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Rarefaction
About: Rarefaction is a research topic. Over the lifetime, 1852 publications have been published within this topic receiving 26943 citations.
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TL;DR: In this article, it was shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law.
Abstract: This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary L 2-energy method. 0. INTRODUCTION We will establish the asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in two or more space dimensions. We consider n-dimensional scalar viscous conservation laws of the form n n (1) ut+EL(f(u))x = E aijux1u x E R, t > 0, i=1 i,j=l where u E R1, A = (aij), called the viscosity matrix, is a constant positive definite matrix, and we assume that all the flux functions are smooth (say in Cn ) and equation (1) is genuinely nonlinear in the x1-direction [8], i.e., for a fixed constant a > 0, (2) Jf'(u) > a . The initial data for equation (1) is (3) u(x, O) = uo(x) satisfying (4) lim |u(x1, *) U? ILx(Rn-I) = 0 where u?, u_ < u+, are two constants. A planar rarefaction wave (in x1 direction) Ur(xl, t) is a solution of the following initial value problem for the Received by the editors September 27, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 35Q99, 35K35.
55 citations
TL;DR: In this article, the effect of various flow param- eters entering into the problem such as Hartmann number, rarefaction parameter, fluid-wall inter-action parameter and wall-ambient temperature difference ratio are discussed with the aid of line graphs.
55 citations
TL;DR: The expansion of a two-electron population plasma into vacuum is investigated in a controlled laboratory experiment and a potential double layer, called a rarefaction shock, develops where the two separate.
Abstract: The expansion of a two-electron population plasma into vacuum is investigated in a controlled laboratory experiment. As the plasma expands, the colder electron population lags behind the energetic tail population, and a potential double layer, called a rarefaction shock, develops where the two separate. Upstream of this double layer, both electron populations exist; but downstream, only the tail electrons do. During the expansion, ions are accelerated to energies well above the tail electron energy.
55 citations
TL;DR: The asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions.
Abstract: The inflow problem of full compressible Navier–Stokes equations is considered on the half-line $(0,+\infty)$. First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state $(\rho_+,u_+,\theta_+)$ belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.
54 citations
TL;DR: In this article, the authors studied the asymptotic behavior of the rarefaction wave in the Burgers equation with viscosity and showed that the solution satisfies supR |u −uR| ∼t−1/2 ast → ∞.
Abstract: This paper is concerned with the asymptotic behavior toward the rarefaction waveuR(x/t) of the solution of the Burgers equation with viscosity. If the initial data are suitably close to constant stateu± atx=±∞, then the solutionu(x, t), roughly speaking, satisfies supR |u −uR| ∼t−1/2 ast → ∞ and, except for the “neighborhoods” of the corners,x=u±t ofuR, sup |u−uR|∼t−1. In the proof the exact forms ofu are available.
54 citations