【作者】 胡丹妮；

【导师】 李刚； 朱江；

【作者基本信息】 南京信息工程大学 ， 应用数学， 2008， 硕士

【摘要】 本文针对较为典型的工程问题及流体力学中常见的一类不可压拟牛顿流不定常热耦合Stokes问题进行研究，在假设初始温度θ₀∈L²（Ω），耦合函数κ∈L^∞（Ω×（0，T）），考虑齐次边界条件并假设耦合函数μ，κ∈C（IR×（0，T））是有界的条件下，主要研究了这类问题方程组弱解的存在性，惟一性和解的爆破的一些结果，模型如下：全文主要包括三部分的内容：第一部分，在引入变分问题的基础上，利用Faedo-Galerkin方法证明了此类不可压拟牛顿流不定常热耦合Stokes问题弱解的存在性。第二部分，针对此类问题，利用嵌入定理的相关知识以及Meyers估计和Schauder不动点定理证明了弱解的存在性。第三部分，通过建立弱解对初边值的估计式，利用相关技巧得到弱解的惟一性。同时在一定的假设条件下得到了解的爆破。各部分中均将微分方程组转化为变分问题，通过研究相应变分问题的不动点来研究原微分方程组解的存在性。在文章的证明过程中，嵌入定理，Meyers估计，Schauder不动点定理和Faedo-Galerkin方法发挥了重要作用。更多还原

【Abstract】 Several typical kinds of PDE models called coupled nonlinear equations have been considered in this thesis and the existence , uniqueness and blowup results of the respective elliptic equations are obtained.The model in this thesis is:This thesis is mainly composed of three parts of contents:In the first part, under the condition of the variational formulation, existence of the unsteady thermally coupled Stokes problem which describing the unsteady flow of a quasi-Newtonian fluid with temperature-dependent viscosity and with a viscous heating is proved under Faedo-Galerkin method.In the second part, by Meyers estimate, Schauder fixed point theorem together we study the boundary value problem for the thermally coupled Stokes problem and the coexistence of existence is obtained.The third part studies the estimate of the weak solution which depends on the initial and boundary conditions is established to prove the uniqueness. Results on blowup of the weak solution is also studied.Differential equations are transformed to variational formulation all the above parts. By the existence of the fixed point for the operaor equation, we obtain the existence for the differential equations, in which the Meyers estimate, Faedo-Galerkin method and Schauder fixed point theorem play an important role.更多还原