Communications in Mathematical Physics

, Volume 296, Issue 2, pp 303–321 | Cite as

Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

  • Jiequan Li
  • Yuxi ZhengEmail author


The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.


Euler Equation Rarefaction Wave Riemann Problem Simple Wave Characteristic Decomposition 
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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsCapital Normal UniversityBeijingPeoples Republic of China
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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