September  2011, 1(3): 353-389. doi: 10.3934/mcrf.2011.1.353

Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil, Brazil

2. 

Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex

Received  November 2010 Revised  April 2011 Published  September 2011

The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. The exponential stability, as well as, the global existence of weak solutions are investigated when the exponent in the nonlinear term ranges over the interval $[1, 4)$. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
Citation: Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
References:
[1]

M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical signifcance, Phys. Rev. Lett., 31 (1973), 125-127. doi: 10.1103/PhysRevLett.31.125.  Google Scholar

[2]

E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4.  Google Scholar

[3]

E. Bisognin, V. Bisognin and G. P. Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469.  Google Scholar

[4]

J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. doi: 10.1007/BF02099010.  Google Scholar

[5]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation," Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

J. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  Google Scholar

[7]

M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. Google Scholar

[8]

J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258.  Google Scholar

[9]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417.  Google Scholar

[10]

F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.  Google Scholar

[11]

C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68.  Google Scholar

[12]

G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Appl. Math., 60 (2002), 111-129.  Google Scholar

[13]

G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping,, Quarterly of Appl. Math., ().   Google Scholar

[14]

S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024.  Google Scholar

[15]

S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600.  Google Scholar

[16]

A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015.  Google Scholar

[17]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102.  Google Scholar

[18]

L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012.  Google Scholar

[19]

L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229.  Google Scholar

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.  Google Scholar

[21]

J. Simon, Compact sets in the $L^p(0,T;B)$ spaces, Analli Mat. Pura Appl., 146 (1987), 65-96.  Google Scholar

[22]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[23]

O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001. Google Scholar

show all references

References:
[1]

M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical signifcance, Phys. Rev. Lett., 31 (1973), 125-127. doi: 10.1103/PhysRevLett.31.125.  Google Scholar

[2]

E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4.  Google Scholar

[3]

E. Bisognin, V. Bisognin and G. P. Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469.  Google Scholar

[4]

J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. doi: 10.1007/BF02099010.  Google Scholar

[5]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation," Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

J. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  Google Scholar

[7]

M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. Google Scholar

[8]

J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258.  Google Scholar

[9]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417.  Google Scholar

[10]

F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.  Google Scholar

[11]

C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68.  Google Scholar

[12]

G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Appl. Math., 60 (2002), 111-129.  Google Scholar

[13]

G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping,, Quarterly of Appl. Math., ().   Google Scholar

[14]

S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024.  Google Scholar

[15]

S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600.  Google Scholar

[16]

A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015.  Google Scholar

[17]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102.  Google Scholar

[18]

L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012.  Google Scholar

[19]

L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229.  Google Scholar

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.  Google Scholar

[21]

J. Simon, Compact sets in the $L^p(0,T;B)$ spaces, Analli Mat. Pura Appl., 146 (1987), 65-96.  Google Scholar

[22]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[23]

O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001. Google Scholar

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