Asymptotic Behavior Of Solutions To A Hyperbolic-Elliptic Coupled System In Radiating Gas

This dissertation is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic-elliptic coupled system in n-dimensional radiatinggaswith initial datau(x,0)=u(x₁,…,x_n,0)=u₀(x₁,…,x_n)→u_±, x₁→±∞, (0.1.2)where u_±are given constant states, a∈Rⁿ is a constant vector and u, q are unknown functions of the spacial variable x = (x₁,x₂,…,x_n)∈Rⁿ and the time variable t. Typically, u=u(x, t) and q=(q₁,…,q_n)(x, t) represent the velocity and radiating heat flux of the gas respectively.The system (0.1.1) is a simplified version of the model for the motion of radiating gas in n-dimensional space. More precisely, in a certain physical situation, the system (0.1.1) gives a good approximation to the fundamental system describing the motion of a radiating gas, which is a quite general model for compressible gas dynamics where heat radiative transfer phenomena are taken into account and given by the hyperbolic-elliptic coupled equationswhereρ,u,p,e andθare respectively the mass density, velocity, pressure, internal energy and absolute temperature of the gas, while q is the radiative heat flux, and a and b are given positive constants depending only on the gas itself. The first three equations are motivated as for the usual Euler system, which describe the inviscid flow of a compressible fluid and express conservation of mass, momentum and energy respectively (see [5]). And the investigation of the Euler equations is a classical topic. However the physical motivation of the fourth equation, which take into account of heat radiation phenomena, is given in [15, 79].First, for the case with the same end states u_- = u₊ = 0, we prove the existence and uniqueness of the global solutions to the Cauchy problem (0.1.1), (0.1.2) by combining some a priori estimates and the local existence based on the continuity argument for 1 < n < 8. Then the L^p-convergence rates of solutions are respectively obtained by applying the L²-energy method for n = 1,2,3 and the L^p-energy method for 3 < n < 8 and interpolation inequality. Furthermore, by semigroup argument, we obtain the optimal 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 n-dimensional space (n = 1,2,3) behaviors like planar rarefaction waves. Its L^∞convergence rate is also obtained by the standard L²-energy method and L¹-estimate.Whether case u_- = u₊ = 0 or u_-< u₊, the underlying obstacles result from the high order L^p-energy estimates. On the other hand, in the case of u_- = u₊ = 0 for 3 < n < 8, the underlying obstacle also result from W^3,p estimates. However, we fortunately find that the system (0.1.1) can be transformed into a high-order scalar equation of conservation law. This is also one of the crucial observations for overcoming our difficulty.