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Existence and Stability of Multidimensional Transonic Shocks for the Euler Equations for Steady Potential Fluids in Unbounded Domains

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

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  • Gui-Qiang Chen

    (Northwestern University, Department of Mathematics)

  • Mikhail Feldman

    (University of Wisconsin, Department of Mathematics)

Abstract

We are concerned with the existence and stability of multidimensional transonic shocks in inviscid compressible fluid dynamics. In this paper, we focus on inviscid steady potential fluid flows, which are governed by the Euler equations consisting of the conservation law of mass and the Bernoulli law for velocity. Then the Euler equations for the velocity potential ϕ: Ω ⊂ Rn → R can be formulated into the following second order nonlinear equation of mixed elliptic-hyperbolic type: (1) $$ div(\rho (|D\varphi {{|}^{2}})D\varphi ) = 0, $$ where the density function ρ(q 2) has the form: (2) $$ \rho ({{q}^{2}}) = {{\left( {1 - \frac{{\gamma - 1}}{2}{{q}^{2}}} \right)}^{{\frac{1}{{\gamma - 1}}}}} $$ with the adiabatic exponent ψ>1.

Suggested Citation

  • Gui-Qiang Chen & Mikhail Feldman, 2003. "Existence and Stability of Multidimensional Transonic Shocks for the Euler Equations for Steady Potential Fluids in Unbounded Domains," Springer Books, in: Thomas Y. Hou & Eitan Tadmor (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 419-432, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55711-8_38
    DOI: 10.1007/978-3-642-55711-8_38
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