【作者】 蒋鹏；

【导师】 王亚光；

【作者基本信息】 上海交通大学 ， 基础数学， 2010， 博士

【摘要】 此博士论文研究辐射流体力学方程组定解问题的苻干数学理论问题,主要包括一维和高维辐射流体力学方程组初边值问题光滑解的局部存在性,三维辐射流体力学方程组解的奇性以及一维辐射流体力学方程组整体弱解的存在性辐射流体力学是描述热辐射在流体中的传播以及该辐射对一般流体运动的影响的学科。热辐射在物理问题中的重要性是随着温度的增加而增大的,这是因为辐射的能量密度是随着温度的四次方来变化的。在适当的温度下(比如几千Kelvin),辐射在流体运动过程中所起的作用是通过热辐射过程来传播能量。在更高的温度环境下(比如几百万Kelvin),辐射场的能量和动量密度将控制相应的流体粒子。在这种情况下,辐射场明显地影响流体的运动。因此在高温环境中,研究流体运动就必须考虑辐射的作用。辐射流体力学理论有着广泛的应用,如天体物理,激光核聚变,超新星爆炸理论等。对于一维和高维辐射流体力学方程组,我们先考虑其初边值问题的适定性。我们将利用Picard迭代,能量估计等方法得到光滑解的局部存在性,唯一性和稳定性。其次对三维等熵辐射流体力学方程组,我们证明该系统的一些C1解无论初值的大小,必在有限时间内产生破裂。最后我们考虑一维辐射流体力学方程组Cauchy问题体熵解。通过补偿列紧的方法,我们得到了在给定任意大的初值下整体的L∞弱熵解的存在性。更多还原

【Abstract】 Some mathematical problems in radiation hydrodynamics equations are considered in this thesis, including the local existence of smooth solutions to the initial boundary value problem for the one-dimensional and multidimensional radiation hydrodynamics equations, the formation of singularities of solutions to the three-dimensional radiation hydrodynamics equations and the global weak solutions to the one-dimensional radiation hydrodynamics equations.The radiation hydrodynamics concerns the propagation of thermal radiation through a fluid, and the effect of this radiation on the hydrodynamics describing the fluid motion. The importance of thermal radiation in physical problems increases as the temperature is raised, because the radiation energy density varies as the fourth power of the temperature. At mod-erate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominating the corre-sponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. Therefore, at higher temperature condition, one must consider the functions of radiation when we study the motion of fluid. The theory of radiation hydrodynamics owns a wide range of application, including in astrophysical, laser fusion, supernove explosions.For one-dimensional and multidimensional radiation hydrodynamics equations, we first study the well-posedness of the initial-boundary value problems. By using the Picard iter-ation, energy estimate we obtain the local existence of smooth solutions. Second for the three-dimensional isentropic radiation hydrodynamics equations, we prove that some C1 so-lutions should blow-up in a finite time regardless of the size of the initial disturbance. Finally, for the Cauchy problem of the one-dimensional radiation hydrodynamics equations, by us-ing the compensated compactness argument, we establish the existence of a global entropy solution in L∞with arbitrarily large initial data.更多还原