Vortex dynamics and blow-up phenomena in two dimensions

在一个区域或黎曼曲面上的N个点涡的动力学是由一阶哈密顿系统描述的，该系统可以追溯到Kirchhoff和Routh（19世纪80年代）。在哈密顿系统和偏微分方程之间有许多有趣的关系。它是由理想流体的二维欧拉方程导出的奇异极限，假设涡度集中在N个不同的点上，类似于天体力学中的点质量概念。基尔霍夫-劳斯哈密顿算子依赖于狄利克雷拉普拉斯算子的格林函数，当涡旋彼此接近或接近区域边界时，它是奇异的。同样类型的Hamilton函数出现在数学物理的其他问题中，例如，在金兹堡-朗道理论中，作为涡旋动力学的重整化能量。大量涡旋的统计力学导致了平均场型偏微分方程，回到了Onsager关于二维湍流的工作。令人惊讶的是，平均场型方程也出现在微分几何中。可以用变分方法得到解，然而，相关的欧拉-拉格朗日泛函对于某些关键参数值可能不是紧致的：可能存在梯度流的轨道在测量意义上收敛于Dirac delta的和，这些delta位于与Kirchhoff-Routh函数相似但比Kirchhoff-Routh函数更复杂的涡哈密顿函数的临界点。因此，关于涡旋哈密顿量的临界点的信息对于理解与平均场型方程相关的泛函的紧性损失是至关重要的。当一个人想要构造爆破解时，这个信息也很重要，即当参数接近临界值时，在狄拉克函数和的度量意义上收敛的解族。作为出现涡旋型汉密尔顿函数的最后一个例子，我们提到了对Toda系统的爆炸现象的研究。这个项目的主要目标是：1。研究点涡动力学在一个域或黎曼曲面上，特别是寻找平衡点作为基尔霍夫-劳斯哈密顿的临界点，并找到周期解。发展平均场型方程的莫尔斯理论，特别是寻找涡旋哈密顿量的临界点，最终目的是证明平均场方程解的存在性、多重性以及爆破解的构造。发展具有Neumann边界条件的SU(3) Toda系统的Morse理论，以建立解的存在性和多重性，并构造爆破解。

The dynamics of N point-vortices in a domain or on a Riemann surface is described by a first order Hamiltonian system that goes back to Kirchhoff and Routh (1880s). There are many fascinating relations between this Hamiltonian system and partial differential equations. It has been derived as a singular limit from the 2D Euler equations for an ideal fluid assuming that the vorticity is concentrated in N different points, analogous to the concept of point-masses in celestial mechanics. The Kirchhoff-Routh Hamiltonian depends on the Green function of the Dirichlet Laplace operator and is singular when vortices come close to each other or approach the boundary of the domain. The same type of Hamilton function appears in other problems from mathematical physics, for instance as renormalized energy for the dynamics of vortices in the Ginzburg-Landau theory.Statistical mechanics for large numbers of vortices leads to partial differential equations of mean field type, going back to work of Onsager on two-dimensional turbulence. Surprisingly, mean field type equations arise also in differential geometry. Solutions can be obtained with variational methods, the associated Euler-Lagrange functionals, however, may not be compact for certain critical parameter values: there may be orbits of the gradient flow that converge in the sense of measures to sums of Dirac deltas which are located at critical points of a vortex Hamiltonian that is similar to but more complicated than the Kirchhoff-Routh function. Therefore information about critical points of vortex Hamiltonians is crucial for understanding the loss of compactness of the functionals associated to mean field type equations. This information is also important when one wants to construct blow-up solutions, i.e. families of solutions that converge in the sense of measures towards sums of Dirac deltas as a parameter approaches a critical value. As a last example where vortex type Hamilton functions appear we mention the study of blow-up phenomena for Toda systems.The principal goals of this project are:1. to investigate point-vortex dynamics in a domain or on a Riemann surface, in particular to find equilibria as critical points of the Kirchhoff-Routh Hamiltonian, and to find periodic solutions.2. to develop Morse theory for mean field type equations, in particular to find critical points of vortex Hamiltonians, the ultimate goal being the proof of existence and multiplicity of solutions of mean field equations as well as the construction of blow-up solutions.3. to develop Morse theory for the SU(3) Toda system with Neumann boundary conditions or on surfaces in order to establish existence and multiplicity of solutions and to construct blow-up solutions.