Abstract
This article is concerned with nonlinear stability together with the corresponding convergence rates of rarefaction waves for a hyperbolic-elliptic coupled system arising in the 1D motion of a radiating gas with large initial perturbation. Compared with former results, although we ask the L2(R)-norm of the initial perturbation to be small but the L2(R)-norm of the first- and second-order derivatives of the initial perturbation with respect to the spatial variable x can be large and consequently, the H2(R)-norm of the initial perturbation can be large. AMS Subject Classifications: 34K25; 35M10.
Highlights
1 Introduction This article is concerned with the large time asymptotic behavior of the global solution to the Cauchy problem for the following hyperbolic-elliptic coupled system ut + f (u)x + qx = 0, −qxx + q + ux = 0 (1:1)
It is worth to pointing out in this step that we ask the initial perturbation to satisfy (1.8). Another purpose of this article is to show that if the Cauchy problem (2.12)-(2.14) admits a unique global solution (j(t, x), ψ(t, x)) = (u(t, x) - w(t, x), q(t, x) + ∂xw(t, x)) and there exists some positive constant E, which is independent of t, x, and δ for sufficiently small δ > 0, such that φ(t) L∞(R) + φx(t) L∞(R) ≤ E, ∀t ≥ 0 (1:16)
Theorem 2.2 (Decay estimates) Let j0 (x) Î H2(R) ∩ L1(R) and (j(t, x), ψ(t, x)) be the unique global solution of the Cauchy problem (2.12)-(2.14) satisfying (1.16) and (1.17), there exists a positive constant T such that for each fixed a > 4, we have for any t ≥ 0 that
Summary
Recall that depending on whether δ = u+ - u-, the strength of the rarefaction waves, and/or certain Sobolev norm, cf ||u0(x) − wR0(x)||L2(R) + ||∂xu0(x)||H1(R), of the initial perturbation u0(x) - w(0, x) are assumed to be small or not, the corresponding stability results are called local (or global) nonlinear stability of strong (or weak) rarefaction waves, respectively For such a problem, when f (u) = 1 u2, by introducing the smooth approximation 2 w (t, x) of the rarefaction wave solution wR(t, x) as the unique solution of the following. It is worth to pointing out in this step that we ask the initial perturbation to satisfy (1.8) Another purpose of this article is to show that if the Cauchy problem (2.12)-(2.14) admits a unique global solution (j(t, x), ψ(t, x)) = (u(t, x) - w(t, x), q(t, x) + ∂xw(t, x)) and there exists some positive constant E, which is independent of t, x, and δ for sufficiently small δ > 0, such that φ(t) L∞(R) + φx(t) L∞(R) ≤ E, ∀t ≥ 0.
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