【作者】 王玉柱；

【导师】 孔德兴；

【作者基本信息】 上海交通大学 ， 应用数学， 2010， 博士

【摘要】 本文主要研究了流体力学方程组的Cauchy问题。本文由以下几章组成：第一章为绪论。在本章中我们简要回顾了一阶拟线性双曲组和与本文相关的一些偏微分方程的研究历史及一些重要结果，并叙述了本文的主要结论。第二章研究了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。在初值是一个常状态的小扰动并且初速度的旋度等于零的假设下，我们证明了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。第三章研究了两维的可压非等熵Euler方程的Cauchy问题。当初值是一个常状态的小扰动时，我们给出光滑球对称解的生命跨度的精确估计。在第四章中我们证明了Minkowski空间R¹⁺³中具有慢衰减初值的极值曲面方程Cauchy问题整体光滑解的存在性和唯一性。在第五章中我们证明了广义Boussinesq方程Cauchy问题整体解的存在性和唯一性。在适当的假设下，我们进一步证明当t趋向无穷大时，广义Boussinesq方程Cauchy问题整体小解的L_∞范数趋向于0。第六章研究了R³空间中的不可压magneto-micropolar流体方程组和Rⁿ（n =2, 3）空间中的具有部分粘性的不可压magneto-micropolar流体方程组的Cauchy问题，得到了光滑解的爆破准则。更多还原

【Abstract】 This thesis mainly concerns the Cauchy problems for fluid flows. This thesis isorganized as follows:In Chapter 1, we review some of the history and important known results on thestudies of first-order quasilinear hyperbolic systems, and other partial differential equationsrelated to the topic of the thesis. We also state the main results in this thesis.In Chapter 2, we investigate the two-dimensional compressible isentropic Euler equationsfor Chaplygin gases. Under the assumption that the initial data is close to a constantstate and the vorticity of the initial velocity vanishes, we prove the global existence of thesmooth solution to the Cauchy problem for two-dimensional flow of Chaplygin gases.In Chapter 3, we study the spherically symmetric Eulerian flows of ideal polytropicgases with variable entropy in two space dimensions. Under the assumption that theinitial data is close to a constant state, we give a precise estimate on the life-span ofsmooth solutions.In Chapter 4, we prove the global existence of classical solutions to the Cauchyproblem for the minimal surface equation in the Minkowski space ?1+3 with slow decayinitial data.In Chapter 5, we show the existence and the uniqueness of global solution for theCauchy problem for the generalized Boussinesq equation. Under suitable assumptions, wealso prove that the L~∞norm of small solution to the Cauchy problem for the generalizedBoussinesq equation decays to zero as ?? tends to the infinity.In Chapter 6, we investigate the Cauchy problem for incompressible magneto-micropolarfluid equations in R~3 and incompressible magneto-micropolar fluid equations with partialviscosity in R~n(n = 2, 3). In this Chapter we derive blow-up criteria of smooth solutions.更多还原