多组分流体中的NS-MS模型和非线性紧性定理

We intend to study two aspects. First, we consider the Navier-Stokes-Maxwell-Stefan(NS-MS) equations which model the dynamics of a multicomponent fluid. Applications arise, for instance, from physics(sedimentation, astrophysics), medicine(dialysis,respiratory airways),and chemistry(electrolysis, ion exchange, chemical reactors). The NS-MS system has been adopted by a number of industries. The understanding of the analytical structure of coupled NS-MS system is of great importance for an accurate modeling and efficient numerical simulation of these applications. Because of the complex mathematical structure such as the singular Maxwell-Stefan diffusion matrix, the cross-diffusion terms, and the Navier-Stokes coupling, standard tools like maximum principle and regularity theory are not available for the general cases of NS-MS system. In view of these difficulties, it is not natural that there exist only partial mathematical results on such system in the literature. The main progresses were made within these three years, which only dealt with some special cases instead a general one. So the NS-MS model is a new topic in the eyes of mathematicans from Europe and USA. We will study the global well-posedness and long time behaviour of impressible and compressible full NS-MS equations, using an entropy-dissipation method. Second, we consider some nonliner compactness theorems in Lp(0,T;B).The Aubin-Lion(-Dubinskii) lemma states criteria under which a set of functions is relatively compact in Lp(0,T;B), which is widely used in the study of evolution partial differential equations. Introduced in 1960s, mainly improved by Simon in 1987, it was revisited by some authors within these three years. A typical application of Dubinskii-type lemma is the study of some evoultion equations with terms of exponential functions. We will investigate some nonliner compactness theorems which can be applied to general funcitons.

本项目拟研究两个方面。第一，刻画多组分流体的Navier-Stokes-Maxwell- Stefan(NS-MS)方程组，在物理学、医学和化学等学科均有应用，是工业界广泛采用的模型。鉴于其复杂的数学结构(如奇异MS矩阵、交叉扩散项、NS方程的强耦合作用)，标准的极值原理和正则性理论很难适用于完整的模型。致使现有理论进展主要集中于近三年，且仅限于某些特殊情形；对其一般形式的研究尚属空白。作为一个新兴的数学研究对象，NS-MS模型受到了多个欧美数学研究团队的关注。我们拟运用熵-耗散方法研究该模型在不可压和可压情形下的整体适定性和大时间行为。第二，Aubin-Lions(-Dubinskii)引理是对发展方程进行紧性讨论的基本工具之一,引入于上世纪60年代，近三年重获关注。现有Dubinskii-型引理主要适用于带有指数非线性项的发展方程，我们拟突破这一限制，探讨适用面更广的非线性紧性定理。

我们主要围绕多组分流体中的NS-MS模型的整体适定性和大时间行为；非线性紧性定理；包含MS在内的反应-交叉-扩散模型的整体适定性等内容进行研究。已经在Commun. Math. Phys.，Arch. Rational Mech. Anal.，M3AS等重要的国际学术期刊上发表或接收SCI期刊论文13篇。代表性成果如下： 第一、化学工程等学科中广泛应用的Maxwell-Stefan方程组是刻画多组分扩散系统的经典模型，又是工业界广泛采用的模型。我们证明了法国科学院院士R.Temam于1995年提出但未证明的结果。完成了某些情况下不可压NS-MS模型的整体弱解存在性证明。此项成果获得了R.Temam本人的多次引用。还被柏林魏尔斯特拉斯研究所教授W.Dreyer，巴黎综合理工大学教授V.Giovangigli，布拉格查理大学教授M.Pokorny和华沙大学教授P.B.Mucha等人多次引用。此外，证明了弱解以指数形式衰减到齐次稳态的特性。第二、利用改进的熵方法和新的逼近格式建立了细致平衡条件下或某种弱交叉扩散条件下多物种人口动力学模型整体弱解的存在性；而且发现了熵单调性与Markov链相关的细致平衡条件之间的内在关系。此外，得到了适合反应-交叉-扩散模型理论分析的非线性紧性定理。被法国科学院院士R.Temam和法国J.L.Lions实验室的A.Moussa等人多次引用。第三、我们证明了一类交叉扩散模型（耦合Poisson方程）弱解的唯一性。这类模型包括多组分流体Maxwell-Stefan方程组，人口动力学模型和用于离子传输的容积填充方程组等模型的某些具有共性的特殊情形。此项成果被法国J.L.Lions实验室的A.Moussa等人引用。第四、我们建立了熵耗散反应-交叉-扩散模型重整化解的弱-强唯一性。

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