Advanced Search
Article Contents
Article Contents

# Radially symmetric stationary wave for two-dimensional Burgers equation

• 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.

Mathematics Subject Classification: Primary: 35A01; Secondary: 35B40, 35K59, 37K40.

 Citation:

•  [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.

## Article Metrics

HTML views(413) PDF downloads(253) Cited by(0)

## Other Articles By Authors

• on this site
• on Google Scholar

### Catalog

/

DownLoad:  Full-Size Img  PowerPoint