February  2021, 26(2): 1205-1221. doi: 10.3934/dcdsb.2020160

Forced oscillation of viscous Burgers' equation with a time-periodic force

1. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA

* Corresponding author: Taige Wang

Received  December 2019 Published  February 2021 Early access  May 2020

This paper is concerned about the existence of periodic solutions of the viscous Burgers' equation when a forced oscillation is prescribed. We establish the existence theory by contraction mapping in $ H^s[0,1] $ with $ s\ge 0 $. Asymptotical periodicity is obtained as well, and the periodic solution is achieved by selecting a suitable function as initial data to generate a solution and passing time limit to infinity. Moreover, uniqueness and global stability is achieved for this periodic solution.

Citation: Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
References:
[1]

J. BonaS. Sun and B. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[2]

J. BonaS. Sun and B. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane, Comm. Cont. Math., 5 (2003), 369-400.  doi: 10.1142/S021919970300104X.

[3]

J. BonaS. Sun and B. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. PDE, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[4]

H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1993), 409-426.  doi: 10.1090/S0273-0979-1983-15105-4.

[5]

H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1-30.  doi: 10.1002/cpa.3160310102.

[6]

S. ChenC. HsiaC. Jung and B. Kwon, Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers' equation, J. Math. Anal. Appl., 445 (2017), 655-676.  doi: 10.1016/j.jmaa.2016.08.018.

[7]

W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.  doi: 10.1002/cpa.3160461102.

[8]

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

[9] R. Grimshaw, Nonlinear Ordinary Differential Equations, Applied Mathematics and Engineering Science Texts. CRC Press, Boca Raton, FL, 1993. 
[10]

J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., 19 (1966), 371-420.  doi: 10.1002/cpa.3160190404.

[11]

G. ŁukaszewiczE. E. Ortega-Torres and M. A. Rojas-Medar, Strong periodic solutions for a class of abstract evolution equations, Nonlinear Anal., 54 (2003), 1045-1056.  doi: 10.1016/S0362-546X(03)00125-1.

[12]

P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68.  doi: 10.1002/cpa.3160310103.

[13]

P. Rabinowitz, Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.  doi: 10.1137/0513027.

[14]

G. R. Sell and Y. You, Inertial manifolds: The nonselfadjoint case, J. Diff. Eqns., 96 (1992), 203-255.  doi: 10.1016/0022-0396(92)90152-D.

[15]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Reginal Confeences Series in Applied Math., 66, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.

[16]

M. Usman and B. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, Jrl. Syst. Sci. & Comp., 20 (2007), 284-292.  doi: 10.1007/s11424-007-9025-2.

[17]

M. Usman and B. Zhang, Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability, Disc. Cont. Dyn. Sys., 26 (2010), 1509-1523.  doi: 10.3934/dcds.2010.26.1509.

[18]

O. Vejvoda, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, 1982.

[19]

C. E. Wayne, Periodic solutions of nonlinear partial differential equations, Notices of Amer. Math. Soc., 44 (1997), 895-902. 

[20]

B. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), 337–357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001.

show all references

References:
[1]

J. BonaS. Sun and B. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[2]

J. BonaS. Sun and B. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane, Comm. Cont. Math., 5 (2003), 369-400.  doi: 10.1142/S021919970300104X.

[3]

J. BonaS. Sun and B. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. PDE, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[4]

H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1993), 409-426.  doi: 10.1090/S0273-0979-1983-15105-4.

[5]

H. Brézis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1-30.  doi: 10.1002/cpa.3160310102.

[6]

S. ChenC. HsiaC. Jung and B. Kwon, Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers' equation, J. Math. Anal. Appl., 445 (2017), 655-676.  doi: 10.1016/j.jmaa.2016.08.018.

[7]

W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498.  doi: 10.1002/cpa.3160461102.

[8]

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

[9] R. Grimshaw, Nonlinear Ordinary Differential Equations, Applied Mathematics and Engineering Science Texts. CRC Press, Boca Raton, FL, 1993. 
[10]

J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., 19 (1966), 371-420.  doi: 10.1002/cpa.3160190404.

[11]

G. ŁukaszewiczE. E. Ortega-Torres and M. A. Rojas-Medar, Strong periodic solutions for a class of abstract evolution equations, Nonlinear Anal., 54 (2003), 1045-1056.  doi: 10.1016/S0362-546X(03)00125-1.

[12]

P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68.  doi: 10.1002/cpa.3160310103.

[13]

P. Rabinowitz, Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.  doi: 10.1137/0513027.

[14]

G. R. Sell and Y. You, Inertial manifolds: The nonselfadjoint case, J. Diff. Eqns., 96 (1992), 203-255.  doi: 10.1016/0022-0396(92)90152-D.

[15]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Reginal Confeences Series in Applied Math., 66, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.

[16]

M. Usman and B. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, Jrl. Syst. Sci. & Comp., 20 (2007), 284-292.  doi: 10.1007/s11424-007-9025-2.

[17]

M. Usman and B. Zhang, Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability, Disc. Cont. Dyn. Sys., 26 (2010), 1509-1523.  doi: 10.3934/dcds.2010.26.1509.

[18]

O. Vejvoda, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, 1982.

[19]

C. E. Wayne, Periodic solutions of nonlinear partial differential equations, Notices of Amer. Math. Soc., 44 (1997), 895-902. 

[20]

B. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), 337–357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001.

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