The effect of dissipation on solutions of the complex KdV equation

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1 Mathematics an Computers in Simulation 69 (25) The effect of issipation on solutions of the complex KV equation Jiahong Wu a,, Juan-Ming Yuan a,b a Department of Mathematics, Oklahoma State University, Stillwater, OK 7478, USA b Department of Applie Mathematics, Provience University, 2 Chung-Chi Roa, Shalu 433, Taichung Hsien, Taiwan Available online 13 June 25 Abstract It is known that some perioic solutions of the complex KV equation with smooth initial ata blow up in finite time. In this paper, we investigate the effect of issipation on the regularity of solutions of the complex KV equation. It is shown here that if the initial atum is comparable to the issipation coefficient in the 2 -norm, then the corresponing solution oes not evelop any finite-time singularity. The solution actually ecays exponentially in time an becomes real analytic as time elapses. Numerical simulations are also performe to provie etaile information on the behavior of solutions in ifferent parameter ranges. 25 IMACS. Publishe by Elsevier B.V. All rights reserve. Keywors: Complex KV Burgers equation; Dissipation; Global well-poseness 1. Introuction Consier the complex KV equation with a perioic bounary conition u t + 2uu x + δu xxx =, u(x, t) = u(x + 1,t), (1.1) where δ> is a parameter an u = u(x, t) is complex-value. It is known that there exists a smooth initial atum u such that the solution of the initial-value problem (IVP) of (1.1) with u(x, ) = u (x) (1.2) Corresponing author. aress: jiahong@math.okstate.eu (J. Wu) /$3. 25 IMACS. Publishe by Elsevier B.V. All rights reserve. oi:1.116/j.matcom

2 59 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) blows up in a finite time. In fact, a class of elliptic solutions represente by the Weierstrass p-function evelop finite-time singularities [1,2]. Numerical computations of [6] also inicate finite-time singularities for a special series-type solution. It is worth pointing out that the complex KV equation is much more sophisticate than its real counterpart. For the real KV equation, the hierarchy of infinite conservation laws provies global in time bouns for its solutions in any Sobolev space H k with k [5]. Thus, no finite-time singularity is possible an the real KV equation is well-pose. In contrast, the complex KV equation is equivalent to a system of two nonlinearly couple equations an the conservation laws no longer allow the euction of global bouns. In fact, we o not know if a solution of the IVP (1.1) an (1.2) has a finite 2 -norm for all time even if it is initially in 2. This is precisely where the problem arises. In this paper, we conuct a theoretical an numerical stuy to investigate the effects of issipation on the regularity of solutions of the complex KV equation. More precisely, we examine solutions of the issipative complex KV equation of the form u t + 2uu x + δu xxx + ν( ) α u =, u(x, t) = u(x + 1,t), (1.3) where ν> an ( ) α is a Fourier multiplier operator, namely for each l Z ( ) α u(l) = 2πl 2α û(l), where the Fourier transform û is efine by û(l, t) = 1 e i2πxl u(x, t)x, an u can be recovere from û through the inverse transform, namely u(x, t) = l= e i2πlx û(l, t). When α = 1, the issipation becomes the Burgers term νu xx. It was shown in a previous work [6] that the 2 -norm of any solution of the complex KV Burgers equation ominates its norms in H k for any k 1. Therefore, any solution u of the complex KV Burgers equation with u H k remains in H k as long as its 2 -norm remains boune. This result suggests that attention be focuse on the 2 -norm. In this paper, we prove rigorously that if ν an the initial atum u satisfy for some constant C ν C u 2, (1.4) then the 2 -norm of the solution of (1.3) with u(x, ) = u (x) is boune uniformly for all time. When (1.4) hols, the solution ecays exponentially in time an becomes real analytic for large time. Systematic numerical computations are also performe on the complex KV Burgers equation to provie etaile information on the behavior of its solutions in ifferent parameter ranges. The numerical results are consistent with our theory.

3 2. Global existence result J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) The goal of this section is to prove a global existence an uniqueness result for the IVP of the complex KV Burgers type equation u t + 2uu x + δu xxx + ν( ) α u =, u(x, t) = u(x + 1,t), u(x, ) = u (x). (2.1) The initial atum u is perioic, square integrable an has mean zero, namely 1 u (x)x =. Theorem 2.1. et δ, ν>an α 1. Assume that u is in H s with s 1, perioic an has mean zero. Assume that ν an u further satisfy ν>c α u 2, (2.2) where C α = 2 1 α /π α 1/2. Then the IVP (2.1) has a unique global solution u satisfying u ([, ); H s ) 2 ([, ); H α+s ). Remark. Note that the conition (2.2) oes not epen on δ. Thus, this theorem also hols for the case δ =, namely the complex Burgers equation. Proof. The tool is the Galerkin approximation [4]. et N 1 an l N. Consier the sequence {û N (l, t)} solving the equations t ûn(l, t) = ν 2πl 2α û N (l, t) + iδ(2πl) 3 û N (l, t) 4πi û N (l 1,t)l 2 û N (l 2,t) (2.3) with the initial conition l 1 +l 2 =l û N (l, ) = û N (l), l N, where the sum in (2.3) is restricte to l 1 N an l 2 N. This is a finite system of orinary ifferential equations (ODE) for û N (l, t). Since the right-han sie of (2.3) is a locally ipschitz function of û N (l, t), the basic ODE theory asserts that there exists a T> such that there is a unique continuous function û N (l, t) on[,t] solving (2.3). To establish global existence for û N (l, t), we now show that û N (l, t) is boune uniformly. Dotting (2.3) with û N (l, t), summing over l an aing the complex conjugate, we obtain t û N (l, t) 2 + 2ν l l 2πl 2α û N (l, t) 2 = K with K = 8πiR l û N (l, t) û N (l 1,t)l 2 û N (l 2,t), l 1 +l 2 =l

4 592 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) where R enotes the real part. Now efine the perioic, mean zero function u N (x, t) by its Fourier transform û N (l, t), u N (x, t) = e i2πlx û N (l, t). (2.4) l N Then K can be written as 1 K = 4R u N (x, t) (u N (x, t)u N x (x, t)) x. Integrating by parts yiels 1 1 K = 4 u N (x, t) 2 Ru N x (x, t)x = 4 ((Ru N ) 2 + (Iu N ) 2 )(Ru N ) x x = 4 (Iu N ) 2 (Ru N ) x x. By the Schwarz inequality an the estimate that f 2 2 f 2 f x 2 for any f with mean zero, K 4 Iu N Iu N 2 (Ru N ) x Iu N 3/2 2 (Iu N ) x 1/2 2 (Ru N ) x 2. Because û N (,t) =, the Poincaré inequality implies that Therefore, Iu N 2 1 2π (IuN ) x 2. K 4 π Iu N 2 (Iu N ) x 2 (Ru N ) x 2 2 π Iu N 2 u N x 2 2. Ientifying u N (,t) 2 2 = l û N (l, t) 2 an ( ) α/2 u N 2 2 = l 2πl 2α û N (l, t) 2, we obtain t un (,t) 2 + 2ν 2 ( )α/2 u N 2 2 Iu N 2 2 u N π x 2 2. (2.5) Applying Poincaré s inequality u N x (2π) α 1 ( )α/2 u N 2 2, (2.6)

5 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) (2.5) can then be written as t un (,t) 2 2(ν C α Iu 2 2) ( ) α/2 u N 2 2, (2.7) where C α = 2 1 α /π α 1/2. When u satisfies (2.2), then ν>c α Iu 2, an (2.7) implies that u N (,t) 2 ecreases as t grows. Thus, u N (,t) 2 u N 2 u 2. That is, u N 2 is boune uniformly in t an in the orer of the approximation N. In aition, the following inequality hols t ( ) α/2 u N (,τ) 2 τ C, (2.8) 2 where C is a constant epening on α an ν only. Thus, for any t>, there is a subsequence u N j an a 2 function u(x, t) such that u N j converges weakly to u(x, t). It is easily verifie that u(x, t) is a weak solution of (2.1), namely u(x, t) satisfies (2.1) in the istributional sense. We now show that u(x, t) actually satisfies (2.1) in the classical sense. The first step is to boun u x. Recall that u N efine in (2.4) satisfies t u N + 2P N (u N u N x ) + δun xxx + ν( )α u N xx =, un (x, ) = u N (x), (2.9) where P N enotes the projection onto the space spanne by the moes {e i2πlx } l N. Now, ifferentiate (2.9) an take the 2 -norm, t un x ν ( )α/2 u N x 2 = 2 ū N 2 x un x 2 x 4 u N x ūn xx IuN x x. (2.1) For notational convenience, we label the terms on the right-han sie as I an II. Applying the Poincaré inequality we obtain u N xx 2 1 (2π) α 1 ( )α/2 u N x 2, I 2 u N x u N 3 x 5/2 u N 2 xx 1/2 2 where C 1 = 2 2/(2π) α 1. By Young s inequality, C 1 u N x 5/2 ( ) α/2 u N 2 x 1/2, 2 I ν 2 ( )α/2 u N x C 2 u N x 1/3 2, (2.11)

6 594 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) where C 2 is a constant epening on α an ν only. II 4 2 u N 1/2 u N 2 x 3/2 u N 2 xx 2 ν 2 ( )α/2 u N x 2 + C 3 u N 2 2 u N x 3 2, (2.12) Inserting the estimates (2.11) an (2.12) in (2.1), we obtain t un x ν ( )α/2 u N x 2 2 C 2 u N x 1/3 2 + C 3 u N 2 u N x 3 2, which, in particular, implies that t un x 2 C 4 u N x 7/3 2 + C 5 u N 2 u N x 2 2. Integrating with respect to t an noticing (2.6) an (2.8), we obtain for any t> sup τ t u N x (,τ) 2 un x 2 + C 6 sup u N x (,τ) 1/3 + C τ t 2 7. This inequality allows us to conclue that u N x (,t) 2 C 8, where C 8 is a constant epening on α, ν an u x 2 only. Thus, we conclue that u, the limit of u N, must also has square integrable erivatives at any time t. The bouneness for the general norm u H s can be prove by iteration. We omit the etails. Global solutions in Theorem 2.1 actually ecay exponentially in time. Theorem 2.2. et δ, ν> an α 1. Assume that u H s (s 1) is perioic an has mean zero. If u further satisfies ν>c α u 2, then the corresponing solution u of (2.1) ecay exponentially in 2 u(,t) 2 e Ct u 2 (2.13) for all t, where C = (2π) α (ν C α Iu 2). In aition, there exist a time t an a constant C such that for any t t, u x (,t) 2 e Ct u x 2. (2.14) Proof. Combining (2.5) with (2.6) an letting N, we fin that u satisfies t u(,t) ν ( )α/2 u 2 2 2C α Iu 2 ( ) α/2 u 2 2.

7 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) Applying Poincaré s inequality yiels t u(,t) C u(,t) t u(,t) (ν C α Iu 2) ( ) α/2 u 2, 2 which immeiately implies the inequality in (2.13). To show (2.14), we recall that there exists a constant C 9 epening on α, ν an u x 2 only such that u x (,t) 2 C 9 an t u xx (,τ) 2 2 τ C 9. Since u satisfies (2.1), u x obeys t u x(,t) ν ( )α/2 u x 2 = 2 2 ū x u x 2 x 4 u x ū xx Iu x x. To boun the terms J 1 an J 2 on the right-han sie, we use the Gagliaro Nirenberg type inequalities followe by Poincaré s inequality J 1 2 u x 3 C 1 u 5/4 3 u 2 xx 7/4 C 2 11 u 2 u xx 2 2, J 2 4 u xx 2 u x 2 C 12 u 4 xx 2 u 1/2 u 2 x 1/2 u 2 xx 2 = C 12 u 1/2 u 2 x 1/2 u 2 xx 2 C 2 13 u 1/2 u 2 xx 2 2. We thus have establishe that t u x(,t) 2 + 2ν 2 (2π) u xx 2 α/2 C 14( u u 1/2 ) u 2 xx 2 2. Since u(,t) 2 ecays exponentially in t, we can choose t such that for any t t, 2ν (2π) C 14( u(,t) α/2 2 + u(,t) 1/2 ) >. (2.15) 2 Therefore, for t t, t u x(,t) C u xx (,t) 2 2, where C enotes the quantity on the left-han sie of (2.15). This inequality then leas to (2.14). The following theorem further asserts that the global solutions become analytic as time elapses. For the sake of a concise presentation, we focus on the case α = 1, namely the complex KV Burgers equation. Theorem 2.3. et δ, ν> an α = 1. et u H s (s 1) be perioic of perio 1 an have mean zero. Assume that u satisfies ν>c α u 2.

8 596 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) Then there exists t > such that the solution u of (2.1) is real analytic for t t. More precisely, u can be expane into the complex plane such that the extension u(z, t) is analytic in S t {z = x + iy : x [, 1],y [ ν(t t ),ν(t t )]}. (2.16) Proof. Accoring to Theorem 2.2, there exists t > such that u x (,t) 2 ecays exponentially in t for t t. Because û(l, t) 1/2 l 2 l 2 û(l, t) 2 1/2 = C u x (,t) 2 2, there exists a time (still enote t ) such that û(l, t) ν. Now, consier the functions (2.17) Y(t) = û(l, t) e νπ l (t t), Z(t) = l û(l, t) e νπ l (t t). Noticing that û(l, t) satisfies the equation t û(l, t) = ν 2πl û(l, t) + iδ(2πl)3 û(l, t) 4πi we obtain after some manipulations that Y an Z satisfies Y(t) t + Z(t)(ν Y(t)). l 1 +l 2 =l û(l 1,t)l 2 û(l 2,t), Because of (2.17), namely Y(t ) ν, this equation then implies that Y(t) is a non-increasing function of t for t t. We thus conclue that for any t t Y(t) = û(l, t) e νπ l (t t) ν. That is, u(z, t), the extension of u(x, t) to the complex plane, is analytic in the set S t (2.16). efine in 3. Numerical results In this section, we present several representative results from our numerical experiments performe on the complex KV Burgers equation, namely (2.1) with α = 1 an δ = 1/4π 2. The initial ata are of the

9 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) Fig. 1. a = 8, ν = 1, n = 124 an k = 1 8. form u (x) = a e 2πix. We have use the spectral metho with an FFT algorithm esigne for a real KV equation [3]. Appropriate moifications have been mae to suit the complex equation. The numerical results are consistent with our theory. We compute the solutions for a range of a s an ν s. When the conition (2.2) is satisfie, the corresponing solutions ecay exponentially. However, if ν is sufficiently small, then the solutions appear to blow up. Table 1 a vs. ν c (n = 124 an k = 1 8 ) a ν c

10 598 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) Fig. 2. a = 8, ν =.2, n = 124 an k = 1 8. The soli curves represent the real part f an the ashe curves represent the imaginary part g. Fig. 3. a vs. ν c (n = 124 an k = 1 8 ). The line is ν c = a.

11 J. Wu, J.M. Yuan / Mathematics an Computers in Simulation 69 (25) In Fig. 1, a = 8, ν = 1, n = 124 an k = 1 8, where n represents the number of moes an k the time step. The plots clearly inicate that the 2 -norms of u, its real part f an imaginary part g all ecay exponentially in time from the beginning. The exponential ecay of u is asserte in Theorem 2.2. In Fig. 2, a = 8, ν =.2, n = 124 an k = 1 8. Obviously, ν C α a. That is, the conition (2.2) in Theorem 2.1 is violate. Both the real part f an the imaginary part g quickly lose shapes an increase very rapily. At t =.2, the numerical instability kicks in an f an g appear to blow up. This computation was then repeate using n = 248 an the same behavior of f an g occurre again. For a ranging from 6 to 2, we compute the corresponing solutions associate with a range of ν. Our purpose has been to fin the critical ν c = ν c (a) such that the solutions appear to blow up for ν<ν c an the solutions are boune for ν ν c. The results are given in Table 1. To see how ν c epens on the corresponing a, Table 1 is plotte in Fig. 3. It is clear that ν c an a, the 2 -norm of the initial ata are linearly relate. References [1] B. Birnir, Singularities of the complex perioic KV flows, Commun. Pure Appl. Math. 39 (1986) [2] B. Birnir, An example of blow-up for the complex KV equation an existence beyon the blow-up, SIAM J. Appl. Math. 47 (1987) [3] T. Chan, T. Kerkhoven, Fourier methos with extene stability intervals for the Korteweg e Vries equation, SIAM J. Numer. Anal. 22 (1985) [4] C.R. Doering, J.D. Gibbon, Applie Analysis of the Navier Stokes Equations, Cambrige University Press, [5] M.D. Kruskal, R.M. Miura, C.S. Garner, N.J. Zabusky, Korteweg e Vries equation an generalizations. V. Uniqueness an nonexistence of polynomial conservation laws, J. Math. Phys. 11 (197) [6] J.-M. Yuan, J. Wu, The complex KV equation with or without issipation, Discrete Contin. Dyn. Syst. Ser. B 5 (25)