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

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

Taige Wang 1,, and  Bing-Yu Zhang 1,

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.

Keywords: Semigroup, contraction mapping, asymptotical periodicity, global stability.
Mathematics Subject Classification: Primary: 35K55, 34K13.
Citation: Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
References:
[1]

J. Bona, S. 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.  Google Scholar

[2]

J. Bona, S. 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.  Google Scholar

[3]

J. Bona, S. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[6]

S. Chen, C. Hsia, C. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[11]

G. Łukaszewicz, E. 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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

show all references

References:
[1]

J. Bona, S. 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.  Google Scholar

[2]

J. Bona, S. 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.  Google Scholar

[3]

J. Bona, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. Chen, C. Hsia, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[11]

G. Łukaszewicz, E. 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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

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