几类偏微分方程的伪抛物粘性化问题

Viscosity is one of the most important parts in the study of partial differential equations, which can reflect the real physical phenomena and become a powerful theoretical study tool. This project intends to study such a kind of viscosity: pseudo-parabolic viscosity. It has a special form of hybrid time-space derivative, can be used to describe the thermodynamic temperature in the heat conduction, viscoelasticity in the string vibration, dispersion of the BBM equation, capillary pressure of oil and water dynamics, and also can be used as an effective regularization for treating the ill-posed models, such as backward heat equation. Via studying the heat equation, fractional diffusion equation and conservation law with pseudo-parabolic viscosity, this project aims to investigate the effect of the viscosity on the behavior of solutions, so as to reveal the difference and relation between the important properties of solutions to that of the corresponding parabolic and hyperbolic models, and thus provide some approximation processes, which are more reasonable and tally with the real situation, for theoretical analysis and numerical calculation. The special structure of the pseudo-parabolic equation will bring essential difficulties and new challenge to the research. Our results can provide important reference information and theoretical basis for practical problems, and also enrich and develop the theory of partial differential equations.

粘性是偏微分方程研究中的重要组成部分，合理的粘性需要既能反映真实的物理现象又能成为有力的理论研究工具。本项目拟研究的就是这样一类粘性：伪抛物粘性。它具有时-空混合导的特殊形式，可用来描述双温热传导的热力学温度，弦振动的粘弹性，BBM方程的色散效应，水油动力学的毛管压力，同时也可作为研究倒向热方程等非适定问题的有效正则化方式。本项目旨在通过研究热方程、分数阶扩散方程及守恒律方程的伪抛物粘性化问题，考察伪抛物粘性对解的性态的影响，从而揭示其继承与发展抛物型及双曲型方程解的反映重要物理意义的性质，为理论分析和数值计算提供一种更合理的更符合真实现象的逼近过程。伪抛物型方程本身特有的结构，会为研究带来本质困难和全新挑战，需要对研究工具和方法的不断拓展和创新。研究成果既可以为实际问题提供重要的参考信息和理论依据，又可以丰富和发展偏微分方程的理论体系。

本项目研究的伪抛物粘性具有时-空混合导的特殊形式，可用来描述双温热传导的热力学温度，弦振动的粘弹性，水油动力学的毛管压力，同时也可作为研究正倒向扩散方程等非适定问题的有效正则化方式。项目研究内容包括对具伪抛物粘性的热方程发现初值在无穷远处的任意小非稀疏性都将导致解在有限时刻的爆破现象发生。对Lane-Emden热流系统的时间周期解问题做出了全面解答，建立了完整的理论结果，确定了两条临界曲线。对于混合伪抛物Kirchhoff方程，采用位势井理论，通过初始能量划分解的整体存在和有限时刻爆破，不但得到了整体存在解的衰减和增长速率的最新估计，还进一步得到解与稳态解的收敛关系。对完全非线性变指标椭圆抛物方程，通过使用几何切线方法，并将平面度方法的精细改进与紧性和伸缩技术相结合，获得了此类方程粘性解的局部C^{1,\alpha}正则性。项目组按照年度研究计划开展了研究工作，基本达到了预期目标，研究工作发表于Journal of Funcional Analysis，Nonlinearity，Bulletin of the London Mathematical Society等期刊上。伪抛物粘性本身特有的结构，会为研究带来本质困难和全新挑战，需要对研究工具和方法不断拓展和创新。研究成果既可以为实际问题提供重要的参考信息和理论依据，又可以丰富和发展偏微分方程的理论体系。