Convergence to steady states in symmetric nonlocal advection-diffusion systems

Let $K$ be a kernel satisfying

$$\lVert K\lVert_{L^{\infty}} < \infty,$$

and let the interaction coefficients satisfy $\gamma_{ij}=\gamma_{ji}$ for all $i,j=1,\dots,N$. For any initial datum $u_0=(u_{1,0},\dots,u_{N,0})\in L^1(\mathbb{T})^N$ that is positive in each component, let $\mathbf{u}$ be the corresponding solution of the system.

Steady-state convergence conjecture. The solution $\mathbf{u}$ converges towards a steady state.

The conjecture concerns the long-time behaviour of symmetric multi-species nonlocal advection-diffusion systems. Energy monotonicity guarantees convergence of the energy to a finite minimum, but does not establish convergence of every solution to a steady state; numerical investigations have observed stable steady states and no perpetually fluctuating solutions. The problem is left open.