July  2017, 16(4): 1455-1470. doi: 10.3934/cpaa.2017069

Damping to prevent the blow-up of the korteweg-de vries equation

Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France

Received  April 2016 Revised  February 2017 Published  April 2017

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.

Citation: Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
References:
[1]

C. J. AmickJ. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

J. L. BonaV. A. DougalisO. A. Karakashian and W. R. McKinney, The effect of dissipation on solutions of the generalized Korteweg-de Vries equation, J. Comput. Appl. Math., 74 (1996), 127-154.  doi: 10.1016/0377-0427(96)00021-0.  Google Scholar

[3]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.  Google Scholar

[4]

J. L. Bona and R. Smith, Existence of solutions to the Korteweg-de Vries initial value problem, Nonlinear wave motion (Proc. AMS-SIAM Summer Sem. , Clarkson Coll. Tech. , Potsdam, N. Y. , 1972), (1974), 179-180. Lectures in Appl. Math. , Vol. 15.  Google Scholar

[5]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[6]

J.-P. Chehab and G. Sadaka, On damping rates of dissipative KdV equations, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487-1506.  doi: 10.3934/dcdss.2013.6.1487.  Google Scholar

[7]

J. -P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, Commun. Pure Appl. Anal., 12 (2013), 519-546.  doi: 10.3934/cpaa.2013.12.519.  Google Scholar

[8]

B. DubrovinT. Grava and C. Klein, Numerical study of breakup in generalized Korteweg-de Vries and Kawahara equations, SIAM J. Appl. Math., 71 (2011), 983-1008.  doi: 10.1137/100819783.  Google Scholar

[9]

J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods: KdV and KP solutions, Lett. Math. Phys., 76 (2006), 249-267.  doi: 10.1007/s11005-006-0068-4.  Google Scholar

[10]

J. -M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finitedimensional dynamical system in the long time, J. Differential Equations, 74 (1988), 369-390.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[11]

J. -M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differential Equations, 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[12]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.   Google Scholar

[13]

O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[14]

T. Grava and C. Klein, Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-Ⅱ equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 733-757.  doi: 10.1098/rspa.2007.0249.  Google Scholar

[15]

Jr. R.J. Iório, KdV, BO and friends in weighted Sobolev spaces, Functional-analytic methods for partial differential equations, 1450 (1989), 104-121.  doi: 10.1007/BFb0084901.  Google Scholar

[16]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine Series 5, 39 (1895), 422-443.   Google Scholar

[17]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2), 155 (2002), 235-280.  doi: 10.2307/3062156.  Google Scholar

[18]

E. Ott and R. N. Sudan, Damping of solitaries waves, Phys. Fluids, 13 (1970), 1432-1435.   Google Scholar

show all references

References:
[1]

C. J. AmickJ. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

J. L. BonaV. A. DougalisO. A. Karakashian and W. R. McKinney, The effect of dissipation on solutions of the generalized Korteweg-de Vries equation, J. Comput. Appl. Math., 74 (1996), 127-154.  doi: 10.1016/0377-0427(96)00021-0.  Google Scholar

[3]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.  Google Scholar

[4]

J. L. Bona and R. Smith, Existence of solutions to the Korteweg-de Vries initial value problem, Nonlinear wave motion (Proc. AMS-SIAM Summer Sem. , Clarkson Coll. Tech. , Potsdam, N. Y. , 1972), (1974), 179-180. Lectures in Appl. Math. , Vol. 15.  Google Scholar

[5]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation, Phys. D, 192 (2004), 265-278.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[6]

J.-P. Chehab and G. Sadaka, On damping rates of dissipative KdV equations, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1487-1506.  doi: 10.3934/dcdss.2013.6.1487.  Google Scholar

[7]

J. -P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations, Commun. Pure Appl. Anal., 12 (2013), 519-546.  doi: 10.3934/cpaa.2013.12.519.  Google Scholar

[8]

B. DubrovinT. Grava and C. Klein, Numerical study of breakup in generalized Korteweg-de Vries and Kawahara equations, SIAM J. Appl. Math., 71 (2011), 983-1008.  doi: 10.1137/100819783.  Google Scholar

[9]

J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods: KdV and KP solutions, Lett. Math. Phys., 76 (2006), 249-267.  doi: 10.1007/s11005-006-0068-4.  Google Scholar

[10]

J. -M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finitedimensional dynamical system in the long time, J. Differential Equations, 74 (1988), 369-390.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[11]

J. -M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differential Equations, 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[12]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.   Google Scholar

[13]

O. Goubet and R. M. S. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. Differential Equations, 185 (2002), 25-53.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[14]

T. Grava and C. Klein, Numerical study of a multiscale expansion of the Korteweg-de Vries equation and Painlevé-Ⅱ equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 733-757.  doi: 10.1098/rspa.2007.0249.  Google Scholar

[15]

Jr. R.J. Iório, KdV, BO and friends in weighted Sobolev spaces, Functional-analytic methods for partial differential equations, 1450 (1989), 104-121.  doi: 10.1007/BFb0084901.  Google Scholar

[16]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine Series 5, 39 (1895), 422-443.   Google Scholar

[17]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2), 155 (2002), 235-280.  doi: 10.2307/3062156.  Google Scholar

[18]

E. Ott and R. N. Sudan, Damping of solitaries waves, Phys. Fluids, 13 (1970), 1432-1435.   Google Scholar

Figure 1.  Initialization
Figure 2.  Dichotomy
Figure 3.  Initialization
Figure 4.  Find the damping
Figure 5.  At left, solution at different times $t=$ 0, 2, 4, 4.9925 and 5.3303. At right, $H^1$-norm and $L^2$-norm evolution without damping and a perturbed soliton as initial datum. Here $p=5$
Figure 6.  At left, solution at different times $t=$ 0, 2, 5, 10, 11 and 11.3253. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma_k=0.0025$ and a perturbed soliton as initial datum. Here $p=5$
Figure 7.  At left, solution at different times $t=$ 0, 2, 5, 10, 15 and 20. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma_k=0.0027$ and a perturbed soliton as initial datum. Here $p=5$
Figure 8.  Example of a build damping. Here the initial datum is the perturbed soliton. Here $p=5$
Figure 9.  At left, solution at different times $t=$ 0, 2, 5, 10, 15 and 20. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma = \gamma_1$ and a perturbed soliton as initial datum. Here $p=5$
Figure 10.  At left, solution at different times $t=$ 0, 2, 5, 7 and 7.928. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma = \gamma_2$ and a perturbed soliton as initial datum. Here $p=5$
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