Unidirectional flocks in hydrodynamic Euler alignment system II: singular models

Global solutions to multi-dimensional topological Euler alignment systems

“…We can assume without loss of generality that all k i > 0. Indeed, for those that are equal to zero, we replace the finite difference by the gradient thereby lowering N , still satisfying (16).…”

“…In this work we aim to provide a systematic treatment of the regularity theory for topological systems. Our first result gives the continuation criterion, a direct analogue of the metric case [16].…”

“…Yet, for the metric (τ = 0) multi-dimensional systems several classes of global smooth solutions have been discovered. Those include unidirectional flocks [16], nearly aligned flocks [22], and flocks with a small spectral gap data in 2D [14]. Moreover, an effective continuation criterion proved in [16] requires only control on the gradients ∇ρ L ∞ ([0,T )×T n ) and ∇u L ∞ ([0,T )×T n ) .…”

“…Those include unidirectional flocks [16], nearly aligned flocks [22], and flocks with a small spectral gap data in 2D [14]. Moreover, an effective continuation criterion proved in [16] requires only control on the gradients ∇ρ L ∞ ([0,T )×T n ) and ∇u L ∞ ([0,T )×T n ) . We refer to the text [23] for a systematic exposition.…”

“…Theorem 1.1 applies to reveal two new classes of global smooth solutions. First is the class of parallel shear flocks -similar to unidirectional class discussed in [16] but with incompressible velocity field u = (U (x 2 , . .…”