American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[2]

J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
[3]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[4]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959
[5]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661
[6]

Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021188
[7]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135
[8]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69
[9]

Kuo-Shou Chiu. Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021060
[10]

Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473
[11]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[12]

Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193
[13]

Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824
[14]

Yuncheng You. Asymptotical dynamics of the modified Schnackenberg equations. Conference Publications, 2009, 2009 (Special) : 857-868. doi: 10.3934/proc.2009.2009.857
[15]

Giuseppe Buttazzo, Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Networks & Heterogeneous Media, 2007, 2 (4) : 761-777. doi: 10.3934/nhm.2007.2.761
[16]

Viorel Nitica, Andrei Török. On a semigroup problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2365-2377. doi: 10.3934/dcdss.2019148
[17]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[18]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333
[19]

J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625
[20]

John Banks. Topological mapping properties defined by digraphs. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (213)
  • HTML views (307)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences