Some Dynamical Questions in Hamiltonian Partial Differential Equations

对于许多物理现象来说，能量耗散的影响非常小，忽略它们可以提供一个很好的近似。 这些现象包括海洋中的大尺度水波、空间中的等离子体和受控聚变装置、天体物理中的恒星和星系动力学。这些现象的数学模型的动力学行为是复杂的，并且包括相干结构的形成，例如海洋中的海啸、旋转的恒星（例如太阳）和观察到的星系结构，仅举几例。该项目的目标之一是帮助理解在实验和自然界中观察到的这种相干结构的形成和持久性。另一个主题是了解和控制许多应用中的不稳定性。一个这样的例子是控制聚变装置中等离子体的不稳定性，以实现用于能量生产的受控核聚变的目标。另一个例子是径向旋转中子星的不稳定性，这与引力波的探测有关。数学分析方法是本研究的主要工具。严格的数学使得进行稳定的数值计算和更好地理解数值和实验研究中发现的现象成为可能。该项目将包括从研究生到博士后水平的学生的研究，从而提供一个有力的指导计划。该方案的一个目标是为具有不定能量泛函的Hamilton偏微分方程找到一个不稳定性指数公式，包括重力水波，离子声波方程和无碰撞等离子体的Vlasov模型。第二个目标是找到几个恒星模型，包括旋转和磁性的恒星，中子星和相对论性星系的稳定性标准（特别是转折点原则）。第三个目标是了解一维Vlasov-Poisson方程的线性稳定Bernstein-Greene-Kruskal（BGK）波附近的长时间动力学。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

For many physical phenomena dissipation of energy effects are very small and ignoring them could provide a good approximation. Such phenomena include large scale water waves in the ocean, the plasmas in space and controlled fusion devices, dynamics of stars and galaxies in astrophysics. The dynamical behavior of mathematical models of these phenomena is complex and includes formation of coherent structures, such as tsunamis in the ocean, rotating stars (e.g. Sun) and the observed galaxy structures, to name a few. One of the objectives of this project is to contribute to the understanding of the formation and persistence of such coherent structures observed in experiments and in nature. Another topic is to understand and control the instability in many applications. One such example is to control the instability of plasmas in fusion devices to achieve the goal of controlled nuclear fusion for energy production. Another example is the instability of radially rotating neutron stars, which is related to the detection of gravitational waves. Methods of mathematical analysis are the primary tools employed in this investigation. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies. The project will incorporate research of students from graduate through postdoctoral levels thus providing for a vigorous mentoring program.Many conservative models have Hamiltonian structures. One goal of this proposal is to find an instability index formula for Hamiltonian PDEs with an indefinite energy functional, including gravity water waves, ion acoustic wave equations and Vlasov models for collisionless plasmas. The second goal is to find stability criteria (particularly turning point principles) for several stellar models including rotating and magnetic stars, neutron stars and relativistic galaxies. The third goal is to understand the long-time dynamics near linearly stable Bernstein–Greene–Kruskal (BGK) waves for the one-dimensional Vlasov-Poisson equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.