修正的两个分量的 Camassa-Holm 系统的若干问题研究

This program mainly studies the Cauchy problems of three modified two-component Camassa-Holm systems with different form. Since all of the systems have strong physical background, the study of them has strong practical significance.. Firstly, we establish local well-posedness for systems in Besov spaces, which optimize the present corresponding results. Secondly, according to the characteristic of our systems, we derive the precise blow-up scenario by the methods of harmonic analysis, then on its basis to study blow-up of strong solutions, global existence of strong solutions of strong solutions. Finally, we study the the existence and the uniqueness of global weak solutions using mollifying the initial data, viscous approximation or Lagrangian coordinates transformation.. The program studies and improves the local well-posedness, provides blow-up of strong solutions, global existence of strong solutions, research the existence and uniqueness of global weak solutions of the Cauchy problems of three modified two-component Camassa-Holm systems.

本项目主要研究三个不同形式的修正的两个分量的 Camassa-Holm 系统的 Cauchy 问题. 这些浅水波系统具有很强的物理背景，从而使得研究的现实意义比较强.. 首先，在Besov空间中研究系统的局部适定性，优化现有的局部适定性结果；其次，结合系统的结构特点，用调和分析的方法推导系统更精确的爆破机制，并以此为基础研究系统的爆破现象和整体强解的存在性；最后，利用磨光初值，粘性消失或者拉格朗日坐标变化的方法研究系统整体弱解的存在性和唯一性.. 本项目将研究改进三个系统Cauchy问题的局部适定性，给出强解的爆破，强解的整体存在性等方面的结果，并研究系统整体弱解的存在性和唯一性.

波动方程在声学，电磁学和流体力学等领域有重要的应用背景. 本项目研究浅水波方程中修正的两个分量的 Camassa-Holm 方程的强解的爆破，强解的整体存在性和弱解的存在唯一性. 主要结果有：弱耗散 μ-Degasperis-Procesi 方程 (Camassa-Holm 方程是它的特殊形式) 强解的爆破，强解的整体存在性和整体弱解的存在唯一性；弱耗散 μ-Hunter–Saxton 方程强解的爆破；变指数微分方程弱解的存在性. 这些结果揭示了 Camassa-Holm 方程解的某些特性，为后续研究奠定了基础.