In this paper, we study the asymptotic decay rates to the planar rarefaction waves to the Cauchy problem for a hyperbolic-elliptic coupled system called as a model system of the radiating gas in R n (n = 3, 4, 5) if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in (H 2 ∩ L 1 ∩ W 2,6 ) (R n ). The analysis is based on the L p -energy method and several special interpolation inequalities.
We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ R over the whole space R n for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ R, we not only obtain the L p -L q time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of smallamplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.
The vortex sheet solutions are considered for the inviscid liquid-gas two-phase flow. In particular, the linear stability of rectilinear vortex sheets in two spatial dimensions is established for both constant and variable coefficients. The linearized problem of vortex sheet solutions with constant coefficients is studied by means of Fourier analysis, normal mode analysis, and Kreiss symmetrizer, while the linear stability with variable coefficients is obtained by Bony-Meyer paradifferential calculus theory. The linear stability is crucial to the existence of vortex sheet solutions of the nonlinear problem. A novel symmetrization and some weighted Sobolev norms are introduced to study the hyperbolic linearized problem with characteristic boundary.
This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic-elliptic coupled system in the multi-dimensional radiating gasFirst, for the case with the same end states u − = u + = 0, we prove the existence and uniqueness of the global solutions to the above Cauchy problem by combining some a priori estimates and the local existence based on the continuity argument. Then L p -convergence rates of solutions are respectively obtained by applying L 2 -energy method for n = 1, 2, 3 and L p -energy method for 3 < n < 8 and interpolation inequality. Furthermore, by semigroup argument, we obtain the decay rates to the diffusion waves for 1 n < 8. Secondly, for the case with the different end states u − < u + , our main concern is that the corresponding Cauchy problem in ndimensional space (n = 1, 2, 3) behaviors like planar rarefaction waves. Its convergence rate is also obtained by L 2 -energy method and L 1 -estimate.
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