流体力学方程的适定性及其相关问题

The application mainly concerns with the well-posedness and the solitary waves of the partial differential equations of hydrodynamics. Based on the published work of our team (Invent.Math., Comm.Math.Phys., Arch.Rat.Mech. Anal.,et.al.), the program will study: (1) The H?lder continuity of the data to solution map of the Benjamin-Ono equation in weak topology of energy space and the optimality of the index by harmonic analysis method, and the possibilty of the low regularity of this solution map in Sobolev space of negative index; (2) For Ostrovsky equation we establish a similar Liouville theorem to elliptic equation and monotone formula to obtain asymptotic stability of their solitary waves; (3) To obtain the existence of multi-bump solutions of two dimension Ostrovsky equation by variational method. We also study the symmetry of the nodal sets of their solitary waves and the uniqueness of the solitary waves with minimum energy in the sense of translation and scaling using the curvature flow method and variational method; (4) The incontinuity of the data-to-solution map of the one dimensional compressible Euler equation with physical vacuum condition in usual Sobolev space and the continuity in suitable weighted function spaces. This will be useful in understanding the difference between the degenerate and the nondegenerate equations from the point of solution maps. There are very little work on the asymptotic stability of high order equations and the continuity of the solution map of degenerate equations now. We will do our best to make a break through on such problems.

本项目研究流体力学中一些偏微分方程的适定性及其相关问题。基于项目组成员已有的工作(Invent.Math.,Comm.Math.Phys.,Arch.Rat.Mech.Anal.等)，(1) 我们找出Benjamin-Ono方程的解映射在能量空间弱拓扑下的连续指标并给出指标最佳的证明，由此探索负指数Sobolev空间中解的存在性和解对初值的连续依赖性；(2) 利用色散性质、刚性定理等证明四阶Ostrovsky方程孤立波的渐近稳定性；（3）利用变分方法和曲率流的分析技巧证明两维Ostrovsky方程多峰孤立波的存在性和一定意义下的唯一性； (4)用构造近似解的方法证明具物理真空一维可压Euler方程组的解映射在任意阶的Sobolev空间中不连续但在适当加权Sobolev空间中连续，从解映射的角度对退化和非退化方程的区别加深理解。退化方程解映射连续性目前只有很少的工作，本项目力求有一点突破。

我们考虑流体力学的几个方程，对以下几方面作了研究： (1). 考虑b-族方程在时间[0, 1]上的解映射，定义域是Sobolev空间指标s大于3/2的有界集，赋予Sobolev空间指标r间于0和s之间的弱拓扑，值域是Sobolev空间指标r，我们得到这种解映射的Holder连续性。这说明b-族方程初值问题的低频部分改变了波速，在今后的研究中有望对相关方程有进一步了解。 (2). 考虑2-分组Camassa-Holm系统在时间区间[0, 1]上的解映射，假设A是指标分别为s和r两个Sobolev空间的乘积，s大于等于1且r非负。考虑A的有界集到A的映射。我们得到这种解映射是非一致连续的。这使我们对非一致连续机制有了进一步的认识，在今后的研究中有望对CH方程和2-分组CH 系统的区别有进一步了解。 (3). 考虑Keldysh方程在直角坐标系和极坐标系下的特解。我们得到Keldysh的特解和已知的Tricomi方程的特解有本质区别，且Keldysh 算子比Tricomi算子在退化线上退化性的强。 当以流体速率为应变量，流函数和势函数为自变量时混合定常气流化成Keldysh型方程。Keldysh的特解对理解音速线为直线，管壁在音速线附近充分平直时的光滑跨音速流有重要作用。 (4). 得到Tricomi算子极点在椭圆区域基本解的解析表达式，以及当极点趋于退化线时基本解的极限行为； 对Chaplygin方程解的存在性进行研究。利用Tricomi方程的基本解等已知知识我们对Chaplygin方程混合边值问题解的存在性的有了进展。