分数阶 Burgers 方程时间周期弱解的唯一性与渐近稳定性

Abstract

Abstract:

The paper is concerned with the time-periodic (T-periodic) problem of fractional Burgers equation on the real line. Based on the Galerkin approximates and Fourier expansion, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution for the nonlinear equation are established by constructing a suitable contraction mapping. Furthermore, we show that the unique T-periodic solution is asymptotically stable. In addition, our method can be extended to the classical forced Burgers equation in a bounded region, which improves the known result.

Key words: Uniqueness, Fractional order, Contraction mapping, Asymptotic stability

CLC Number:

O175

Xu Fei, Zhang Yong. Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation[J].Acta mathematica scientia,Series A, 2023, 43(6): 1710-1722.

Trendmd

References

[1]	Alibaud N, Andreianov B. Non-uniqueness of weak solutions for the fractal Burgers equation. Ann Inst Henri Poincar Anal Non Linéaire, 2010, 27: 997-1016
[2]	Bouchard J P, Georges A. Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications. Phys Rep, 1990, 195: 127-293
[3]	Burgers J M. Correlation problem in a one-dimensional model of turbulence. Nederl Akad Wetensch Proc, 1950, 53: 247-260
[4]	Caffarelli L, Silvestre L. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann of Math, 2010, 171(3): 1903-1930
[5]	Chen S H, Hsia C H, Jung C Y, Kwon B. Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burger's equation. J Math Anal Appl, 2017, 445: 655-676
[6]	Cole J D. On a quasi-linear parabolic equation occurring in aerodynamics. Quart Appl Math, 1951, 9: 225-236
[7]	Defterli O, D'Elia M, Du Q, et al. Fractional diffusion on bounded domains. Fract Calc Appl Anal, 2015, 18: 342-360
[8]	Dong H, Du D, Li D. Finite time singularities and global well-posedness for fractal Burgers equations. Indiana Univ Math J, 2009, 58: 807-821
[9]	Droniou J, Gallouet T, Vovelle J. Global solution and smoothing effect for a non-local regularization of a hyperbolic equation. J Evol Equ, 2003, 3: 499-521
[10]	Galdi G P, Sohr H. Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body. Arch Ration Mech Anal, 2004, 172: 363-406
[11]	Hopf E. The partial differential equation $u_{t}+uu_{x}=\mu u_{xx}$. Comm Pure Appl Math, 1950, 3: 201-230
[12]	Hsia C H, Shiue M C. On the asymptotic stability analysis and the existence of time-periodic solutions of the primitive equations. Indiana Univ Math J, 2013, 62: 403-441
[13]	Iwabuchi T. Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation. Ann Inst Henri Poincar Anal Non Linéaire, 2020, 37: 855-876
[14]	Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm Pure Appl Math, 1988, 41: 891-907
[15]	Kiselev A, Nazarov F, Shterenberg R. Blow up and regularity for fractal Burgers euation. Dyn Partial Differ Equ, 2008, 5: 211-240
[16]	Kobayashi T. Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow under (GOC) for a symmetric perturbed channel in $R^{2}$. J Math Soc Japan, 2015, 67: 1023-1042
[17]	Mellet A, Mischler S, Mouhot C. Fractional diffusion limit for collisional kinetic equations. Arch Ration Mech Anal, 2011, 199: 493-525
[18]	Miao C, Wu G. Global well-posedness of the critical Burgers equation in critiacal Besov spaces. J Differential Equations, 2009, 247: 1673-1693
[19]	Wang T, Zhang B. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete Contin Dyn Syst Ser B, 2021, 26: 1205-1221
[20]	Xu F, Zhang Y, Li F. Uniqueness and stability of steady-state solution with finite energy to the fractal Burgers equation. arXiv:2107.11761v1

Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Junyuan Deng, Junhao Zhang. Stability to a Hyperbolic System with Cattaneo's Law in the Half Space [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1444-1462.
[2]	Lv Dongting. Globally Asymptotic Stability of 2-Species Reaction-Diffusion Systems of Spatially Inhomogeneous Models [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1171-1183.
[3]	Liang Qing. Existence, Uniqueness and Stability of the Global Solutions to Two Classes of Pantogragh Stochastic Functional Differential Equations with Markovian Switching [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1184-1205.
[4]	Cao Lei,Chen Xiao. The Existence of Domain Wall Solution Arising in Abelian Higgs Model Subject to Born-Infeld Theory [J]. Acta mathematica scientia,Series A, 2025, 45(2): 567-575.
[5]	Pan Lijun, Lv Shun, Weng Shasha. Riemann Solution and Stability of Coupled Aw-Rascle-Zhang Model [J]. Acta mathematica scientia,Series A, 2024, 44(4): 885-895.
[6]	Fan Shishi, Li Haixia, Lu Yindou. Dynamics Analysis of a Diffusive Predator-Prey Model with Beddington-DeAngelis Function Response and Harvesting [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1929-1942.
[7]	He Xuyang,Mao Mingzhi,Zhang Tengfei. Existence and Stability of a Class of Impulsive Neutral Stochastic Functional Differential Equations with Poisson Jump [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1221-1243.
[8]	Li Jize,Qiu Jixiu,Zhou Yonghui. Anticipative Nonlinear Filtering Equations Affected by Observation Noises and Stability of Linear Filtering [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1244-1254.
[9]	Nan Deng,Meiqiang Feng. Positive Doubly Periodic Solutions To Telegraph Equations: Existence, Uniqueness, Multiplicity and Asymptotic Behavior [J]. Acta mathematica scientia,Series A, 2022, 42(5): 1360-1380.
[10]	Shengliang Zhang,Junxian Huang. A High Order MQ Quasi-Interpolation Method for Time Fractional Black-Scholes Model [J]. Acta mathematica scientia,Series A, 2022, 42(5): 1496-1505.
[11]	Hongxing Wu,Dengbin Yuan,Shenghua Wang. Asymptotic Stability Analysis of Solutions to Transport Equations in Structured Bacterial Population Growth [J]. Acta mathematica scientia,Series A, 2022, 42(3): 807-817.
[12]	Shaowen Yao,Wenjie Li,Zhibo Cheng. Nondegeneracy and Uniqueness of Periodic Solution for Third-Order Nonlinear Differential Equations [J]. Acta mathematica scientia,Series A, 2022, 42(2): 454-462.
[13]	Jianzhong Min,Xiangao Liu,Zixuan Liu. Serrin's Type Solutions of the Incompressible Liquid Crystals System [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1671-1683.
[14]	Zaiyong Feng,Ning Chen,Yongpeng Tai,Zhengrong Xiang. On the Observer Design for Fractional Singular Linear Systems [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1529-1544.
[15]	Qing Cao,Xiaochuan Xu. Inverse Spectral Problem for the Diffusion Operator from a Particular Set of Eigenvalues [J]. Acta mathematica scientia,Series A, 2021, 41(3): 577-582.