Large-time behavior of solutions to the inflow problem of full compressible NavierStokes equations is investigated on the half line \mathbf{R}_+=(0,+\infty). The wave structure, which contains four wavesthe transonic (or degenerate) boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave, and the 3-rarefaction wave to the inflow problemis described, and the asymptotic stability of the superposition of the above four wave patterns to the inflow problem of full compressible NavierStokes equations is proven under some smallness conditions. The proof is given by the elementary energy analysis based on the underlying wave structure. The main points in the proof are the treatments of the degeneracies in the transonic boundary layer solution and the wave interactions in the superposition wave.

In this article, we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, properly separating entropy fuxes and production rates, and determining a dissipation matrix. Our approach takes advantage of both extended irreversible thermodynamics and GENERIC formalisms and shows a direct correspondence with Levermore’s moment-closure hierarchies for the Boltzmann equation. As a direct application, a new ten-moment model beyond the classical hierarchies is constructed and is shown to recover the Euler equations in the equilibrium state. These interesting results may put various macroscopic modeling approaches, starting from the general principles of non-equilibrium thermodynamics, on a solid microscopic foundation based on the Boltzmann equation.

We consider the self-adjoint Smilansky Hamiltonian H in L2 R2) associated with the formal differential expression -x2-1/2(y2+y2)-2y(x) in the sub-critical regime, (0, 1). We demonstrate the existence of resonances for H on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small > 0. In addition, we refine the previously known results on the bound states of H in the weak coupling regime (0+). In the proofs we use Birman-Schwinger principle for H, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

METHODS AND APPLICATIONS OF ANALYSIS. c 2009 International Press Vol. 16, No. 4, pp. 507520, December 2009 006

We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of the equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a finite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis.