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 c 1 and c 2 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 c 1 ≠c 2 that if the initial data satisfy (ρ 0 ,u 0 , n 0 , v 0 ) ∈ 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 c 1 =c 2 (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.
Acta Mathematica Scientia
Stability analysis of the compressible Euler-Euler system around planar Couette flow
29-Oct-2025