Swarming: hydrodynamic alignment with pressure
Eitan Tadmor
Abstract
We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering toward a weighted average heading. We consider the class of so-called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -alignment hydrodynamics, based on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 p"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacians and weighted by a general family of symmetric communication kernels. The main new aspect here is the long-time emergence behavior for a general class of pressure tensors <italic>without</italic> a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as “entropic”, and prove the flocking of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -alignment hydrodynamics, driven by singular kernels with a general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.