We study the two main types of trajectories of the ABC ow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also nd and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of ows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC ows.

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For a Lagrangian scheme solving the compressible Euler equations in cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-preserving in cartesian coordinates, the design of a Lagrangian scheme in cylindrical coordinates, which is high order in one-dimension and second order in two-dimensions, and can maintain both spherical symmetry-preservation and positivity-preservation simultaneously, is challenging. In this paper we design such a Lagrangian scheme and provide numerical results to demonstrate its good behavior.

We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and  $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.