May  2021, 41(5): 2167-2185. doi: 10.3934/dcds.2020357

Radially symmetric stationary wave for two-dimensional Burgers equation

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

2. 

Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

Received  July 2020 Published  October 2020

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that the stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric initial perturbation.

Citation: Huijiang Zhao, Qingsong Zhao. Radially symmetric stationary wave for two-dimensional Burgers equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (5) : 2167-2185. doi: 10.3934/dcds.2020357
References:
[1]

L.-L. FanH.-X. LiuT. Wang and H.-J. Zhao, Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation, J. Differential Equations, 257 (2014), 3521-3553.  doi: 10.1016/j.jde.2014.07.001.  Google Scholar

[2]

I. Hashimoto, Asymptotic behavior of radially symmetric solutions for Burgers equation in several space dimensions, Nonlinear Anal, 100 (2014), 43-58.  doi: 10.1016/j.na.2014.01.004.  Google Scholar

[3]

I. Hashimoto, Behavior of solutions for radially symmetric solutions for Burgers equation with a boundary corresponding to the rarefaction wave, Osaka J. Math., 53 (2016), 799-811.   Google Scholar

[4]

I. Hashimoto, Stability of the radially symmetric stationary wave of the Burgers equation with multi-dimensional initial perturbation in exterior domain, Mathematische Nachrichten, (2020), 1-15. https: //doi.org/10.1002/mana.201900233. doi: 10.1002/mana.201900233.  Google Scholar

[5]

I. Hashimoto and A. Matsumura, Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation, J. Differential Equations, 266 (2019), 2805-2829.  doi: 10.1016/j.jde.2018.08.045.  Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.  doi: 10.1007/BF01212358.  Google Scholar

[7]

T.-P. LiuA. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal., 29 (1998), 293-308.  doi: 10.1137/S0036141096306005.  Google Scholar

[8]

T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[9]

T.-P. Liu and S.-H. Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Rational Mech. Anal., 139 (1997), 57-82.  doi: 10.1007/s002050050047.  Google Scholar

[10]

T.-P. Liu and S.-H. Yu, Multi-dimensional wave propagation over a Burgers shock profile, Arch. Ration. Mech. Anal., 229 (2018), 231-337.  doi: 10.1007/s00205-018-1217-5.  Google Scholar

[11]

A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, IMS Conference on Differential Equations from Mechanics (Hong Kong, 1999),, Methods Appl. Anal., 8 (2001), 645-666.  doi: 10.4310/MAA.2001.v8.n4.a14.  Google Scholar

[12]

K. NakamuraT. Nakamura and S. Kawashima, Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws, Kinet. Relat. Models, 12 (2019), 923-944.  doi: 10.3934/krm.2019035.  Google Scholar

[13]

K. Nishihara, Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws, J. Math. Anal. Appl., 255 (2001), 535-550.  doi: 10.1006/jmaa.2000.7255.  Google Scholar

[14]

K. Nishihara, Asymptotic behaviors of solutions to viscous conservation laws via $L^2-$energy method, Adv. Math. (China), 30 (2001), 293-321.   Google Scholar

[15]

T. Yang, H.-J. Zhao and Q.-S. Zhao, Asymptotics of radially symmetric solutions for the exterior problem of multidimensional Burgers equation (in Chinese), Sci. Sin. Math., 51 (2021), 1–16, See also arXiv: 1908.03354. Google Scholar

[16]

H. Yin and H.-J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space,, Kinetic and Ralated Models, 2 (2009), 521-550.  doi: 10.3934/krm.2009.2.521.  Google Scholar

show all references

References:
[1]

L.-L. FanH.-X. LiuT. Wang and H.-J. Zhao, Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation, J. Differential Equations, 257 (2014), 3521-3553.  doi: 10.1016/j.jde.2014.07.001.  Google Scholar

[2]

I. Hashimoto, Asymptotic behavior of radially symmetric solutions for Burgers equation in several space dimensions, Nonlinear Anal, 100 (2014), 43-58.  doi: 10.1016/j.na.2014.01.004.  Google Scholar

[3]

I. Hashimoto, Behavior of solutions for radially symmetric solutions for Burgers equation with a boundary corresponding to the rarefaction wave, Osaka J. Math., 53 (2016), 799-811.   Google Scholar

[4]

I. Hashimoto, Stability of the radially symmetric stationary wave of the Burgers equation with multi-dimensional initial perturbation in exterior domain, Mathematische Nachrichten, (2020), 1-15. https: //doi.org/10.1002/mana.201900233. doi: 10.1002/mana.201900233.  Google Scholar

[5]

I. Hashimoto and A. Matsumura, Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation, J. Differential Equations, 266 (2019), 2805-2829.  doi: 10.1016/j.jde.2018.08.045.  Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.  doi: 10.1007/BF01212358.  Google Scholar

[7]

T.-P. LiuA. Matsumura and K. Nishihara, Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal., 29 (1998), 293-308.  doi: 10.1137/S0036141096306005.  Google Scholar

[8]

T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[9]

T.-P. Liu and S.-H. Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Rational Mech. Anal., 139 (1997), 57-82.  doi: 10.1007/s002050050047.  Google Scholar

[10]

T.-P. Liu and S.-H. Yu, Multi-dimensional wave propagation over a Burgers shock profile, Arch. Ration. Mech. Anal., 229 (2018), 231-337.  doi: 10.1007/s00205-018-1217-5.  Google Scholar

[11]

A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, IMS Conference on Differential Equations from Mechanics (Hong Kong, 1999),, Methods Appl. Anal., 8 (2001), 645-666.  doi: 10.4310/MAA.2001.v8.n4.a14.  Google Scholar

[12]

K. NakamuraT. Nakamura and S. Kawashima, Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws, Kinet. Relat. Models, 12 (2019), 923-944.  doi: 10.3934/krm.2019035.  Google Scholar

[13]

K. Nishihara, Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws, J. Math. Anal. Appl., 255 (2001), 535-550.  doi: 10.1006/jmaa.2000.7255.  Google Scholar

[14]

K. Nishihara, Asymptotic behaviors of solutions to viscous conservation laws via $L^2-$energy method, Adv. Math. (China), 30 (2001), 293-321.   Google Scholar

[15]

T. Yang, H.-J. Zhao and Q.-S. Zhao, Asymptotics of radially symmetric solutions for the exterior problem of multidimensional Burgers equation (in Chinese), Sci. Sin. Math., 51 (2021), 1–16, See also arXiv: 1908.03354. Google Scholar

[16]

H. Yin and H.-J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space,, Kinetic and Ralated Models, 2 (2009), 521-550.  doi: 10.3934/krm.2009.2.521.  Google Scholar

[1]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[2]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[3]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[4]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[5]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[6]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[7]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[8]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[9]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[10]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[11]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[12]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[13]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[14]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[15]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[16]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389

[17]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[18]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[19]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[20]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

2019 Impact Factor: 1.338

Article outline

[Back to Top]