Stability analysis of the compressible Euler-Euler system around planar Couette flow

This work is studied by Prof. H.-L. Li, Mr. J. Sun, Dr. D. Zhang and Dr. S. Zhao (School of Mathematical Sciences and Academy for Multidisciplinary Studies, Capital Normal University).  The linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for (ρ, u) and (n, v) with the sound speeds c₁ and c₂ respectively coupled each other through the drag force on T×R (where T is a periodic torus) is investigated.

In view of the mature results for incompressible flows, the velocities (u, v) can be decomposed into the irrotational component (compressible component) (Q[u], Q[v]) and the rotational component (incompressible component) (P[u], P[v]) by the Helmholtz projection operator. After a rigorously Fourier-based weighted energy estimates, the team derived precise decay/growth rates:

It is shown in general for the different sound speeds c₁≠c₂ that if the initial data satisfy (ρ₀,u₀, n₀, v₀) ∈ H^7/2(T×R) and the zero mode (i.e. the x-average) of the initial data are zero, then the densities (ρ, n) and the irrotational component of the velocities (Q[u], Q[v]) grow at O(t^1/2), while the rotational component of the velocities (P[u], P[v]) decay at O(t^-1/2) horizontally and O(t^-3/2) vertically.

For the case of the same sound speeds c₁=c₂ (same sound speeds), by virtue of the drag force, it is proved that the density difference ρ-n and the relative velocity u − v satisfy a stronger stability results than those of the densities ρ, n and the velocities u, v. Precisely, (ρ-n, Q[u-v]) remains stable and P[u-v] decays rapidly at O(t^-1) horizontally and O(t^-2) vertically. Moreover, the growth rate of order O(t^1/2) for (ρ, n, Q[u], Q[v]) is shown to be sharp in this special case.

The results highlight the drag force's role as a stabilizer when sound speeds align, offering new mathematical ideas for turbulence control.