open access

https://doi.org/10.14943/83514

https://hdl.handle.net/2115/69118

BV-solutions of a hyperbolic-elliptic system for a radiating gas

英語

1997-01-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

368

1-33

This paper is concerned with the initial value problem of a system owning a hyperbolic equation with respect to unknown function u(x, t) and an elliptic one with respect to unknown function q(x, t) in one space dimension. This system originates in the dynamics of a radiating gas. The purpose in the present paper is to give the results on global existence and asymptotic behaviour of EV-solutions to the present system for the two cases: the first case is that initial data uo(x) decay as lxl → ∞ and the second one is that initial data uo(x) tend to two given constants 'U± with u_ < u+ as x→ ±∞. In the first case, we prove that the present problem is well-posed in EV, that is, for any EV-initial datum, there exists a unique EV-solution. We also show that if initial data uo are small in a certain sence, then the solutions u(·, t) decay in the order O(t-(I-I/p)/2) as t→ ∞ in V(R) with p E [1, ∞]. Furthermore, in the second case, we prove that the present problem is well-posed in ro+EV, where ro(x) is u_ when x < 0 and u+ when x > 0. Finally, we prove the main result of the present paper that if both iu+ - u-1 and initial EV-perturbations to ro are small in a certain sense, then the rarefaction waves of the inviscid Burgers equation are stable for the present system and the convergence rates of u( ·, t) to the rarefaction waves as t→ ∞ are O(c{I-I/p)/2) in V(R) with p E (1, ∞].

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