Continuum limit of the lattice Lohe group model and emergent dynamics

We study the emergent dynamics and global well-posedness of the matrix-valued integro-differential equation which can be derived from the continuum limit of the lattice Lohe group model. The lattice Lohe group model corresponds to the generalized high-dimensional Kuramoto model. The solution to the lattice Lohe group model can be cast as a simple function-valued solution to the continuum Lohe group model. We first construct a local classical solution to the continuum Lohe group model, and then we find an invariant set and derive a global well-posedness in some sufficient frameworks formulated in terms of initial data, system functions, and system parameters. We also show that phase-locked states can emerge from the admissible class of initial data in a large coupling regime. Moreover, we show that sequence of simple functions obtained from the solutions of the lattice Lohe group model converges to a local classical solution to the continuum Lohe group model in supremum norm.