# American Institute of Mathematical Sciences

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  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.

Citation: Huijiang Zhao, Qingsong Zhao. Radially symmetric stationary wave for two-dimensional Burgers equation. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] K. Nishihara, Asymptotic behaviors of solutions to viscous conservation laws via $L^2-$energy method, Adv. Math. (China), 30 (2001), 293-321. [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. [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.

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