Structures and singular perturbations limits of solutions to the systems of non-strictly hyperbolic nonlinear partial differential equations for the conservation laws in the phase transition dynamics

(1)我们证明了具有脐点的2×2非线性双曲守恒律方程组存在经典解和激波，也存在非经典解和超压缩激波. (2)本文给出了一类具有高阶项的双曲型守恒方程的奇摄动极限收敛于守恒律解的一个充分条件。(3)利用补偿紧性方法证明了相对论性欧拉方程含真空解的存在性。(4)在一般Lp空间的背景下，我们证明了Boltzmann方程在真空附近解的稳定性。

(1) We show that, for the 2×2 nonlinear hyperbolic systems of conservation laws with an umbilic point, there exists classical solutions, shock waves and also non-classical solutions, overcompressive shock waves. (2) We give a sufficient condition for guarantee that the singular perturbation limits of the hyperbolic conservation conservation equation with a higher order terms converge to the solution to the conservation law. (3) We prove the existence of solutions including vacuum to the relativistic Euler equation by using the compensated compactness method. (4) We show the stability of solution to the Boltzmann equation near vacuum in the context of general Lp spaces.