the latter inequality. Recalling, however, that ul is an exact solution of the k-abc-equation and using Lemma 9, it is straightforward to show that δ 1 ∥∂t ul (t)∥H σ . n( 2 − k )(k+1) . Combining this estimate with inequalities (4.8) and (4.9) we conclude that k

∥E1 ∥H σ . nσ−s−1+kδ + nσ−s−1+( 2 −1)δ ≃ nσ−s−1+kδ ,

n ≫ 1.

(4.10)

Estimate for E2 . By the definitions of E2 and uh , we have     δ δ E2 = uk−2 (∂x ul )2 n−δ · n− 2 −s ϕ′ nxδ cos (nx − ωt) − uk−2 (∂x ul )2 n− 2 −s ϕ nxδ n sin (nx − ωt) . l l 2

Moreover, note that the definitions of ϕ and ϕ˜ imply that (ϕ˜′ (x)) · ϕ(x) = 0 for all x ∈ R, thus the quantity  2  (∂x ul (x, 0)) ϕ nxδ vanishes. This allows us to express E2 in the form      3 δ 2 E2 = uk−2 (∂x ul )2 n− 2 δ−s ϕ′ nxδ cos (nx − ωt) − uk−2 (∂x ul (t))2 − (∂x ul (0)) n1− 2 −s ϕ nxδ sin (nx − ωt) , l l which via the triangle inequality implies        δ ∥E2 ∥H σ 6 uk−2 ∂x ul (t) + ∂x ul (0) n1− 2 −s ϕ nxδ sin (nx − ωt) ∂x ul (t) − ∂x ul (0) H σ l     3 + uk−2 (∂x ul )2 n− 2 δ−s ϕ′ nxδ cos (nx − ωt)H σ . l Then, using the algebra property in H σ and Lemmas 9 and 10 we obtain k

∥E2 ∥H σ . nσ−s−1+( 2 −1)δ + nσ−s−1+kδ ≃ nσ−s−1+kδ ,

n ≫ 1.

(4.11)

The remaining terms E3 , . . . , E9 that are present on the right-hand side of Eq. (4.5) can be estimated in a similar fashion. The only additional estimate required is the inequality ∥f g∥H σ . ∥f ∥L∞ ∥g∥H σ + ∥f ∥H σ ∥g∥L∞ , whose proof can be found in [53]. Overall, we have the following lemma for the error E ω,n .

192

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

Lemma 11. Suppose that s > 52 , 32 < σ < s − 1 and 0 < δ < k2 with k > 3. Then, the error E ω,n associated with the approximate solutions (4.2) of the k-abc-equation on the line admits the estimate ∥E ω,n ∥H σ . nσ−s−1+kδ ,

n ≫ 1.

Let uω,n (x, t) be the exact solution of the k-abc-equation corresponding to the initial condition uω,n (x, 0). As in the case of the circle, the difference v = uω,n − uω,n satisfies the Cauchy problem (3.13) and can be shown via Lemma 11 to admit the following analogue to estimate (3.22): ∥v∥H σ . nσ−s−1+kδ ,

n ≫ 1, s > 52 ,

3 2

< σ < s − 1, 0 < δ < k2 .

(4.12)

The norm ∥v∥H s can now be estimated by combining estimate (4.12) and Sobolev interpolation. First, note that for any r > s > 52 we have ∥v(t)∥H r . ∥uω,n (t)∥H r + ∥uω,n (0)∥H r 6 ∥ul (t)∥H r + ∥uh (t)∥H r + ∥ul (0)∥H r + ∥uh (0)∥H r , where the first inequality above is due to the size estimate (1.12) for the exact solution uω,n . Next, using Lemma 9 for ul and Lemma 10 for uh , we find δ

∥v(t)∥H r . n( 2 − k )r + nr−s . nr−s . 1

(4.13)

Interpolating between s1 = σ and σ2 = r by means of Lemma 7 then yields r−s

s−σ

r−s

∥v(t)∥H s . n(σ−s−1+kδ) r−σ · n(r−s) r−σ = n(kδ−1) r−σ ,

n ≫ 1.

Then, since r > s > σ, imposing the restriction δ < k1 implies that ∥uω,n (t) − uω,n (t)∥H s vanishes as n → ∞ for all t ∈ [0, T ] and for any ω given by (3.8). Hence, for ω1 and ω2 given by (3.8) we find ∥uω1 ,n (t) − uω2 ,n (t)∥H s > ∥uω1 ,n (t) − uω2 ,n (t)∥H s − ∥uω1 ,n (t) − uω1 ,n (t)∥H s − ∥uω2 ,n (t) − uω2 ,n (t)∥H s     r−s δ  1 ,n  2 ,n & uω (t) − uω (t) + n− 2 −s ϕ nxδ cos (nx − ω1 t) − cos (nx − ω2 t)  s − n(kδ−1) r−σ . l l H

 θ+φ 

 θ−φ 

Moreover, the identity cos θ − cos φ = 2 sin 2 sin 2 together with Lemmas 10 and 9 yields    r−s δ 1   2 t ∥uω1 ,n (t) − uω2 ,n (t)∥H s & sin ω1 −ω  − n( 2 − k )s − n(kδ−1) r−σ ∀t ∈ [0, T ]. 2 Therefore, for n sufficiently large and some t∗ > 0 we have      2 ∥uω1 ,n (t∗ ) − uω2 ,n (t∗ )∥H s & sin ω1 −ω t  ≃ 1. ∗ 2 The proof of Theorem 2 in the case of the line is complete.



5. H¨ older continuity on the line and on the circle In Sections 3 and 4, we showed that the dependence of the solution of the k-abc-equation on the initial data u0 ∈ H s , s > 25 , is non-uniform from any bounded subset of H s into C([0, T ]; H s ), both on the circle and on the line. We now prove Theorem 3, which states that this dependence is uniform, and more precisely of H¨ older type, when measured in a larger space H r , r < s, for (s, r) inside the regions A1 , A2 , A3 and A4 defined by Eq. (1.15) and depicted in Fig. 1.2. The H¨older exponent in each of these regions is specified by Eq. (1.14).

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

193

5.1. Lipschitz continuity in the region A1 We consider the k-abc system (2.69), which was also used in the proof of continuity of the data-to-solution map and reads ∂t u = auk−2 w3 − uk w − F1 (u, w), k−2

∂t w = 3au

2

k

(5.1a) k−1

w ∂x w − u ∂x w + (c − k) u

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

2

w +

b k+1

u

k+1

− F2 (u, w),

w(x, 0) = ∂x u0 (x),

(5.1b) (5.1c)

with the nonlocal terms F1 and F2 defined by Eq. (2.4). Moreover, we let (˜ u, w) ˜ satisfy the system ∂t u ˜ = a˜ uk−2 w ˜3 − u ˜k w ˜ − F1 (˜ u, w), ˜ k−2

∂t w ˜ = 3a˜ u

2

k

(5.2a) k−1

w ˜ ∂x w ˜−u ˜ ∂x w ˜ + (c − k) u ˜

u ˜(x, 0) = u ˜0 (x),

2

w ˜ +

b k+1

u ˜

k+1

− F2 (˜ u, w), ˜

w(x, ˜ 0) = ∂x u ˜0 (x).

Subtracting system (5.2) from system (5.1) and setting ϕ = u − u ˜, ψ = w − w ˜ yields the system  k−2    ∂t ϕ = au g2 − uk ψ + af3 w ˜ 3 − f1 w ˜ ϕ − F1 (u, w) + F1 (˜ u, w), ˜  k−2    ∂t ψ = au g2 − uk ∂x ψ + auk−2 ∂x g2 + (c − k) uk−1 g1 ψ   b f0 ϕ − F2 (u, w) + F2 (˜ u, w), ˜ + af3 ∂x (w ˜ 3 ) + (c − k) f2 w ˜ 2 − f1 ∂x w ˜ + k+1 ϕ(x, 0) = u0 (x) − u ˜0 (x),

ψ(x, 0) = ∂x [u0 (x) − u ˜0 (x)] ,

(5.2b) (5.2c)

(5.3a)

(5.3b) (5.3c)

where the functions fj and gj are defined as in (2.51) but with v, z replaced by u ˜ and w. ˜ r Applying the operator D on Eqs. (5.3a) and (5.3b), multiplying on the right by Dr ϕ and Dr ψ respectively, and integrating the resulting expressions with respect to x, we obtain the energy equations    1 d 2 ∥ϕ∥H r = Dr auk−2 g2 − uk ψ · Dr ϕ dx (5.4a) 2 dt    + Dr af3 w ˜ 3 − f1 w ˜ ϕ · Dr ϕ dx (5.4b)    − Dr F1 (u, w) − F1 (˜ u, w) ˜ · Dr ϕ dx (5.4c) and 1 d 2 ∥ψ∥H r = 2 dt



  Dr auk−2 g2 − uk ∂x ψ · Dr ψ dx    + Dr auk−2 ∂x g2 + (c − k) uk−1 g1 ψ · Dr ψ dx    b + Dr af3 ∂x (w ˜ 3 ) + (c − k) f2 w ˜ 2 − f1 ∂x w ˜ + k+1 f0 ϕ · Dr ψ dx    − Dr F2 (u, w) − F2 (˜ u, w) ˜ · Dr ψ dx.

We first estimate the right-hand side of Eq. (5.4). By the Cauchy–Schwarz inequality, we have        |(5.4a)| 6 Dr auk−2 g2 − uk ψ L2 ∥Dr ϕ∥L2 =  auk−2 g2 − uk ψ H r ∥ϕ∥H r .

(5.5a) (5.5b) (5.5c) (5.5d)

(5.6)

Our goal is now to separate the product appearing inside the H r norm on the right-hand side of (5.6). Depending on the range of r, the available tools towards this goal are the algebra property, the “negative multiplier” estimate of Lemma 5, and the following lemma which is proved in [27].

194

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

Lemma 12. If −1 6 r 6 0, s >

3 2

and r + s > 1, then ∥f g∥H r 6 cr,s ∥f ∥H s−1 ∥g∥H r .

For 12 < r 6 s − 1, the algebra property in H r and the well-posedness estimate (2.25) imply    k−2   k−2  2 2  k  au g2 − uk ψ H r . ∥u∥H r ∥w∥H r + ∥w∥H r ∥w∥ ˜ H r + ∥w∥ ˜ H r + ∥u∥H r ∥ψ∥H r    k−2  2 2 k 6 ∥u∥H s−1 ∥w∥H s−1 + ∥w∥H s−1 ∥w∥ ˜ H s−1 + ∥w∥ ˜ H s−1 + ∥u∥H s−1 ∥ψ∥H r   k−2  2 2  k . ∥u0 ∥H s ∥u0 ∥H s + ∥u0 ∥H s ∥˜ u0 ∥H s + ∥˜ u0 ∥H s + ∥u0 ∥H s ∥ψ∥H r . ϱk ∥ψ∥H r .

(5.7)

For − 12 < r 6 12 , the algebra property in H r is not available; however, we can employ Lemma 5 with σ = r + 1 and the algebra property in H r+1 to obtain  k−2      au g2 − uk ψ H r . auk−2 g2 − uk H r+1 ∥ψ∥H r    k−2  2 2 k 6 ∥u∥H s−1 ∥w∥H s−1 + ∥w∥H s−1 ∥w∥ ˜ H s−1 + ∥w∥ ˜ H s−1 + ∥u∥H s−1 ∥ψ∥H r . ϱk ∥ψ∥H r ,

(5.8)

where we note that the second inequality above is due to the fact that r 6 s − 2 since r 6 21 and s > 25 . Finally, for −1 6 r 6 − 12 we employ Lemma 12. Noting that r + s > 1 since s > 52 and using also the algebra property in H s−1 , we have  k−2      au (5.9) g2 − uk ψ H r . auk−2 g2 − uk H s−1 ∥ψ∥H r . ϱk ∥ψ∥H r ∥ϕ∥H r . Hence, for the three ranges of r specified above inequality (5.6) yields the estimate |(5.4a)| . ϱk ∥ψ∥H r ∥ϕ∥H r .

(5.10)

Repeating the arguments leading to estimate (5.10), we obtain for the same three ranges of r the estimate 2

|(5.4b)| . ϱk ∥ϕ∥H r .

(5.11)

Regarding the nonlocal term (5.4c), analogously to (2.55) we express F1 (u, w) − F1 (˜ u, w) ˜ in the form F1 (u, w) − F1 (˜ u, w) ˜     3 = D−2 k (k + 2) − 8a − b − c (k + 1) w ˜ f3 − 43 a (k − 2) f4 ∂x (w ˜4 ) ϕ    + k (k + 2) − 8a − b − c (k + 1) uk−2 g2 ψ − 34 a (k − 2) uk−3 ∂x (g3 ψ)      b + D−2 ∂x k+1 f0 + cf2 w ˜ 2 − a(k − 2)f4 w ˜ 4 ϕ + c uk−1 g1 − a (k − 2) uk−3 g3 ψ and use the Cauchy–Schwarz inequality as well as the fact that uk−3 ∂x (g3 ψ) = ∂x (uk−3 g3 ψ) − ∂x (uk−3 )g3 ψ to arrive at the inequality   3          |(5.4c)| . w ˜ f3 ϕH r + f4 ∂x (w ˜ 4 )ϕH r−2 + uk−2 g2 ψ H r + uk−3 g3 ψ H r + ∂x (uk−3 )g3 ψ H r−2         b    2 4 k−1 k−3 +  k+1 f0 + cf2 w ˜ − a(k − 2)f4 w ˜ ϕ r +  c u g1 − a (k − 2) u g3 ψ  r ∥ϕ∥H r . H

H

From the point of view of estimates, all the terms on the right-hand side of the above inequality are similar with the two local terms (5.4a) and (5.4b) with the exception of the second term, which involves ∂x (w) ˜ 4. We now provide the details for this term.

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

If

3 2

< r 6 s − 1, then we employ the algebra property in H r−1 and the size estimate (2.25) to obtain   4 f4 ∂x (w ˜ 4 )ϕH r−2 . ∥f4 ∥H r−1 ∥w∥ ˜ H r ∥ϕ∥H r−1 . ϱk ∥ϕ∥H r .

If

1 2

195

(5.12)

< r 6 32 , then Lemma 5 with σ = r and the algebra property in H r yield     4 f4 ∂x (w ˜ 4 )ϕH r−2 . ∂x (w) ˜ 4 H r−1 ∥f4 ϕ∥H r . ∥w∥ ˜ H r ∥f4 ∥H r . ϱk ∥ϕ∥H r .

(5.13)

If − 21 < r 6 12 , then we employ again Lemma 5 but now with σ = r + 1. Using also the algebra property in H r+1 together with the condition r 6 s − 3, which implies that ∥∂x (w) ˜ 4 ∥H r+1 6 ∥∂x (w) ˜ 4 ∥H s−2 and thus allows us to use the well-posedness estimate (2.25), we find     f4 ∂x (w ˜ 4 H r+1 ∥ϕ∥H r ˜ 4 )ϕH r−2 . f4 ∂x (w)   ˜ 4 H r+1 ∥ϕ∥H r . ∥f4 ∥H r+1 ∂x (w)   . ∥f4 ∥H s ∂x (w) ˜ 4 H s−2 ∥ϕ∥H r . ϱk ∥ϕ∥H r . (5.14) If −1 6 r 6 − 21 , then we use Lemma 12 but with s replaced by s − 1. Note, however, that the third condition of the lemma, namely that r + s > 2, does not necessarily hold for all −1 6 r 6 − 21 and s > 25 . Thus, we have to impose this condition by requiring that r > −s + 2. Then, using also the algebra property in H s−2 we find     f4 ∂x (w ˜ 4 H s−2 ∥ϕ∥H r ˜ 4 )ϕH r−2 . f4 ∂x (w)   . ∥f4 ∥H s−2 ∂x (w) ˜ 4 H s−2 ∥ϕ∥H r  4 . ∥f4 ∥H s−2 w ˜ H s−1 ∥ϕ∥H r . ϱk ∥ϕ∥H r . (5.15) Overall, in the three regions of the sr-plane { 21 < r 6 s − 1}, {− 21 < r 6 {−1 6 r 6 − 12 } ∩ {r > −s + 2} the following inequality is valid:

1 2}

∩ {r 6 s − 3} and

  d ∥ϕ∥H r . ϱk ∥ψ∥H r + ∥ϕ∥H r . dt

(5.16)

Next, we proceed to the energy equation (5.5). Actually, comparing Eqs. (5.4) and (5.5) we realize that the only term of (5.5) which is fundamentally different in terms of estimates is the local term (5.5a). Indeed, this term involves ∂x ψ, which has been encountered up to this point only as part of the nonlocal term (5.4c) and thus was easy to handle due to the smoothness provided by Dr−2 . For term (5.5a), however, such help is not available and hence we need to employ the Calderon–Coifman–Meyer commutator estimate of Lemma 6.   In particular, we first commute auk−2 g2 − uk with ∂x to write            Dr auk−2 g2 − uk ∂x ψ = Dr ∂x , auk−2 g2 − uk ψ − auk−2 g2 − uk Dr ∂x ψ − Dr ∂x auk−2 g2 − uk ψ . Then, we have              |(5.5a)| 6  Dr ∂x , auk−2 g2 − uk ψ · Dr ψ dx +  auk−2 g2 − uk Dr ∂x ψ · Dr ψ dx         +  Dr ∂x auk−2 g2 − uk ψ · Dr ψ dx.

(5.17)

Using the Cauchy–Schwarz inequality and Lemma 6 with σ = r, ρ = s − 1 and −1 6 r 6 s − 2, we estimate the first term of (5.17) as follows:             Dr ∂x , auk−2 g2 − uk ψ · Dr ψ dx 6  Dr ∂x , auk−2 g2 − uk ψ L2 ∥ψ∥H r   2 2 . auk−2 g2 − uk H s−1 ∥ψ∥H r . ϱk ∥ψ∥H r . (5.18)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

196

For the second term of (5.17), integrating by parts and using the Sobolev embedding yields    1           2  auk−2 g2 − uk Dr ∂x ψ · Dr ψ dx =  ∂x auk−2 g2 − uk (Dr ψ) dx 2  1  2 6 ∂x auk−2 g2 − uk L∞ ∥Dr ψ∥L2 2   2 2 . auk−2 g2 − uk H s−1 ∥ψ∥H r . ϱk ∥ψ∥H r .

(5.19)

Regarding the third term of (5.17), we first apply the Cauchy–Schwarz inequality to obtain              Dr ∂x auk−2 g2 − uk ψ · Dr ψ dx 6 ∂x auk−2 g2 − uk ψ H r ∥ψ∥H r . The first term on the right-hand side of the above inequality requires different treatment for different values of r. For 21 < r 6 s − 2, by the algebra property in H r we have   k−2     2 2 ∂x au (5.20) g2 − uk H r 6 ∂x auk−2 g2 − uk H s−2 ∥ψ∥H r . ϱk ∥ψ∥H r . For − 21 < r 6

1 2

with r 6 s − 3, using Lemma 5 with σ = r + 1 we find   k−2       ∂x au g2 − uk ψ H r . ∂x auk−2 g2 − uk H r+1 ∥ψ∥H r   6 auk−2 g2 − uk H r+2 ∥ψ∥H r   6 auk−2 g2 − uk  s−1 ∥ψ∥ r . ϱk ∥ψ∥ H

H

Hr

.

Finally, for −1 6 r 6 − 21 with r > −s + 2 we use Lemma 12 with s replaced by s − 1 to obtain     k−2     ∂x au g2 − uk ψ H r . ∂x auk−2 g2 − uk H s−2 ∥ψ∥H r . ϱk ∥ψ∥H r . Hence, in the three regions of the sr-plane { 12 < r 6 s − 2}, {− 21 < r 6 {−1 6 r 6 − 12 } ∩ {r > −s + 2} we have the estimate

1 2}

(5.21)

(5.22)

∩ {r 6 s − 3} and

2

|(5.5a)| . ϱk ∥ψ∥H r .

(5.23)

The remaining local terms (5.5b) and (5.5c) can be handled similarly in the same three regions of the sr-plane and admit the estimates 2

|(5.5b)| . ϱk ∥ψ∥H r ,

2

|(5.5c)| . ϱk ∥ψ∥H r .

(5.24)

Moreover, expressing the difference F2 (u, w) − F2 (˜ u, w) ˜ analogously to Eq. (2.55) we observe that the nonlocal term (5.5d) is similar to the nonlocal term (5.4c) from the point of view of estimates. Indeed, the only     terms that could require special handling are those involving D−2 ∂x f4 ∂x (w ˜ 4 )ϕ and D−2 ∂x uk−3 ∂x (g3 ψ) . However, the first of these two terms can be estimated via the Cauchy–Schwarz inequality and the same techniques required for estimates (5.20)–(5.22) over the same three regions of the sr-plane that arose in   the derivation of estimates (5.24) and (5.23). Furthermore, the term involving D−2 ∂x uk−3 ∂x (g3 ψ) can also be estimated using the arguments that led to estimates (5.23) and (5.24) after writing uk−3 ∂x (g3 ψ) as ∂x (uk−3 g3 ψ) − ∂x (uk−3 )g3 ψ. Overall, in the three regions of the sr-plane { 12 < r 6 s − 2}, {− 12 < r 6 12 } ∩ {r 6 s − 3} and {−1 6 r 6 − 12 } ∩ {r > −s + 2} we have the inequality   d ∥ψ∥H r . ϱk ∥ψ∥H r + ∥ϕ∥H r , dt which combined with inequality (5.16) implies    d ∥ϕ∥H r + ∥ψ∥H r . ϱk ∥ϕ∥H r + ∥ψ∥H r dt over the regions { 21 < r 6 s − 2}, {− 12 < r 6 12 } ∩ {r 6 s − 3} and {−1 6 r 6 − 21 } ∩ {r > −s + 2}.

(5.25)

(5.26)

A. Alexandrou Himonas, D. Mantzavinos / Nonlinear Analysis 133 (2016) 161–199

197

Integrating inequality (5.26) and substituting for ϕ and ψ in terms of u and u ˜ then yields      k ∥u − u ˜∥H r + ∥∂x (u − u ˜)∥H r . ∥u0 − u ˜0 ∥H r + ∂x u0 − u ˜0 H r eϱ T , from which we deduce the estimate k

∥u − u ˜∥H r+1 . ∥u0 − u ˜0 ∥H r+1 eϱ

T

.

Relabeling r to r − 1, we equivalently have k

∥u − u ˜∥H r . ∥u0 − u ˜ 0 ∥ H r eϱ

T

(5.27)

with the relevant regions now reading { 32 < r 6 s − 1}, { 21 < r 6 23 } ∩ {r 6 s − 2} and {0 6 r 6 12 } ∩ {r > −s + 3}. As these three regions combine to the region A1 defined by Eq. (1.15), the proof of Lipschitz continuity in the region A1 is complete.  5.2. H¨ older continuity in the regions A2 , A3 and A4 We now use the Lipschitz continuity inside the region A1 in combination with the Sobolev interpolation Lemma 7 in order to extract H¨ older continuity inside the regions A2 , A3 and A4 defined by Eq. (1.15) and with exponents specified by Eq. (1.14). ˜∥H r 6 ∥u − u In the case of the region A2 , we have 52 < s < 3 and 0 6 r 6 −s + 3. Thus, ∥u − u ˜∥H 3−s and, since 0 6 3 − s 6 s, we can use the Lipschitz estimate (5.27) with r = 3 − s to infer k

∥u − u ˜∥H r 6 ∥u − u ˜∥H 3−s . ∥u0 − u ˜0 ∥H 3−s eϱ

T

.

Setting then σ1 = r, σ = 3 − s and σ2 = s in Lemma 7 and using the well-posedness size estimate (2.25), we find 2s−3

3−s−r

∥u0 − u ˜0 ∥H 3−s 6 ∥u0 − u ˜0 ∥Hs−r ∥u0 − u ˜0 ∥Hs−r .ϱ r s

3−s−r s−r

2s−3

∥u0 − u ˜0 ∥Hs−r . r

Therefore, we conclude that ∥u − u ˜∥H r . eϱ 5 2

For the region A3 , we have s >

k

T

ϱ

3−s−r s−r

2s−3

∥u0 − u ˜0 ∥Hs−r , r

for (s, r) ∈ A2 .

and s − 2 6 r 6 32 . Thus, since s − 2 6 r < s, Lemma 7 yields r−s+2

s−r

∥u − u ˜∥H r 6 ∥u − u ˜∥H2s−2 ∥u − u ˜∥H s2

.

It then follows by the size estimate (2.25) and the Lipschitz estimate (5.27) for r = s−2 (recall that r = s−2 with s > 52 is included in A1 ) that for s − 2 6 r < s, which contains the region A3 , we have k

∥u − u ˜ ∥ H r . eϱ

T

ϱ

r−s+2 2

s−r

∥u0 − u ˜0 ∥H2s−2 6 eϱ

Finally, in the case of the region A4 we have s > σ1 = s − 1, σ = r and σ2 = s to obtain

5 2

k

T

ϱ

r−s+2 2

s−r

∥u0 − u ˜0 ∥H2r .

and s − 1 6 r < s, hence we apply Lemma 7 with

s−r

r−s+1

∥u − u ˜∥H r 6 ∥u − u ˜∥H s−1 ∥u − u ˜∥H s

.

Then, the Lipschitz continuity (5.27) for r = s − 1 (recall that r = s − 1 with s > the well-posedness estimate (2.25) imply k

∥u − u ˜∥H r . eϱ

T r−s+1

ϱ

The proof of Theorem 3 is complete.

s−r

∥u0 − u ˜0 ∥H s−1 6 eϱ 

k

T r−s+1

ϱ

s−r

∥u0 − u ˜0 ∥H r ,

5 2

is included in A1 ) and

(s, r) ∈ A4 .

198

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