Eulerian dynamics with a commutator forcing III. Fractional diffusion of order $0<\alpha <1$

Abstract

We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel $\varphi (x)=|x{|}^{-(1+\alpha )}$. Following our works \cite{ST2017a,ST2017b} which focused on the range $1\le \alpha <2$, and Do et. al. \cite{DKRT2017} which covered the range $0<\alpha <1$, in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in $({\rho }_0,u_0)\in H^{2+\alpha }\times H^3$, the solution approaches exponentially fast to a flocking state solution consisting of a wave $\bar{\rho }={\rho }_{\infty }(x-t\bar{u}))$ traveling with a constant velocity determined by the conserved average velocity $\bar{u}$. The convergence is accompanied by exponential decay of all higher order derivatives of $u$.

References in corpus (3)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Critical thresholds in flocking hydrodynamics\\with nonlocal alignment
• Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models

Cited by in corpus (9)

• Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods
• Singular Cucker-Smale Dynamics
• Global existence and stability of nearly aligned flocks
• Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems
• Flocking with short-range interactions
• Weak and Strong Solutions to the Forced Fractional Euler Alignment System
• Global well-posedness for the Euler alignment system with mildly singular interactions
• Global regularity for 1D Eulerian dynamics with singular interaction forces
• Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system