| Abstract: |
| We discuss a unique existence and an asymptotic stability of a stationary solution for a dissipative system of hyperbolic balance laws over one-dimensional half space. Assuming the existence of an entropy function, we rewrite the system to a symmetric dissipative hyperbolic system. One difficulty of this problem occurs from the fact that the number of boundary conditions for the unique existence of the stationary solution does not coincide with the number of boundary conditions for well posedness of the time-dependent equations. To handle this problem, we derive a formula to calculate the former number. Then we see the former is less than or equal to the latter. So we may add boundary conditions for the time-dependent equations by some values of the stationary solution on the boundary to make the time-dependent problem well-posed. Then we discuss the asymptotic stability of the stationary solution. It is proved by deriving the a-priori estimate in Sobolev space. To derive a basic estimate, we use an equation satisfied by an energy form, defined through the entropy function. For higher order estimates we utilize the symmetric system. Finally, we discuss concrete models, such as the discrete Boltzmann equations and a system of thermal non-equilibrium flow. |
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