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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 |
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.
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. |
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. |
[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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