Acta Appl Math DOI 10.1007/s10440-016-0049-2
Convergence of the Kuramoto–Sinelshchikov Equation to the Burgers One Giuseppe Maria Coclite1 · Lorenzo di Ruvo2
Received: 19 July 2015 / Accepted: 16 March 2016 © Springer Science+Business Media Dordrecht 2016
Abstract We consider the Kuramoto–Sinelshchikov equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting. Keywords Singular limit · compensated compactness · Kuramoto–Sinelshchikov equation · entropy condition Mathematics Subject Classification (2000) 35G25 · 35L65 · 35L05
1 Introduction The Kuramoto–Sivashinsky (KS) equation reads as 2 4 u + c∂xxxx u = 0, ∂t u + u∂x u + a∂xx
a, c ∈ R,
(1.1)
a is called the anti-diffusion parameter. Equation (1.1) was derived independently by Kuramoto [28–30] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky [40] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
B G.M. Coclite
[email protected]; url: http://www.dm.uniba.it/Members/coclitegm/
L. di Ruvo
[email protected]; url: http://www.dm.uniba.it/Members/coclitegm/ 1
Department of Mathematics, University of Bari, via E. Orabona 4, 70125 Bari, Italy
2
Department of Science and Methods for Engineering, University of Modena and Reggio Emilia, via G. Amendola 2, 42122 Reggio Emilia, Italy
G.M. Coclite, L. di Ruvo
Equation (1.1) also describes incipient instabilities in a variety of physical and chemical systems (see [5, 22, 31]). The well-posedness and dynamical properties of KS equations have been investigated in [20, 27, 36, 37]. Moreover, in [1, 4], the control problem of (1.1) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [7], the problem of global exponential stabilization of (1.1) with periodic boundary conditions is analyzed. In [23], it is proposed a generalization of optimal control theory for (1.1), while in [33], the problem of global boundary control of (1.1) is considered. A generalization of (1.1) is 2 3 4 u + b∂xxx u + c∂xxxx u = 0, ∂t u + u∂x u + a∂xx
a, b, c ∈ R,
(1.2)
which is known as Kuramoto–Sivashinsky–Korteweg–de Vries equation, (see [25, 44]), or the Benney–Lin equation (see [2, 34]). Equation (1.2) has a prominent position in describing the evolution of small but finite amplitude long waves in fluid dynamics. It also arises as a model for long waves in a viscous fluid flowing down an inclined plane and also describes drift waves in a plasma (see [16, 43]). Moreover, (1.2) was derived by Kuramoto in the study of phase turbulence in the Belousov–Zhabotinsky reaction (see [6]). In [25, 45], the authors proved the existence of the solution for (1.2). In [38], the existence of the solitonic solution for (1.2) is proven. In [21], the authors prove that every L-periodic, mean-zero solution of (1.1) is on average o(L) for L 1. In this paper, we consider two cases for (1.2). Choosing a = c = 1, b = k 2 ≥ 0 in (1.2), we have 2 3 4 u + k 2 ∂xxx u + ∂xxxx u = 0. ∂t u + u∂x u + ∂xx
(1.3)
Choosing a = 0, b = 0, c = 1, we have 4 u = 0. ∂t u + u∂x u + ∂xxxx
(1.4)
Arguing as in [14], we re-scale the two equations as follows 3 4 2 ∂t u + u∂x u + k 2 β 2 ∂xxx u + β 3 ∂xxxx u = −β∂xx u, 3 2
4 ∂t u + u∂x u + β ∂xxxx u = 0,
(1.5) (1.6)
where β is the diffusion parameter. We are interested in the no high frequency limit, we send β → 0 in (1.5). In this way, we pass from (1.5) to the Burgers equation ∂t u + u∂x u = 0.
(1.7)
The uniqueness of bounded entropy solutions for (1.7) has been proved in [26]. Here we work with L2 ∩ L4 solutions and we use the uniqueness result of [42]. Let us consider (1.5). Choosing the following approximation 3 4 2 2 u + β 3 ∂xxxx u = −β∂xx u + ε∂xx u, ∂t u + u∂x u + k 2 β 2 ∂xxx
(1.8)
we prove that the solution of (1.5) converges to the unique entropy solution of (1.7), under the following assumption β = O ε2 . (1.9) u0 ∈ L2 (R),
Convergence of Kuramoto–Sinelshchikov Type Equation
For (1.6), consider the following approximation 3
4 2 u = ε∂xx u, ∂t u + u∂x u + β 2 ∂xxxx
(1.10)
we can choose the initial datum and β in two different ways. Following [17, Theorem 7.1] and [10, Appendices A and B], the first choice is the following (see Theorem 4.1): β = o ε4 . (1.11) u0 ∈ L2 (R), Since · L4 is a conserved quantity for (1.6), the second choice is (see. Theorem 3.1) β = O ε2 . (1.12) u0 ∈ L2 (R) ∩ L4 (R), We note that (1.12) is the same assumption that is used to prove that the solutions of the Korteweg–de Vries equation converge to the unique entropy solution of (1.7) (see [18, 32, 39]), while (1.11) is the same assumption that is used to prove that the solutions of the Rosenau equation converge to the unique entropy solution of (1.7) (see [10]). Moreover, arguing as [8, 9], we have the same result for the following equation 3 4 u + ∂xxxx u = 0. ∂t u + u∂x u + k 2 ∂xxx
(1.13)
From the mathematical point of view, assuming (1.12), we do not need to prove a L∞ -estimate (see Lemma 3.2), which is needed under the assumption (1.11) (see Lemma 4.1). It is interesting to observe that (1.6) is a fourth order equation, like the viscous Camassa– Holm equation (see [15]). In [15], the authors only proved that the solutions of the Camassa– Holm equation converge to a distributional solution of (1.7), with (1.14) β = O ε4 , while, using the kinetic method, in [24], the convergence to the unique entropy solution of (1.7) is proven in [24], with (1.15) β = o ε4 . Following [18], in [15], and choosing (1.15), the dissipation of energy is proven for the Camassa–Holm equation. Finally, in [13] we improve the result of [24] showing that the convergence to the unique entropy solution of (1.7) can be proved using the compensated compactness and (1.14). Here, using the approximation (1.10), we prove that, under the assumption (1.16) β = O ε2 , the solution of (1.6) converges to the distributional solution of (1.7). Moreover, we prove the dissipation of energy, this is 2 u 1 ∂t + ∂x u3 ≤ 0, in weak sense in R+ × R. 2 3 The paper is organized in five sections. In Sect. 2, we prove that the solution of (1.3) converges to the unique entropy solution of (1.7), in Lp setting, with 1 ≤ p < 2. In Sect. 3, we prove that the solution of (1.6) converges to unique entropy solution of (1.7), in Lp ,
G.M. Coclite, L. di Ruvo
with 1 ≤ p < 4. In Sect. 4, we prove that the solution of (1.6) converges to unique entropy solution of (1.7), in Lp , with 1 ≤ p < 2. Section 4 is an appendix, where we prove that the solution of the Korteweg–de Vries–Burgers equation converges to the unique entropy solution of (1.7), in Lp , with 1 ≤ p < 2.
2 The Kuramoto–Sivashinsky Equation: u0 ∈ L2 (R), β = O(ε 2 ) In this section, we consider (1.5), and assume (1.9) on the initial datum. We study the dispersion-diffusion limit for (1.5). Therefore, we fix two small numbers 0 < ε, β < 1 and consider the following fourth order problem ⎧ 2 2 3 3 4 ⎪ ⎨∂t uε,β + uε,β ∂x uε,β + k β ∂xxx uε,β + β ∂xxxx uε,β 2 2 (2.1) = −β∂xx uε,β + ε∂xx uε,β , t > 0, x ∈ R, ⎪ ⎩ x ∈ R, uε,β (0, x) = uε,β,0 (x), where uε,β,0 is a C ∞ approximation of u0 such that uε,β,0 → u0
p
in Lloc (R), 1 ≤ p < 2, as ε, β → 0,
uε,β,0 2L2 (R) + β∂x uε,β,0 2L2 (R) ≤ C0 ,
(2.2)
ε, β > 0,
and C0 is a constant independent on ε and β. The main result of this section is the following theorem. Theorem 2.1 Assume that (1.9) and (2.2) hold. Fix T > 0, if β = O ε2 ,
(2.3)
then, there exist two sequences {εn }n∈N , {βn }n∈N , with εn , βn → 0, and a limit function u ∈ L∞ (0, T ); L2 (R) , such that uεn ,βn → u
p strongly in Lloc R+ × R , for each 1 ≤ p < 2,
u is an entropy solution of (1.7).
(2.4) (2.5)
Let us prove some a priori estimates on uε,β , denoting with C0 the constants which depend only on the initial datum. Instead of (2.3), we assume directly β≤
ε2 . 2
(2.6)
It is interesting to observe that (2.6) does not depend on the initial datum. We do not have the same condition in [8–10]. Lemma 2.1 Assume (2.3) holds. For each t > 0, t t
uε,β (t, ·) 2 2 + ε ∂x uε,β (s, ·) 2 2 ds + 2β 3 ∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 . (2.7) xx L (R) L (R) L (R) 0
0
Convergence of Kuramoto–Sinelshchikov Type Equation
Proof Multiplying (2.1) by uε,β , an integration on R gives
d
uε,β (t, ·) 2 2 = 2 L (R) dt
R
= −2
uε,β ∂t uε,β dx
R
u2ε,β ∂x uε,β dx − 2β 3
− 2k 2 β + 2ε
R
R
R
4 uε,β ∂xxxx uε,β dx
3 uε,β ∂xxx uε,β dx − 2β
R
2 uε,β ∂xx uε,β dx
2 uε,β ∂xx uε,β dx
2
2
2 uε,β (t, ·) L2 (R) + 2β ∂x uε,β (t, ·) L2 (R) = −2β 3 ∂xx
2 − 2ε ∂x uε,β (t, ·) L2 (R) . Hence,
d
uε,β (t, ·) 2 2 + 2ε ∂x uε,β (t, ·) 2 2 (R) L L (R) dt
2 2 2 + 2β 3 ∂xx uε,β (t, ·) L2 (R) = 2β ∂x uε,β (t, ·) L2 (R) .
(2.8)
Since 0 < ε < 1, it follows from (2.6) that
2
2
2 2β ∂x uε,β (t, ·) L2 (R) ≤ ε 2 ∂x uε,β (t, ·) L2 (R) ≤ ε ∂x uε,β (t, ·) L2 (R) .
(2.9)
Equation (2.8) and (2.9) give
d
uε,β (t, ·) 2 2 + ε ∂x uε,β (t, ·) 2 2 + 2β 3 ∂ 2 uε,β (t, ·) 2 2 = 0. xx L L L (R) (R) (R) dt
(2.10)
Equation (2.7) follows from (2.2), and (2.10).
Lemma 2.2 Fix T > 0. Assume (2.3) holds. There exists C0 > 0, independent on ε, β such that 1
uε,β L∞ ((0,T )×R) ≤ C0 β − 2 .
(2.11)
2
β 2ε t
∂ 2 uε,β (s, ·) 2 2 ds
β ∂x uε,β (t, ·) L2 (R) + L (R) 2 0 xx t
3
2 + 2β 5 ∂xxx uε,β (s, ·) L2 (R) ds ≤ C0 .
(2.12)
Moreover 2
0
2 Proof Let 0 < t < T . Multiplying (2.1) by −2β∂xx uε,β , we have 2 2 2 3 − 2β∂xx uε,β ∂t uε,β − 2βuε,β ∂x uε,β ∂xx uε,β − 2k 2 β 2 ∂xx uε,β ∂xxx uε,β 2 2 2 3 2 2 − 2β 4 ∂xx uε,β ∂xxx uε,β = −2βε ∂xx uε,β + 2β 2 ∂xx uε,β .
(2.13)
G.M. Coclite, L. di Ruvo
Since −2β
R
−2k 2 β 2 −2β
R
4 R
2 ∂xx uε,β ∂t uε,β dx = β
d
∂x uε,β (t, ·) 2 2 , L (R) dt
2 3 ∂xx uε,β ∂xxx uε,β dx = 0,
3
2 2 4 ∂xx uε,β ∂xxxx uε,β dx = 2β 4 ∂xxx uε,β (t, ·) L2 (R) ,
integrating (2.13) on R, we get β
d
∂x uε,β (t, ·) 2 2 + 2βε ∂ 2 uε,β (t, ·) 2 2 + 2β 4 ∂ 3 uε,β (t, ·) 2 2 xx xxx L L L (R) (R) (R) dt
2
2 2 = 2β 2 ∂xx uε,β (t, ·) L2 (R) + 2β uε,β ∂x uε,β ∂xx uε,β dx. (2.14) R
Since 0 < ε < 1, due to (2.6),
2
2
2
2 2β 2 ∂xx uε,β (t, ·) L2 (R) ≤ βε 2 ∂xx uε,β (t, ·) L2 (R)
2
2 ≤ βε ∂xx uε,β (t, ·) L2 (R) .
(2.15)
Thanks to (2.6) and the Young inequality 2 2β |uε,β ||∂x uε,β | ∂xx uε,β dx R
2uε,β ∂x uε,β 1 2 ε 2 ∂ uε,β dx xx 1 R ε2
2β βε
∂ 2 uε,β (t, ·) 2 2 ≤ u2ε,β (∂x uε,β )2 dx + xx L (R) ε R 2
2
βε
∂ 2 uε,β (t, ·) 2 2 . ≤ εuε,β 2L∞ ((0,T )×R) ∂x uε,β (t, ·) L2 (R) + xx L (R) 2
=β
(2.16)
It follows from (2.14), (2.15), and (2.16) that β
d
∂x uε,β (t, ·) 2 2 + βε ∂ 2 uε,β (t, ·) 2 2 xx L L (R) (R) dt 2
2 + 2β 4 ∂ 3 uε,β (t, ·) 2 ≤ εuε,β 2 ∞ xxx
L (R)
L ((0,T )×R)
∂x uε,β (t, ·) 2 2 . L (R)
Integrating on (0, t), from (2.2), we have
2 βε β ∂x uε,β (t, ·) L2 (R) + 2
t
0
∂ 2 uε,β (s, ·) 2 2 ds + 2β 4 xx L (R)
≤
C0 1 + uε,β 2L∞ ((0,T )×R)
t
0
∂ 3 uε,β (s, ·) 2 2 ds xxx L (R)
t
≤ C0 + εuε,β 2L∞ ((0,T )×R)
0
.
∂x uε,β (s, ·) 2 2 ds L (R) (2.17)
Convergence of Kuramoto–Sinelshchikov Type Equation
We prove (2.11). Due to (2.14), (2.17), and the Hölder inequality, u2ε,β (t, x) = 2
x
−∞
uε,β ∂x uε,β dy ≤ 2
R
|uε,β ∂x uε,β |dx
≤ 2 uε,β (t, ·) L2 (R) ∂x uε,β (t, ·) L2 (R) C0 ≤ 1 1 + uε,β 2L∞ ((0,T )×R) , β2 that is uε,β 4L∞ ((0,T )×R) ≤
C0 1 + uε,β 2L∞ ((0,T )×R) . β
Arguing as [11, Lemma 2.3], we have (2.11). It follows from (2.11) and (2.17) that
2
βε t
∂ 2 uε,β (s, ·) 2 2 ds β ∂x uε,β (t, ·) L2 (R) + xx L (R) 2 0 t
3
2 + 2β 4 ∂xxx uε,β (s, ·) L2 (R) ds ≤ C0 β −1 , 0
which gives (2.12). To prove Theorem 2.1. The following technical lemma is needed [35].
Lemma 2.3 Let Ω be a bounded open subset of R2 . Suppose that the sequence {Ln }n∈N of distributions is bounded in W −1,∞ (Ω). Suppose also that Ln = L1,n + L2,n , −1 where {L1,n }n∈N lies in a compact subset of Hloc (Ω) and {L2,n }n∈N lies in a bounded subset −1 (Ω). of Mloc (Ω). Then {Ln }n∈N lies in a compact subset of Hloc
Moreover, we consider the following definition. Definition 2.1 A pair of functions (η, q) is called an entropy–entropy flux pair if η : R → R is a C 2 function and q : R → R is defined by q(u) =
1 2
u
ξ η (ξ )dξ.
0
An entropy-entropy flux pair (η, q) is called convex/compactly supported if, in addition, η is convex/compactly supported. Now, we are ready for the proof of Theorem 2.1. Proof of Theorem 2.1. Let us consider a compactly supported entropy–entropy flux pair (η, q). Multiplying (2.1) by η (uε,β ), we have
G.M. Coclite, L. di Ruvo 2 2 ∂t η(uε,β ) + ∂x q(uε,β ) = εη (uε,β )∂xx uε,β − βη (uε,β )∂xx uε,β 3 4 − k 2 β 2 η (uε,β )∂xxx uε,β − β 3 η (uε,β )∂xxxx uε,β
= I1,ε,β + I2,ε,β + I3,ε,β + I4,ε,β + I5,ε,β + I6,ε,β + I7,ε,β + I8,ε,β , where I1,ε,β = ∂x εη (uε,β )∂x uε,β , I2,ε,β = −εη
(uε,β )(∂x uε,β )2 , I3,ε,β = −∂x βη (uε,β )∂x uε,β , I4,ε,β = βη
(uε,β )(∂x uε,β )2 , 2 I5,ε,β = −∂x k 2 β 2 η (uε,β )∂xx uε,β ,
(2.18)
2 uε,β , I6,ε,β = k 2 β 2 η
(uε,β )∂x uε,β ∂xx 3
3 I7,ε,β = −∂x β η (uε,β )∂xxx uε,β , 3 uε,β . I8,ε,β = β 3 η
(uε,β )∂x uε,β ∂xxx
Fix T > 0. Arguing as [12, Lemma 3.2], we have that I1,ε,β → 0 in H −1 ((0, T ) × R), and {I2,ε,β }ε,β>0 is bounded in L1 ((0, T ) × R). We claim that in H −1 (0, T ) × R , T > 0, as ε → 0.
I3,ε,β → 0 Due to (2.6) and Lemma 2.1,
βη (uε,β )∂x uε,β 2 2 L ((0,T )×R)
2 2 ≤ β η L∞ (R) ∂x uε,β 2L2 ((0,T )×R)
ε4
η 2 ∞ ∂x uε,β 2 2 L ((0,T )×R) L (R) 4
2 ≤ C0 η L∞ (R) ε 3 → 0. ≤
We have that I4,ε,β → 0
in L1 (0, T ) × R , T > 0, as ε → 0.
Again by (2.6) and Lemma 2.1,
βη (uε,β )(∂x uε,β )2 1 L ((0,T )×R)
≤ β η L∞ (R) ∂x uε,β 2L2 ((0,T )×R)
ε2
η
∞ ∂x uε,β 2 2 L ((0,T )×R) L (R) 2
≤ C0 η ∞ ε → 0. ≤
L (R)
Convergence of Kuramoto–Sinelshchikov Type Equation
We show that
in H −1 (0, T ) × R , T > 0, as ε → 0.
I5,ε,β → 0
From (2.6) and Lemma 2.2,
2 2
k β η (uε,β )∂ 2 uε,β 2 2 xx L ((0,T )×R)
2
2 4 4 2 ≤ k β η L∞ (R) ∂xx uε,β L2 ((0,T )×R)
β 4ε
η 2 ∞ ∂ 2 uε,β 2 2 xx L (R) L ((0,T )×R) ε
2
2
2 ≤ C0 η ∞ ∂ uε,β 2 ε 3 → 0. ≤ k4
xx
L (R)
We obtain that I6,ε,β → 0
L ((0,T )×R)
in L1 (0, T ) × R , T > 0, as ε → 0.
Thanks to (2.6), Lemmas 2.1, 2.2, and the Hölder inequality,
2 2
k β η (uε,β )∂x uε,β ∂ 2 uε,β 1 xx
≤ k 2 β 2 η
L∞ (R)
T
L ((0,T )×R)
R
0
2 |∂x uε,β | ∂xx uε,β dtdx
β 2ε
η ∞ ∂x uε,β L2 ((0,T )×R) ∂ 2 uε,β 2 ≤ k2 xx L (R) L ((0,T )×R) ε
≤ C0 η L∞ (R) ε → 0. We get
in H −1 (0, T ) × R , T > 0, as β → 0.
I7,ε,β → 0 By (2.6) and Lemma 2.2,
3
β η (uε,β )∂ 3 uε,β 2 2 xxx L ((0,T )×R)
3
2 6 2 ≤ β η 2 ∂ uε,β 2 L (R)
xxx
2 ≤ C0 η L2 (R) β → 0. We have I8,ε,β → 0
L ((0,T )×R)
in L1 (0, T ) × R , T > 0, as ε → 0.
Due to (2.6), Lemmas 2.1, 2.2, and the Hölder inequality,
3
β η (uε,β )∂x uε,β ∂ 3 uε,β 1 xxx
≤ β 3 η
L∞ (R)
T 0
L ((0,T )×R)
R
2 |∂x uε,β | ∂xx uε,β dtdx
β ε
η
∞ ∂x uε,β L2 ((0,T )×R) ∂ 2 uε,β 2 xx 1 L (R) L ((0,T )×R) ε2
1 ≤ C0 η ∞ ε 2 → 0.
≤
3 12
L (R)
G.M. Coclite, L. di Ruvo
Therefore, (2.4) follows from Lemmas 2.1, 2.3 and the Lp compensated compactness of [39]. We conclude by proving that u is the unique entropy solution of (1.7). Fix T > 0. Let us consider a compactly supported entropy–entropy flux pair (η, q), and φ ∈ Cc∞ ((0, ∞) × R) a non-negative function. We have to prove that ∞ (2.19) ∂t η(u) + ∂x q(u) φdtdx ≤ 0. 0
R
We define uεn ,βn := un .
(2.20)
We have that ∞ ∂t η(un ) + ∂x q(un ) φdtdx 0
R
∞
= εn 0
R
0
−k
∞
∞
≤ −εn
R
0
0
R
0
R
R
2 (un )∂x un ∂xx un φdtdx
∞ R
∞ R
0
R
0
η
(un )(∂x un )2 φdtdx
0
∞
η
(un )(∂x un )2 φdtdx + k 2 βn2 η
∞
+ βn3
0
R
0
∞
3 ∂x η (un )∂xxx un φdtdx + βn3
∞
βn2
2 ∂x η (un )∂xx un φdtdx + k 2 βn2
η (un )∂x un ∂x φdtdx + βn
∞
+ βn +k
R
0
η
(un )(∂x un )2 φdtdx
R
0
R
0
∞
∂x η (un )∂x un φdtdx + βn
∞
βn2
− βn3
2
R
0
∞
− βn 2
∂x η (un )∂x un φdtdx − εn
2 η
(un )∂x un ∂xx un φdtdx
3 η
(un )∂x un ∂xxx un φdtdx
η (un )∂x un ∂x φdtdx
∞
+ βn3 0
R
2 η (un )∂xx un ∂x φdtdx
∞ R
3 η (un )∂xxx un ∂x φdtdx
3 η
(un )∂x un ∂xxx un φdtdx
≤ εn η L∞ (R) ∂x un L2 (supp(∂x φ)) ∂x φL2 (supp(∂x φ))
+ βn η L∞ (R) ∂x un L2 (supp(∂x φ)) ∂x φL2 (supp(∂x φ))
+ βn η
L∞ (R) φL∞ (R+ ×R) ∂x uε,β 2L2 (supp(∂x φ))
2
+ k 2 βn2 η L∞ (R) ∂xx un L2 (supp(∂x φ)) ∂x φL2 (supp(∂x φ))
2 + k 2 βn2 η
L∞ (R) φL∞ (R+ ×R) ∂x un ∂xx un L1 (supp(φ))
3
+ βn3 η L∞ (R) ∂xxx un L2 (supp(∂x φ)) ∂x φL2 (supp(∂x φ))
3 + βn3 η
L∞ (R) φL∞ (R+ ×R) ∂x un ∂xxx un L1 (supp(φ))
≤ εn η L∞ (R) ∂x un L2 ((0,T )×R) ∂x φL2 ((0,T )×R)
Convergence of Kuramoto–Sinelshchikov Type Equation
+ βn η L∞ (R) ∂x un L2 ((0,T )×R) ∂x φL2 ((0,T )×R)
+ βn η
L∞ (R) φL∞ (R+ ×R) ∂x uε,β 2L2 ((0,T )×R)
2
+ k 2 βn2 η L∞ (R) ∂xx un L2 ((0,T )×R) ∂x φL2 ((0,T )×R)
+ k 2 β 2 η
∞ φL∞ (R+ ×R) ∂x un ∂ 2 un 1 n
L (R)
xx
L ((0,T )×R)
3 + βn3 η L∞ (R) ∂xxx un L2 ((0,T )×R) ∂x φL2 ((0,T )×R)
3 + βn3 η
L∞ (R) φL∞ (R+ ×R) ∂x un ∂xxx un L1 ((0,T )×R) .
Equation (2.19) follows from (2.2), (2.4), Lemmas 2.1 and 2.2.
3 The Kuramoto–Sivashinsky Equation: u0 ∈ L2 (R) ∩ L4 (R), β = o(ε 2 ) In this section, we consider (1.6), and assume (1.12) on the initial datum. We study the dispersion-diffusion limit for (1.6). Therefore, we fix two small numbers ε, β and consider the following fourth order problem 3 4 2 uε,β = ε∂xx uε,β , t > 0, x ∈ R, ∂t uε,β + uε,β ∂x uε,β + β 2 ∂xxxx (3.1) uε,β (0, x) = uε,β,0 (x), x ∈ R, where uε,β,0 is a C ∞ approximation of u0 such that uε,β,0 → u0
p
in Lloc (R), 1 ≤ p < 2, as ε, β → 0,
uε,β,0 4L4 (R) + uε,β,0 2L2 (R) + ε 2 ∂x uε,β,0 2L2 (R) ≤ C0 ,
(3.2) ε, β > 0,
and C0 is a constant independent on ε and β. The main result of this section is the following theorem. Theorem 3.1 Assume that (1.12) and (3.2) hold. Fix T > 0, if β = O ε2 ,
(3.3)
then, there exist two sequences {εn }n∈N , {βn }n∈N , with εn , βn → 0, and a limit function u ∈ L∞ R+ ; L2 (R) ∩ L4 (R) , such that (i) uεn ,βn → u strongly in Lloc (R+ × R), for each 1 ≤ p < 4, (ii) u is a distributional solution of (1.7). p
In particular, we have (iii) dissipation of energy, that is, ∂t
u2 2
1 + ∂x u3 ≤ 0, 3
in weak sense on R+ × R.
(3.4)
G.M. Coclite, L. di Ruvo
Remark 3.1 Thanks to the L4loc summability of u and the energy dissipation stated in (iii), [18] guarantees that u is the unique entropy solution of (1.7). Let us prove some a priori estimates on uε,β , denoting with C0 the constants which depend only on the initial data. Lemma 3.1 For each t > 0,
uε,β (t, ·) 2 + 2ε
t
0
∂x uε,β (s, ·) 2 2 ds + 2β 32 L (R)
t
0
∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 . xx L (R)
(3.5)
Proof Multiplying (3.1) by 2uε,β , an integration on R gives
d
uε,β (t, ·) 2 2 = 2 L (R) dt
R
= −2
uε,β ∂t uε,β dx
R
u2ε,β ∂x uε,β dx + 2ε
3
− 2β 2
R
R
2 uε,β ∂xx uε,β dx
4 uε,β ∂xxxx uε,β dx
2
2 3 2 = −2ε ∂x uε,β (t, ·) L2 (R) − 2β 2 ∂xx uε,β (t, ·) L2 (R) . Hence,
d
uε,β (t, ·) 2 2 + 2ε ∂x uε,β (t, ·) 2 2 + 2β 32 ∂ 2 uε,β (t, ·) 2 2 = 0. xx L (R) L (R) L (R) dt
Integrating on (0, t), from (3.2), we have (3.5). Lemma 3.2 Fix T > 0. Then:
(i) the family {uε,β }ε,β is bounded in L∞ (R+ ; L4 (R)); (ii) the family {ε∂x uε,β }ε,β is bounded in L∞ (R+ ; L2 (R)); 3 1 3 3 3 2 2 uε,β }ε,β , {ε 2 uε,β ∂x uε,β }ε,β , {ε 2 ∂xx uε,β }ε,β , {β 4 uε,β ∂xx uε,β }ε,β are (iii) the families {β 4 ε∂xxx 2 + bounded in L (R × R). 2 Proof Multiplying (3.1) by u3 − 2ε 2 ∂xx uε,β , we have
2 2 u3 − 2ε 2 ∂xx uε,β ∂t uε,β + u3 − 2ε 2 ∂xx uε,β uε,β ∂x uε,β 4 2 3 2 2 + β 2 u3 − 2ε 2 ∂xx uε,β ∂xxxx uε,β = ε u3 − 2ε 2 ∂xx uε,β ∂xx uε,β .
Since
4
2 1
2
uε,β (t, ·) L4 (R) + ε ∂x uε,β (t, ·) L2 (R) , u 4 R 3 2 2 uε,β uε,β ∂x uε,β dx = −2ε 2 uε,β ∂x uε,β ∂xx uε,β dx, u − 2ε 2 ∂xx
R
3
2 − 2ε 2 ∂xx uε,β
d ∂t uε,β dx = dt
R
(3.6)
Convergence of Kuramoto–Sinelshchikov Type Equation 3
β2 R
4 2 uε,β ∂xxxx uε,β dx u3 − 2ε 2 ∂xx
3
= −3β 2 ε R
R
3
2 3 3 u2ε,β ∂x uε,β ∂xxx uε,β dx + 2β 2 ε 2 ∂xxx uε,β (t, ·) L2 (R) ,
3 2 2 uε,β ∂xx uε,β dx u − 2ε 2 ∂xx
2
2
2 = −3ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R) − 2ε 3 ∂xx uε,β (t, ·) L2 (R) ,
an integration of (3.6) on R gives
d dt
1
uε,β (t, ·) 4 4 + ε 2 ∂x uε,β (t, ·) 2 2 L (R) L (R) 4
3
2
2 3 + 2β 2 ε 2 ∂xxx uε,β (t, ·) L2 (R) + 3ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R)
2
2 + 2ε 3 ∂xx uε,β (t, ·) L2 (R) 3 2 3 = 2ε 2 uε,β ∂x uε,β ∂xx uε,β dx + 3β 2 u2ε,β ∂x uε,β ∂xxx uε,β dx. R
(3.7)
R
Observe that 3β
3 2
R
3 u2ε,β ∂x uε,β ∂xxx uε,β dx
3
= −6β 2 3
= 2β 2
R
R
2 3
2 2 uε,β (∂x uε,β )2 ∂xx uε,β dx − 3β 2 uε,β (t, ·)∂xx uε,β (t, ·) L2 (R)
2 3
2 (∂x uε,β )4 dx − 3β 2 uε,β (t, ·)∂xx uε,β (t, ·) L2 (R) .
Therefore, from (3.7), we get d dt
1
uε,β (t, ·) 4 4 + ε 2 ∂x uε,β (t, ·) 2 2 L (R) L (R) 4
3
2
2 3 + 2β 2 ε 2 ∂xxx uε,β (t, ·) L2 (R) + 3ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R)
2
2
2 3
2 + 2ε 3 ∂xx uε,β (t, ·) L2 (R) + 3β 2 uε,β (t, ·)∂xx uε,β (t, ·) L2 (R) 3 2 = 2ε 2 uε,β ∂x uε,β ∂xx uε,β dx + 2β 2 (∂x uε,β )4 dx. R
(3.8)
R
Due to the Young inequality, 2ε 2 R
2 |uε,β ∂x uε,β | ∂xx uε,β dx = 2
R
1 ε 2 uε,β ∂x uε,β ε 32 ∂ 2 uε,β dx xx
2
2
2 ≤ ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R) + ε 3 ∂xx uε,β (t, ·) L2 (R) .
G.M. Coclite, L. di Ruvo
Then, from (3.8),
d 1
uε,β (t, ·) 4 4 + ε 2 ∂x uε,β (t, ·) 2 2 L (R) L (R) dt 4
2 3 2 3 + 2β 2 ε 2 ∂xxx uε,β (t, ·) L2 (R) + 2ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R)
2
2
2 3
2 + ε 3 ∂xx uε,β (t, ·) L2 (R) + 3β 2 uε,β (t, ·)∂xx uε,β (t, ·) L2 (R) 3 ≤ 2β 2 (∂x uε,β )4 dx.
(3.9)
R
To estimate the second term of (3.9), we use the following inequality (see [15, Lemma 4.2])
2
2
2 (∂x uε,β )4 dx ≤ c1 uε,β (t, ·) L2 (R) ∂xx uε,β (t, ·) L2 (R) , (3.10) R
where c1 is some universal constant. Due to (3.5) and (3.10),
2
2
2 3 3 β 2 (∂x uε,β )4 dx ≤ β 2 c1 uε,β (t, ·) L2 (R) ∂xx uε,β (t, ·) L2 (R) R
2
2 3 ≤ β 2 C0 ∂xx uε,β (t, ·) L2 (R) .
Hence, it follows from (3.9) that
d 1
uε,β (t, ·) 4 4 + ε 2 ∂x uε,β (t, ·) 2 2 L (R) L (R) dt 4
2 3 2 3 + 2β 2 ε 2 ∂xxx uε,β (t, ·) L2 (R) + 2ε uε,β (t, ·)∂x uε,β (t, ·) L2 (R)
2
2
2 3
2 + ε 3 ∂xx uε,β (t, ·) L2 (R) + 3β 2 uε,β (t, ·)∂xx uε,β (t, ·) L2 (R)
2
2 3 ≤ β 2 C0 ∂xx uε,β (t, ·) L2 (R) . Equations (3.2), (3.5) and an integration on (0, t) give
1
uε,β (t, ·) 4 4 + ε 2 ∂x uε,β (t, ·) 2 2 L L (R) (R) 4 t t
3
2
2 3 + 2β 2 ε 2 ∂xxx uε,β (s, ·) L2 (R) ds + 2ε uε,β (s, ·)∂x uε,β (s, ·) L2 (R) ds 0
t
+ ε3
∂ 2 uε,β (s, ·) 2 2 ds + 3β 32 xx L (R)
0 3 2
0
t
0
uε,β (s, ·)∂ 2 uε,β (s, ·) 2 2 ds xx L (R)
t
≤ C0 + β C0 0
∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 . xx L (R)
Hence,
uε,β (t, ·) 4 ≤ C0 , L (R)
ε ∂x uε,β (t, ·) L2 (R) ≤ C0 ,
Convergence of Kuramoto–Sinelshchikov Type Equation
3
t
∂ 3 uε,β (s, ·) 2 2 ds ≤ C0 , xxx L (R)
β 2 ε2 0
t
uε,β (s, ·)∂x uε,β (s, ·) 2 2 ds ≤ C0 , L (R)
ε 0
ε3
3
0
t
∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 , xx L (R)
t
β2
uε,β (s, ·)∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 . xx L (R)
0
Therefore, the proof is concluded. We prove the main result.
Proof of Theorem 3.1. Let us consider a compactly supported entropy–entropy flux pair (η, q). Multiplying (3.1) by η (uε,β ), we have 3
2 4 ∂t η(uε,β ) + ∂x q(uε,β ) = εη (uε,β )∂xx uε,β − β 2 η (uε,β )∂xxxx uε,β
= I1,ε,β + I2,ε,β + I3,ε,β + I4,ε,β , where I1,ε,β = ∂x εη (uε,β )∂x uε,β , I2,ε,β = −εη
(uε,β )(∂x uε,β )2 , 3 3 I3,ε,β = −∂x β 2 η (uε,β )∂xxx uε,β ,
(3.11)
3
3 uε,β . I4,ε,β = β 2 η
(uε,β )∂x uε,β ∂xxx
Fix T > 0. Arguing as [12, Lemma 3.2], we have that I1,ε,β → 0 in H −1 ((0, T ) × R), and {I2,ε,β }ε,β>0 is bounded in L1 ((0, T ) × R). We claim that I3,ε,β → 0
in H −1 (0, T ) × R , T > 0, as ε → 0.
By (3.3) and Lemma 3.2,
3
β 2 η (uε,β )∂ 3 uε,β 2 2 xxx L ((0,T )×R)
2 3 ≤ β 3 η L∞ (R) ∂xxx uε,β L2 ((0,T )×R)
2
β 3 ε2 3 = η L∞ (R) 2 ∂xxx uε,β L2 ((0,T )×R) ε
β 2 β 2 ε2
∂ 3 uε,β 2 2 = η L∞ (R) ≤ C0 η L∞ (R) ε → 0. xxx ((0,T )×R) L 2 ε 3
We have that {I4,ε,β }ε,β>0
3
is bounded in L1 (0, T ) × R , T > 0.
G.M. Coclite, L. di Ruvo
Thanks to (3.3), Lemmas 3.1, 3.2 and the Hölder inequality,
3
β 2 η (uε,β )∂x uε,β ∂ 3 uε,β 1 xxx L ((0,T )×R) T
3
∂x uε,β ∂ 3 uε,β dtdx ≤ β 2 η
L∞ (R) xxx R
0
3
β ε = η
L∞ (R) 3 ∂x uε,β L2 ((0,T )×R) ∂xxx uε,β L2 ((0,T )×R) ε2 3 2
3 2
3
β4β4ε2 = η
L∞ (R) ∂x uε,β L2 ((0,T )×R) ∂xxx uε,β L2 ((0,T )×R) ≤ C0 η
L∞ (R) . 3 ε2 3
3
3
Therefore, (2.4) follows from Lemmas 3.1, 3.2, 2.3 and the Lp compensated compactness of [39]. We begin by proving that u is a distributional solution of (1.7). Let φ ∈ C ∞ (R2 ) be a test function with compact support. We have to prove that
∞
R
0
u2 u∂t φ + ∂x φ dtdx + u0 (x)φ(0, x)dx = 0. 2 R
(3.12)
Using the notation of (2.20), from (3.1), we have that u2n un ∂t φ + ∂x φ dtdx + u0,n (x)φ(0, x)dx 2 R R 0 ∞ 2 + εn un ∂xx φdtdx + εn
∞
0 3 2
− βn εn
R
∞ R
0
4 un ∂xxxx φdtdx = 0.
Therefore, (3.12) follows from (3.2) and (i). We prove (3.4). Multiplying (3.1) by uε,β , we have ∂t
u2ε,β
2
1 3 2 4 uε,β − β 2 uε,β ∂xxxx uε,β . + ∂x u3ε,β = εuε,β ∂xx 3
(3.13)
Let φ ∈ Cc∞ ((0, ∞) × R) be a non-negative test function. Fix T > 0. Multiplying (3.13) by φ, we get
2 uε,β 1 3 ∂t + ∂x uε,β φdtdx 2 3 R 0 ∞ 3 2 uε,β ∂xx uε,β φdtdx − β 2 =ε ∞
R
0
∞
= −ε 0 3
+β2 0
∞
(∂x uε,β ) φdtdx − ε 0
∞
R
0
2
R
∞
R
uε,β ∂x uε,β ∂x φdtdx 3
R
4 uε,β ∂xxxx uε,β φdtdx
3 ∂x uε,β ∂xxx uε,β φdtdx + β 2 0
∞ R
3 uε,β ∂xxx uε,β ∂x φdtdx
Convergence of Kuramoto–Sinelshchikov Type Equation ∞
= −ε 0
−β
3 2
0
3
+β2
R
∞
≤ −ε 0 3
+β2
R
R
∞ R
2 2 uε,β φdtdx ∂xx
−β
3 2
0
3 2
∞ R
∞ 0
R
uε,β ∂x uε,β ∂x φdtdx 2 ∂x uε,β ∂xx uε,β ∂x φdtdx
3 uε,β ∂xxx uε,β ∂x φdtdx
uε,β ∂x uε,β ∂x φdtdx − β
∞
0
R
0
∞
0
∞
(∂x uε,β )2 φdtdx − ε
R
2 ∂x uε,β ∂xx uε,β ∂x φdtdx
3 uε,β ∂xxx uε,β ∂x φdtdx
≤ ε∂x φL∞ (R+ ×R) uε,β L2 (supp(∂x φ)) ∂x uε,β L2 (supp(∂x φ))
2
3 + β 2 ∂x φL∞ (R+ ×R) ∂x uε,β L2 (supp(∂x φ)) ∂xx uε,β L2 (supp(∂x φ))
3
3 + β 2 ∂x φL∞ (R+ ×R) uε,β L2 (supp(∂x φ)) ∂xxx uε,β L2 (supp(∂x φ)) ≤ ε∂x φL∞ (R+ ×R) uε,β L2 ((0,T ))×R ∂x uε,β L2 ((0,T )×R)
2
3 + β 2 ∂x φL∞ (R+ ×R) ∂x uε,β L2 ((0,T )×R) ∂xx uε,β L2 ((0,T )×R)
3
3 + β 2 ∂x φL∞ (R+ ×R) uε,β L2 ((0,T )×R) ∂xxx uε,β L2 ((0,T )×R) .
(3.14)
We have that ε∂x φL∞ (R+ ×R) uε,β L2 ((0,T ))×R ∂x uε,β L2 ((0,T )×R) → 0.
(3.15)
Due to Lemma 3.1, 1
1
ε∂x φL∞ (R+ ×R) uε,β L2 ((0,T ))×R ∂x uε,β L2 ((0,T )×R) ≤ ε 2 ∂x φL∞ (R+ ×R) C0 T 2 → 0. We get
2
3 uε,β L2 ((0,T )×R) → 0. β 2 ∂x φL∞ (R+ ×R) ∂x uε,β L2 ((0,T )×R) ∂xx
(3.16)
Thanks to (3.3), and Lemma 3.1,
2
3 β 2 ∂x φL∞ (R+ ×R) ∂x uε,β L2 ((0,T )×R) ∂xx uε,β L2 ((0,T )×R) 3
=
1
β2ε2 ε
1 2
2
∂x φL∞ (R+ ×R) ∂x uε,β L2 ((0,T )×R) ∂xx uε,β L2 ((0,T )×R) 3
≤ C0 ∂x φL∞ (R+ ×R) ε 2 → 0. We obtain that
3
3 β 2 ∂x φL∞ (R+ ×R) uε,β L2 ((0,T )×R) ∂xxx uε,β L2 ((0,T )×R)
(3.17)
G.M. Coclite, L. di Ruvo
Due to (3.3), and Lemmas 3.1, 3.2,
3
3 uε,β L2 ((0,T )×R) β 2 ∂x φL∞ (R+ ×R) uε,β L2 ((0,T )×R) ∂xxx
3
β2ε ∂x φL∞ (R+ ×R) uε,β L2 ((0,T )×R) ∂xxx uε,β L2 ((0,T )×R) ε 3
=
1
≤ C0 ∂x φL∞ (R+ ×R) ε 2 → 0. It follows from (i), (3.14), (3.15), (3.16), and (3.17) that ∞ 2 1 u ∂t φ + ∂x u3 ∂x φ dtdx ≥ 0, 3 R 2 0
that is (3.4).
4 The Kuramoto–Sivashinsky Equation: u0 ∈ L2 (R), β = O(ε 4 ) In this section, we consider (1.6), and assume (1.11) on the initial datum. We consider the approximate problem (3.1), where uε,β,0 is a C ∞ approximation of u0 such that uε,β,0 → u0
p
in Lloc (R), 1 ≤ p < 2, as ε, β → 0,
uε,β,0 2L2 (R) + β∂x uε,β,0 2L2 (R) ≤ C0 ,
(4.1)
ε, β > 0,
and C0 is a constant independent on ε and β. The main result of this section is the following theorem. Theorem 4.1 Assume that (1.9) and (4.1) hold. Fix T > 0, if β = O ε4 ,
(4.2)
then, there exist two sequences {εn }n∈N , {βn }n∈N , with εn , βn → 0, and a limit function u ∈ L∞ (0, T ); L2 (R) , such that (2.4) and (2.5) hold. Let us prove some a priori estimates on uε,β , denoting with C0 the constants which depend only on the initial data. Lemma 4.1 Fix T > 0. Assume (4.2). There exists C0 > 0, independent on ε, β, such that 1
uε,β L∞ ((0,T )×R) ≤ C0 β − 4 .
(4.3)
Moreover, for every 0 < t < T ,
2 β ∂x uε,β (t, ·) L2 (R) + βε + 2β
5 2
t
0
t
0
∂ 2 uε,β (s, ·) 2 2 ds xx L (R)
∂ 3 uε,β (s, ·) 2 2 ds ≤ C0 . xxx L (R)
(4.4)
Convergence of Kuramoto–Sinelshchikov Type Equation 1
2 Proof Let 0 < t < T . Multiplying (3.1) by −2β 2 ∂xx uε,β , integrating on R, we have that
2 1 d
1 2 β 2 ∂x uε,β (t, ·) L2 (R) = −2β 2 ∂xx uε,β ∂t uε,β dx dt R 1 2 2 4 = 2β 2 uε,β ∂x uε,β ∂xx uε,β dx + 2β 2 ∂xx uε,β ∂xxxx uε,β dx R
R
2
2 − 2β ε ∂xx uε,β (t, ·) L2 (R)
3
2 1 2 2 = 2β uε,β ∂x uε,β ∂xx uε,β dx − 2β 2 ∂xxx uε,β (t, ·) L2 (R) 1 2
R
2 1 2 − 2β 2 ε ∂xx uε,β (t, ·) L2 (R) . Hence, 1
β2
d
∂x uε,β (t, ·) 2 2 + 2β 12 ε ∂ 2 uε,β (t, ·) 2 2 xx L (R) L (R) dt
1 2 3 2 + 2β 2 ∂xxx uε,β (t, ·) L2 (R) = 2β 2 uε,β ∂x uε,β ∂xx uε,β dx.
(4.5)
R
Due (4.2) and the Young inequality, 2 1 |uε,β ∂x uε,β | ∂xx uε,β dx 2β 2 R
uε,β ∂x uε,β 1 2 ε 2 ∂ uε,β dx = 2β xx 1 R ε2 1
2 β2 1 2 u2ε,β (∂x uε,β )2 dx + β 2 ε ∂xx uε,β (t, ·) L2 (R) ≤ ε R
2
2 1 2 ≤ C0 εuε,β 2L∞ ((0,T )×R) ∂x uε,β (t, ·) L2 (R) + β 2 ε ∂xx uε,β (t, ·) L2 (R) . 1 2
Therefore, from (4.5), we gain 1
β2
d
∂x uε,β (t, ·) 2 2 + β 12 ε ∂ 2 uε,β (t, ·) 2 2 xx L L (R) (R) dt
2 2 3 + 2β 2 ∂xxx uε,β (t, ·) L2 (R) ≤ C0 εuε,β 2L∞ ((0,T )×R) ∂x uε,β (t, ·) L2 (R) .
Integrating on (0, t), from (4.1) and (3.5), we get 1
β2
t
d
∂x uε,β (t, ·) 2 2 + β 12 ε ∂ 2 uε,β (s, ·) 2 2 ds xx L L (R) (R) dt 0 t
3
2 + 2β 2 ∂xxx uε,β (s, ·) L2 (R) ds 0
≤ C0 + C0 εuε,β 2L∞ ((0,T )×R) ≤
C0 1 + uε,β 2L∞ ((0,T )×R)
0
.
t
∂x uε,β (s, ·) 2 2 ds L (R) (4.6)
G.M. Coclite, L. di Ruvo
We prove (4.3). Due to (3.5), (4.6) and the Hölder inequality, x u2ε,β (t, x) = 2 uε,β ∂x uε,β dy ≤ 2 |uε,β ||∂x uε,β |dx −∞
R
2
≤ 2 uε,β (t, ·) L2 (R) ∂x uε,β (t, ·) L2 (R) C 0 ≤ 1 1 + uε,β 2L∞ ((0,T )×R) , 4 β that is uε,β 4L∞ (R) ≤
C0 β
1 2
1 + uε,β 2L∞ ((0,T )×R) .
Arguing as [10, Lemma 2.2], we have (4.3). It follows from (4.3) and (4.6) that 1
β2
t
d
∂x uε,β (t, ·) 2 2 + β 12 ε ∂ 2 uε,β (s, ·) 2 2 ds xx L L (R) (R) dt 0 t
3
2 1 + 2β 2 ∂xxx uε,β (s, ·) L2 (R) ds ≤ C0 β − 2 , 0
which gives (4.4). We are ready for the proof of Theorem 4.1.
Proof of Theorem 4.1 Let us consider a compactly supported entropy–entropy flux pair (η, q). Multiplying (3.1) by η (uε,β ), we have 3
2 4 ∂t η(uε,β ) + ∂x q(uε,β ) = εη (uε,β )∂xx uε,β − β 2 η (uε,β )∂xxxx uε,β
= I1,ε,β + I2,ε,β + I3,ε,β + I4,ε,β , where I1,ε,β , I2,ε,β , I3,ε,β , I4,ε,β are defined in (3.11). Fix T > 0. Arguing as [12, Lemma 3.2], we have that I1,ε,β → 0 in H −1 ((0, T ) × R), and {I2,ε,β }ε,β>0 is bounded in L1 ((0, T ) × R). We claim that I3,ε,β → 0 in H −1 (0, T ) × R , T > 0, as β → 0. By Lemma 4.1,
3
β 2 η (uε,β )∂ 3 uε,β 2 2 xxx L ((0,T )×R)
2 3 ≤ β 3 η L∞ (R) ∂xxx uε,β L2 ((0,T )×R)
2 1 5
= β 2 β 2 η ∞ ∂ 3 uε,β 2 L (R)
xxx
L ((0,T )×R)
1 ≤ C0 η L∞ (R) β 2 → 0.
We continue by showing that I4,ε,β → 0
in L1 (0, T ) × R , T > 0, as ε → 0.
Convergence of Kuramoto–Sinelshchikov Type Equation
By (4.2), Lemmas 3.1 4.1, and The Hölder inequality,
3
β 2 η (uε,β )∂x uε,β ∂ 3 uε,β 1 xxx L ((0,T )×R) T
3
∂x uε,β ∂ 3 uε,β ≤ β 2 η
L∞ (R) xxx 0
≤
R
β ε
η
∞ ∂x uε,β L2 ((0,T )×R) ∂ 3 uε,β 2 xxx 1 L (R) L ((0,T )×R) ε2 3 2
1 2
β4 1 ≤ C0 η
L∞ (R) 1 ≤ C0 η L∞ (R) ε 2 → 0. ε2 1
Therefore, (2.4) follows from Lemmas 3.1, 2.3 and the Lp compensated compactness of [39]. Arguing as Theorem 2.1, we have (2.5).
Appendix: The Korteweg–de Vries–Burgers Equation In this appendix, we consider the Korteweg–de Vries–Burgers equation, 3 2 ∂t u + u∂x u + ∂xxx u = ∂xx u,
(A.1)
that describes the evolution of non-linear shallow-water waves [3, 19, 41]. Hence, the function u(t, x) is the amplitude of an appropriate linear long wave mode, with linear long wave speed C0 . Arguing as in [14], we re-scale (A.1) as follows 3 2 ∂t u + u∂x u + β 2 ∂xxx u = β∂xx u,
(A.2)
where β is the diffusion parameter. We are interested in the no high frequency limit, we send β → 0 in (A.2). In this way, we pass from (A.2) to (1.7). We augment (A.2) with the initial datum u0 , on which we assume (1.9). We consider the following third order approximation 3 2 2 uε,β = β∂xx uε,β + ε∂xx uε,β , t > 0, x ∈ R, ∂t uε,β + uε,β ∂x uε,β + β 2 ∂xxx (A.3) uε,β (0, x) = uε,β,0 (x), x ∈ R, where uε,β,0 is an C ∞ approximation of u0 , on which (2.2). The main result of this section is the following theorem. Theorem A.1 Assume (1.9) and (2.2) hold. Fix T > 0, if (2.3), then, there exist two sequences {εn }n∈N , {βn }n∈N , with εn , βn → 0, and a limit function u ∈ L∞ (0, T ); L2 (R) , such that (2.4) and (2.5) hold. Let us prove some a priori estimates on uε,β , denoting with C0 the constants which depend only on the initial datum.
G.M. Coclite, L. di Ruvo
Lemma A.1 For each t > 0,
uε,β (t, ·) 2 2 + 2(β + ε) L (R)
t
∂x uε,β (s, ·) 2 2
L (R)
0
ds ≤ C0 .
(A.4)
Proof Multiplying (A.3) by uε,β , an integration on R gives
d
uε,β (t, ·) 2 2 = 2 uε,β ∂t uε,β dx L (R) dt R 3 uε,β dx = −2 u2ε,β ∂x uε,β dx + 2β 2 uε,β ∂xxx R
+ 2(β + ε)
R
R
2 uε,β ∂xx uε,β dx
2
= −2(β + ε) ∂x uε,β (t, ·) L2 (R) . Hence,
d
uε,β (t, ·) 2 2 + 2(β + ε) ∂x uε,β (t, ·) 2 2 = 0. L L (R) (R) dt Integrating on (0, t), from (2.2), we have (A.4).
Lemma A.2 Let T > 0. There exists C0 > 0, independent on ε, β, such that (2.11) holds. In particular, we have t
2
2
2 2
2
uε,β (s, ·) L2 (R) ds β ∂x uε,β (t, ·) L2 (R) + β ε ∂xx 0
t
+ 2β 3 0
∂ 2 uε,β (s, ·) 2 2 ds ≤ C0 . xx L (R)
(A.5)
2 uε,β , an integration on R gives Proof Let 0 < t < T . Multiplying (A.3) by −2β∂xx
2 d
2 uε,β ∂t uε,β dx β ∂x uε,β (t, ·) L2 (R) = −2β ∂xx dt R 2 2 3 uε,β dx + 2β 3 ∂xx uε,β ∂xxx uε,β dx = −2β uε,β ∂x uε,β ∂xx R
R
2
2
2
2 uε,β (t, ·) L2 (R) − 2β 2 ∂xx uε,β (t, ·) L2 (R) − 2βε ∂xx
2
2 2 = −2β uε,β ∂x uε,β ∂xx uε,β dx − 2βε ∂xx uε,β (t, ·) L2 (R) R
2 2 − 2β 2 ∂xx uε,β (t, ·) L2 (R) .
Hence, β
d
∂x uε,β (t, ·) 2 2 + 2βε ∂ 2 uε,β (t, ·) 2 2 xx L L (R) (R) dt
2
2 2 + 2β 2 ∂xx uε,β (t, ·) L2 (R) = −2β uε,β ∂x uε,β ∂xx uε,β dx. R
(A.6)
Convergence of Kuramoto–Sinelshchikov Type Equation
Due to (2.3), and the Young inequality, 2 uε,β dx 2β |uε,β ∂x uε,β | ∂xx R
uε,β ∂x uε,β 1 2 ε 2 ∂ uε,β dx = 2β xx 1 R ε2
2
2 β ≤ u2ε,β (∂x uε,β )2 dx + βε ∂xx uε,β (t, ·) L2 (R) ε R
2
2
2 ≤ D 2 εuε,β 2L∞ ((0,T )×R) ∂x uε,β (t, ·) L2 (R) + βε ∂xx uε,β (t, ·) L2 (R) . Therefore, from (A.6), β
d
∂x uε,β (t, ·) 2 2 + βε ∂ 2 uε,β (t, ·) 2 2 xx L L (R) (R) dt
2 + 2β 2 ∂ 2 uε,β (t, ·) 2 ≤ D 2 εuε,β 2 ∞ xx
L ((0,T )×R)
L (R)
∂x uε,β (t, ·) 2 2 . L (R)
Equations (2.2), (A.4) and an integration on (0, t) give t
2
2
2 β ∂x uε,β (t, ·) L2 (R) + βε ∂xx uε,β (s, ·) L2 (R) ds 0
t
+ 2β 2 0
≤ C0 + D ≤
2
∂ 2 uε,β (s, ·) 2 2 ds xx L (R)
t
εuε,β 2L∞ ((0,T )×R)
C0 1 + uε,β 2L∞ ((0,T )×R)
0
∂x uε,β (s, ·) 2 2 ds L (R) (A.7)
.
We prove (2.11). Due to (A.4), (A.7), and the Hölder inequality, x u2ε,β (t, x) = 2 uε,β ∂x uε,β dy ≤ 2 |uε,β ∂x uε,β |dx R
−∞
≤ 2 uε,β (t, ·) L2 (R) ∂x uε,β (t, ·) L2 (R) C0 ≤ 1 1 + uε,β 2L∞ ((0,T )×R) , β2 that is uε,β 2L∞ ((0,T )×R) ≤
C0 1 + uε,β 2L∞ ((0,T )×R) . β
Arguing as [11, Lemma 2.3], we have (2.11). It follows from (2.11), and (A.7) that
2
β ∂x uε,β (t, ·) L2 (R) + βε
t
0
2
2 + 2β 2 ∂xx uε,β (t, ·) L2 (R) which gives (A.5).
∂ 2 uε,β (s, ·) 2 2 ds xx L (R)
≤ C0 β −1 ,
G.M. Coclite, L. di Ruvo
Proof of Theorem A.1. Let us consider a compactly supported entropy–entropy flux pair (η, q). Multiplying (A.3) by η (uε,β ), we have 2 2 3 uε,β + βη (uε,β )∂xx uε,β − β 2 η (uε,β )∂xxx uε,β ∂t η(uε,β ) + ∂x q(uε,β ) = εη (uε,β )∂xx
= I1,ε,β + I2,ε,β + I3,ε,β + I4,ε,β + I5,ε,β + I6,ε,β , where I1,ε,β = ∂x εη (uε,β )∂x uε,β , I2,ε,β = −εη
(uε,β )(∂x uε,β )2 , I3,ε,β = ∂x βη (uε,β )∂x uε,β , I4,ε,β = −βη
(uε,β )(∂x uε,β )2 , 2 I5,ε,β = −∂x β 2 η (uε,β )∂xx uε,β ,
(A.8)
2 uε,β . I6,ε,β = β 2 η
(uε,β )∂x uε,β ∂xx
Arguing as Theorem 2.1, we have I1,ε,β → 0 in H −1 ((0, T ) × R), {I2,ε,β }ε,β>0 is bounded in L1 ((0, T ) × R), I3,ε,β → 0 in H −1 ((0, T ) × R), I4,ε,β → 0 in L1 ((0, T ) × R), I5,ε,β → 0 in H −1 ((0, T ) × R), and I6,ε,β → 0 in L1 ((0, T ) × R). Arguing as Theorem 2.1, the proof is concluded.
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