Following the authors’ earlier work in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations," and "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," we show that the nonlinear eigenvalue problem introduced in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," can be recast in the language of bifurcation theory as a perturbation of a linearized eigenvalue problem. Solutions of this nonlinear eigenvalue problem correspond to time periodic solutions of the compressible Euler equations that exhibit the simplest possible periodic structure identified in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations." By a Liapunov-Schmidt reduction we establish and refine the statement of a new infinite dimensional KAM type small divisor problem in bifurcation theory, whose solution will imply the existence of exact time-periodic solutions of the compressible Euler equations. We then show that solutions exist to within an arbitrarily high Fourier mode cutoff. The results introduce a new small divisor problem of quasilinear type, and lend further strong support for the claim that the time-periodic wave pattern described at the linearized level in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," is physically realized in nearby exact solutions of the fully nonlinear compressible Euler equations.

The pressure gradient system is a sub-system of the compressible Euler system. It can be obtained either through a flux splitting or an asymptotic expansion. In both derivations, the velocity field is treated as a small remnant of the original velocity of the Euler system. As such, the boundary conditions for the velocity do not necessarily follow the original ones and careful consideration is needed for the validity, integrity, and completeness of the model. We provide numerical simulations as well as basic characteristic analysis and physical considerations for the Riemann problems of the model to find out appropriate internal conditions at the origin. The study reveals subtle structures of the velocity: Both components exhibit discontinuities at the origin at traditional levels of numerical resolutions around 400×400 cells on the unit square, but they vanish at the origin with possible square root type singularity when we increase the resolutions. Comparing to the roll-up of shear waves or vortex-sheets of the Euler system, these singularities are mild and occur only along rays from the origin. The numerics is done at higher resolutions than traditionally possible via the automated clawpack that contains adaptive mesh refinement (AMR) and message passing interface (MPI), for which we provide the Riemann solvers in both the normal and transversal directions, where the Roe’s approximation has the elegant 1/2 average in the model’s original variables.

In this paper we investigate the asymptotic stability of viscous shock wave for a onedimensional isentropic model of viscous gas with density dependent viscosity by a weighted energy method developed in the papers of Matsumura-Mei (1997) and Hashimoto-Matsumura (2007). Under the condition that the viscosity coefficient is given as a function of the absolute temperature which is determined by the Chapman-Enskog expansion theory in rarefied gas dynamics, any viscous shock wave is shown to be asymptotically stable for small initial perturbations with integral zero. This generalizes the previous result of Matsumua-Nishihara (1985) where the viscosity coefficient is given by a constant and a restriction on the strength of the viscous shock wave is assumed. This also analytically assures the spectral stability in the Zumbrun’s theory for any viscous shock wave in our specific case.