Perturbational blowup solutions to the compressible Euler equations with damping

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Background: The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions. Method: In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases γ>1 and γ=1, where γ is the adiabatic constant. Results: With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen.  Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points. Conclusion: The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications. Copyright © 2016 Cheung.
Original languageEnglish
Article number196
Publication statusPublished - Feb 2016


Compressible Euler Equations
Blow-up Solution
Exact Solution
Euler System
Well-posed Problem
Damping Term
Fluid Dynamics
Finite Volume Method
General Solution
Numerical Algorithms
Global Existence
Difference Method
Initial Value Problem
Finite Difference
Ordinary differential equation
Finite Element Method


Cheung, K. L. (2016). Perturbational blowup solutions to the compressible Euler equations with damping. SpringerPlus. Retrieved March 2, 2016, from


  • Blowup
  • Global existence
  • Euler equations
  • Perturbational method
  • Damping
  • Singularity