American Institute of Mathematical Sciences

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

Radially symmetric stationary wave for two-dimensional Burgers equation

Huijiang Zhao 1,2, and  Qingsong Zhao 1,

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  May 2021 Early access  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.

Keywords: Radially stationary wave, Burgers equation, exterior problem, time-asymptotically nonlinear stability.
Mathematics Subject Classification: Primary: 35A01; Secondary: 35B40, 35K59, 37K40.
Citation: Huijiang Zhao, Qingsong Zhao. Radially symmetric stationary wave for two-dimensional Burgers equation. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2167-2185. doi: 10.3934/dcds.2020357
References:
[1]

L.-L. Fan, H.-X. Liu, T. 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. Liu, A. 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. Nakamura, T. 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. Fan, H.-X. Liu, T. 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. Liu, A. 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. Nakamura, T. 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

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