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Nonlinear analysis and PDEs seminar

Time: Tuesday, May 14, 2013, 10h30-11h30

Location: IHP, room 201

Convex entropy, Hopf bifurcation and viscous and inviscid shock stability

Kevin Zumbrun (Indiana University - Personal website)

We discuss relations between one-dimensional inviscid and viscous stability/bifurcation of shock waves in continuum-mechanical systems and existence of a convex entropy. In particular, we show that the equations of gas dynamics admit equations of state satisfying all of the usual assumptions of an ideal gas, along with thermodynamic stability- i.e., existence of a convex entropy- yet for which there occur unstable inviscid shock waves. For general 3?3 systems (but not up to now gas dynamics), we give numerical evidence showing that viscous shocks can exhibit Hopf bifurcation to pulsating shock solutions. Our analysis of inviscid stability in part builds on the analysis of R. Smith characterizing uniqueness of gas dynamical Riemann solutions in terms of the equation of state of the gas, giving an analogous criterion for stability of individual shocks.

 

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