Browse > Article
http://dx.doi.org/10.4134/BKMS.2010.47.1.029

SELF-SIMILAR SOLUTIONS FOR THE 2-D BURGERS SYSTEM IN INFINITE SUBSONIC CHANNELS  

Song, Kyung-Woo (Department of Mathematics, Kyung Hee University)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.1, 2010 , pp. 29-37 More about this Journal
Abstract
We establish the existence of weak solutions in an infinite subsonic channel in the self-similar plane to the two-dimensional Burgers system. We consider a boundary value problem in a fixed domain such that a part of the domain is degenerate, and the system becomes a second order elliptic equation in the channel. The problem is motivated by the study of the weak shock reflection problem and 2-D Riemann problems. The two-dimensional Burgers system is obtained through an asymptotic reduction of the 2-D full Euler equations to study weak shock reflection by a ramp.
Keywords
changing-type equations; degenerating quasilinear elliptic equations; self-similar solutions; 2-D full Euler equations;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 M. Brio and J. Hunter, Mach reflection for the two-dimensional Burgers equation, Physica D 60 (1992), no. 1-4, 194–207.   DOI   ScienceOn
2 S. Canic, B. L. Keyfitz, and E. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: regular reflection of weak shocks, Comm. Pure Appl. Math. 55 (2002), no. 1, 71–92.   DOI   ScienceOn
3 Y. Zheng, Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1849–1868.   DOI
4 Y. Zheng, Two-dimensional Riemann problems for systems of conservation laws, Birkhauser, 2001.
5 S. Canic, B. L. Keyfitz, and G. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), no. 4, 484–511.   DOI   ScienceOn
6 S. Canic and E. Kim, A class of quasilinear degenerate elliptic problems, J. Differential Equations 189 (2003), no. 1 , 71–98.   DOI   ScienceOn
7 L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990.
8 C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math. 47 (1994), no. 5, 593–624.   DOI
9 K. Song, A pressure-gradient system on non-smooth domains, Comm. Partial Differential Equations 28 (2003), no. 1-2, 199–221.   DOI   ScienceOn
10 E. Tabak and R. Rosales, Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection, Phys. Fluids A 60 (1994), no. 5, 1874–1892.   DOI   ScienceOn