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)证明了具有脐点的2×2非线性双曲型守恒律组存在经典解、超压激波和非经典解。(2)给出了带高阶项的双曲型守恒方程的奇摄动极限收敛于守恒律的解的充分条件。(3)利用补偿紧性方法证明了相对论欧拉方程解的存在性。(4)证明了一般Lp空间中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.