Steady-state bifurcation and regularity of nonlinear Burgers equation with mean value constraint

Qingming Hao; Wei Chen; Zhigang Pan; Chao Zhu; Yanhua Wang; Qingming Hao; Wei Chen; Zhigang Pan; Chao Zhu; Yanhua Wang

doi:10.3934/era.2025130

Electronic Research Archive

2025, Volume 33, Issue 5: 2972-2988. doi: 10.3934/era.2025130

Previous Article Next Article

Research article Topical Sections

Steady-state bifurcation and regularity of nonlinear Burgers equation with mean value constraint

1.
Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2.
Department of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, China
3.
Chengdu Jincheng College, Chengdu 611731, China

Received: 17 February 2025 Revised: 12 April 2025 Accepted: 27 April 2025 Published: 15 May 2025

In this paper, we focus on the steady-state bifurcation problem of the nonlinear Burgers equation within a bounded domain, considering both homogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions with a mean value constraint. Unlike previous studies, we develop an enhanced turbulence model by incorporating nonlinear higher-order terms (such as $ u^2 $ and $ u^3 $) and linear source terms $ \lambda u $ into the one-dimensional Burgers equation. Our steady-state bifurcation analysis establishes for the first time how the coupled forward energy cascade and inverse energy transfer mechanisms collectively govern the dynamics of initial flow instability. By combining the spectral theorem for a linear compact operator with the normalized Lyapunov–Schmidt reduction method and the implicit function theorem, we derive the complete criterion for the critical bifurcation condition, the explicit form of the bifurcation solution, and its regularity.
- nonlinear Burgers equation,
- mean value constraint,
- Lyapunov–Schmidt reduction,
- steady-state bifurcation
Citation: Qingming Hao, Wei Chen, Zhigang Pan, Chao Zhu, Yanhua Wang. Steady-state bifurcation and regularity of nonlinear Burgers equation with mean value constraint[J]. Electronic Research Archive, 2025, 33(5): 2972-2988. doi: 10.3934/era.2025130

Related Papers:

Abstract

In this paper, we focus on the steady-state bifurcation problem of the nonlinear Burgers equation within a bounded domain, considering both homogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions with a mean value constraint. Unlike previous studies, we develop an enhanced turbulence model by incorporating nonlinear higher-order terms (such as $ u^2 $ and $ u^3 $) and linear source terms $ \lambda u $ into the one-dimensional Burgers equation. Our steady-state bifurcation analysis establishes for the first time how the coupled forward energy cascade and inverse energy transfer mechanisms collectively govern the dynamics of initial flow instability. By combining the spectral theorem for a linear compact operator with the normalized Lyapunov–Schmidt reduction method and the implicit function theorem, we derive the complete criterion for the critical bifurcation condition, the explicit form of the bifurcation solution, and its regularity.

References

[1]	H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev., 43 (1915), 163–170. https://doi.org/10.1175/1520-0493(1915)43 doi: 10.1175/1520-0493(1915)43
[2]	J. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1 (1948), 171–199. https://doi.org/10.1016/S0065-2156(08)70100-5 doi: 10.1016/S0065-2156(08)70100-5
[3]	J. D. Murray, On Burgers' model equations for turbulence, J. Fluid Mech., 59 (1973), 263–279. https://doi.org/10.1017/S0022112073001564 doi: 10.1017/S0022112073001564
[4]	N. T. Eldabe, E. M. Elghazy, A. Ebaid, Closed form solution to a second order boundary value problem and its application in fluid mechanics, Phys. Lett. A, 363 (2007), 257–259. https://doi.org/10.1016/j.physleta.2006.11.010 doi: 10.1016/j.physleta.2006.11.010
[5]	K. Fujisawa, A. Asada, Nonlinear parametric sound enhancement through different fluid layer and its application to noninvasive measurement, Measurement, 94 (2016), 726–733. https://doi.org/10.1016/j.measurement.2016.09.004 doi: 10.1016/j.measurement.2016.09.004
[6]	Y. L. Chen, T. Zhang, A weak Galerkin finite element method for Burgers' equation, Comput. Appl. Math., 348 (2019), 103–119. https://doi.org/10.1016/j.cam.2018.08.044 doi: 10.1016/j.cam.2018.08.044
[7]	M. A. Shallal, A. H. Taqi, B. F. Jumaa, H. Rezazadeh, M. Inc, Numerical solutions to the 1-D Burgers' equation by a cubic Hermite finite element method, Indian J. Phys., 96 (2022), 3831–3836. https://doi.org/10.1007/s12648-022-02304-4 doi: 10.1007/s12648-022-02304-4
[8]	Y. S. Uçar, N. Yağmurlu, İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Meth. Partial Differ. Equations, 35 (2019), 478–492. https://doi.org/10.1002/num.22309 doi: 10.1002/num.22309
[9]	L. Wang, H. Li, Y. Meng, Numerical solution of coupled Burgers' equation using finite difference and sinc collocation method, Eng. Lett., 29 (2021), 668–674.
[10]	J. Zhang, Q. Yang, The finite volume element method for time fractional generalized Burgers' equation, Fractal Fract., 8 (2024), 1–17. https://doi.org/10.3390/fractalfract8010053 doi: 10.3390/fractalfract8010053
[11]	K. Kaur, G. Singh, An efficient non-standard numerical scheme Coupled with a compact finite difference method to solve the one-dimensional Burgers' equation, Axioms, 12 (2023), 593–609. https://doi.org/10.3390/axioms12060593 doi: 10.3390/axioms12060593
[12]	J. Zhang, J. Yu, A Multi-Quadrics quasi-interpolation scheme for numerical solution of Burgers' equation, Appl. Numer. Math., 208 (2025), 38–44. https://doi.org/10.1016/j.apnum.2024.09.025 doi: 10.1016/j.apnum.2024.09.025
[13]	Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., 62 (2024), 1323–1356.
[14]	R. D. Ortiz, O. M. Nunez, A. M. M. Ramirez, Solving viscous Burgers' equation: Hybrid approach combining Boundary Layer theory and physics-informed neural networks, Mathematics, 12 (2024), 3430. https://doi.org/10.3390/math12213430 doi: 10.3390/math12213430
[15]	T. Mouktonglang, S. Yimnet, N. Sukantamala, B. Wongsaijai, Dynamical behaviors of the solution to a periodic initial-boundary value problem of the generalized Rosenau-RLW-Burgers equation, Math. Comput. Simul., 196 (2022), 114–136. https://doi.org/10.1016/j.matcom.2022.01.004 doi: 10.1016/j.matcom.2022.01.004
[16]	L. Li, K. W. Ong, Dynamic transitions of generalized Burgers equation, J. Math. Fluid Mech., 18 (2016), 89–102. https://doi.org/10.1007/s00021-015-0240-7 doi: 10.1007/s00021-015-0240-7
[17]	Q. Zhang, D. Yan, Z. Pan, Bifurcation from double eigenvalue for nonlinear equation with third-order nondegenerate singularity, Appl. Math. Comput., 220 (2013), 549–559. https://doi.org/10.1016/j.amc.2013.06.024 doi: 10.1016/j.amc.2013.06.024
[18]	D. Wei, S. Guo, Steady-state bifurcation of a nonlinear boundary problem, Appl. Math. Lett., 128 (2022), 107902. https://doi.org/10.1016/j.aml.2021.107902 doi: 10.1016/j.aml.2021.107902
[19]	G. Guo, X. Wang, X. Lin, M. Wei, Steady-state and Hopf bifurcations in the Langford ODE and PDE systems, Nonlinear Anal. Real World Appl., 34 (2017), 343–362. https://doi.org/10.1016/j.nonrwa.2016.09.008 doi: 10.1016/j.nonrwa.2016.09.008
[20]	Z. Pan, L. Jia, Y. Mao, Q. Wang, Transitions and bifurcations in couple stress fluid saturated porous media using a thermal non-equilibrium model, Appl. Math. Comput., 415 (2022), 126727. https://doi.org/10.1016/j.amc.2021.126727 doi: 10.1016/j.amc.2021.126727
[21]	Y. Wang, The Progress of Chemotaxis-fluid Models, Journal of Xihua University (Natural Science Edition), 35 (2016), 30–34, 38. https://doi.org/10.3969/j.issn.1673-159X.2016.04.006
[22]	J. Chen, Y. Han, Global Weak Solvability for a Chemotaxis-Fluid Model with Low Regular Initial Data, Journal of Xihua University (Natural Science Edition), 43 (2024), 97–105. https://doi.org/10.12198/j.issn.1673-159X.5354
[23]	W. Kuang, Y. Dong, Global Well-posedness of 2D Chemotaxis-fluid System, Journal of Xihua University (Natural Science Edition), 43 (2024), 103–108. https://doi.org/10.12198/j.issn.1673-159X.5233
[24]	R. Ma, L. Wei, Z. Chen, Evolution of bifurcation curves for one-dimensional Minkowski-curvature problem, Appl. Math. Lett., 103 (2020), 106176. https://doi.org/10.1016/j.aml.2019.106176 doi: 10.1016/j.aml.2019.106176
[25]	T. Sengul, B. Tiryakioglu, Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The non-self-adjoint case, Math. Anal. Appl., 523 (2023), 127114. https://doi.org/10.1016/j.jmaa.2023.127114 doi: 10.1016/j.jmaa.2023.127114
[26]	Z. Feng, Y. Su, Traveling wave phenomena of inhomogeneous half-wave equation, J. Differ. Equations, 400 (2024), 248–277. https://doi.org/10.1016/j.jde.2024.04.029 doi: 10.1016/j.jde.2024.04.029
[27]	Y. Fan, L. Li, Z. Pan, Q. Wang, On dynamics of double-diffusive magneto-convection in a non-Newtonian fluid layer, Math. Methods Appl. Sci., 46 (2023), 14596–14621. https://doi.org/10.1002/mma.9337 doi: 10.1002/mma.9337
[28]	S. Li, J. Wu, H. Nie, Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie–Gower predator–prey model, Comput. Math. Appl., 70 (2015), 3043–3056. https://doi.org/10.1016/j.camwa.2015.10.017 doi: 10.1016/j.camwa.2015.10.017
[29]	M. Wei, J. Wu, G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction, Nonlinear Anal. Real World Appl., 22 (2015), 155–175. https://doi.org/10.1016/j.nonrwa.2014.08.003 doi: 10.1016/j.nonrwa.2014.08.003
[30]	L. Ma, Z. Feng, Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay, Nonlinear Anal. Real World Appl., 61 (2021), 103327. https://doi.org/10.1016/j.nonrwa.2021.103327 doi: 10.1016/j.nonrwa.2021.103327
[31]	T. Ma, S. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, 1$^{nd}$ edition, Science Press, China, 2007.
[32]	T. Ma, S. Wang, Bifurcation theory and applications, World Sci, Singapore, 2005.
[33]	W. Cao, X. Shan, S. Tang, W. Ouyang, W. Zhang, Solving parametric high-Reynolds-number wall-bounded turbulence around airfoils governed by Reynolds-averaged Navier-Stokes equations using time-stepping-oriented neural network, Phys. Fluids, 37 (2025), 015151. https://doi.org/10.1063/5.0245918 doi: 10.1063/5.0245918
[34]	B. Tripathi, P. W. Terry, A. E. Fraser, E. G. Zweibel, M. J. Pueschel, Three-dimensional shear-flow instability saturation via stable modes, Phys. Fluids, 35 (2023), 105151. https://doi.org/10.1063/5.0167092 doi: 10.1063/5.0167092
[35]	N. Ciola, P. D. Palma, J. C. Robinet, S. Cherubini, Large-scale coherent structures in turbulent channel flow: a detuned instability of wall streaks, Fluid Mech., 997 (2024), A18. https://doi.org/10.1017/jfm.2024.726 doi: 10.1017/jfm.2024.726
[36]	J. Zhu, Y. Peng, W. Zhang, S. Niu, Y. Huang, Y. Li, et al. Research on the high-integration vortex-shape flow channel of a hydraulic servo valve based on additive manufacturing, in CSAA/IET International Conference on Aircraft Utility Systems (AUS 2024), 2024 (2025), 1700–1705. https://doi.org/10.1049/icp.2024.3137
[37]	B. Eckhardt, H. Faisst, A. Schmiegel, T. M. Schneider, Dynamical systems and the transition to turbulence in linearly stable shear flows, Phil. Trans. R. Soc. A, 366 (2008), 1297–1315. https://doi.org/10.1098/rsta.2007.2132 doi: 10.1098/rsta.2007.2132

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)