退化耗散型双曲系统的整体适定性与稳定性研究

This project is devoted to the global well-posedness and asymptotic stability for a class of hyperbolic systems with degenerate dissipation (including the equations of balance laws and the classical Boltzmann equation), with an emphasis on using the harmonic analysis tools. By employing the high-frequency and low-frequency decomposition methods, a new decay framework for solutions is introduced, which improves the classical spectral analysis framework which was well developed by famous PDE experts, for instance, Kawashima, Hoff-Zumbrum and Bianchini,etc..The framework enable us to obtain the optimal decay estimates of solutions to the hyperbolic systems with degenerate dissipation in spatially Besov spaces with critical regularity.It is worth noting that the space-time-velocity mixed spaces are introduced for Boltzmann equation, furthermore, the intricate collision operator is dealt with by using the Bony's para-differential calculus techniques. The study of this project contains the hot topics in the field of PDE, which contain some questions of international frontier and challenging difficulty. The most original results will be obtained.

本项目侧重利用调和分析工具研究一类退化耗散型双曲系统（包括平衡律方程组和经典的Boltzmann方程）的整体适定性与渐近稳定性问题。通过高低频分解方法,引入一个全新的衰减框架,改进了 Kawashima, Hoff-Zumbrum, Bianchini等著名偏微分方程专家的经典谱分析框架，在关于空间变量具有临界正则性指标的Besov空间中获得退化耗散型双曲系统解的最佳衰减估计。值得一提地是，对于Boltzmann方程的研究，本项目引进了包含微观速度的时空速度混合型Besov空间并运用Bony仿微分演算技巧来处理复杂的碰撞算子。本项目的研究内容包含了当今偏微分方程领域的研究热点，具有国际前沿性与挑战性，所得结果具有原创性。

本项目利用调和分析工具研究一类退化耗散型双曲系统（包括平衡律方程组、Boltzmann 方程和Navier-Stokes方程组等）的整体适定性与渐近稳定性问题。通过低高频分解方法，引入一个全新的衰减框架，改进了 Kawashima, Hoff-Zumbrum, Bianchini 等著名偏微分方程专家的经典谱分析框架，在关于空间变量具有临界正则性指标的Besov 空间中获得退化耗散型双曲系统解的最佳衰减估计。对于Boltzmann 方程的研究，本项目引进了包含微观速度的时空速度混合型Besov 空间并运用Bony 仿微分演算技巧发展碰撞算子的非线性估计。对于一类“正则性损失”的耗散方程组，本项目建立了一般形式的Lp-Lq-Lr衰减估计，在临界空间中获得最小衰减性。对于Navier-Stokes 方程组，本项目在负正则的Besov空间中发展非标准的乘积估计建立时间加权衰减不等式。这些结果具有创新性。