在這兩年期計畫裏，我們研究應用科學中非線性守恆律解的存在唯一性及解的行為。我們要研究 的模型最主要有下面的偏微分方程系統: 1. 天文物理中大氣散逸問題有關的可壓縮尤拉方程，此方程帶有重力項及廣域熱源。 2. 有關於生物趨向性帶有小參數的 Keller-Segel 方程。 3. 重力場理論中的可壓縮 Euler-Poisson 方程。 在第一年的計畫中，我們最主要是考慮大氣散逸問題中的解的穩定性，我們最主要是用新的有限 差分格式及對廣域熱源項的迭代方法。我們也會考慮大氣散逸問題中的多型態粒子模型解的適定性及 穩態解的存在唯一性，我們也會提供解的數值模擬。 在第二年的計畫中，我們考慮Keller-Segel 方程的脈衝波的穩定性，我們期待證明，唯一的穩 定脈衝波是穩態解。對於可壓縮Euler-Poisson 方程，我們期待找到邊界動量的條件，使得重力塌陷 的現象不會發生。我們期待得到此類型問題的分析及數值結果。 ;In this two-years project, we study the existence, uniqueness and behavior of solutions to several models of nonlinear systems of balance laws arise in areas of Applied Sciences. The models are governed by the following PDE systems: (1)Compressible Euler equations with gravitational and global heating source in hydrodynamic escape problem (HEP in short) of astrophysics. (2) Keller-Segel systems with small parameters in the chemotaxis of Biology. (3)Compressible Euler-Poisson equations in gravitation theory. In the first year, we focus on the stability of solutions to the HEP model by a new version of finite difference scheme that involves an iteration of global heating source. The multiple-phases model of HEP is also studied on the steady states and the global well-posedness of time-evolutionary solutions. Numerical simulations are also provided. In the second year, we focus on the stability issue of traveling pulses for Keller-Segel systems with small parameters. We wish to show that the only asymptotically stable traveling pulse is the steady state. For the compressible Euler-Poisson equations in gravitations, we find the condition of boundary momentum to prevent the gravitational collapse of the gas-like stars. We wish to obtain both analytic and numerical results.
