In the presence of the interaction between DE and DM, the Layer-Irvine equation can tell us how the coupling between DE and DM affects the process of attaining equilibrium as well as the respective configuration. Our derivation will be based on the perturbation theory given in Sec. II. For DM, the rate of change of the peculiar velocity is given by Eqs.(7) Neglecting the influence of DE and the couplings, the above equation is nothing but the rate of change of the peculiar velocity of the DM particle in the expanding universe described by the Newton’s law which was the starting point in [38]. To derive the energy equation for local irregularities, we follow the same procedure as [38] by forming a product of Eqs.(12) with aV$_{m}$ρ$_{m}$ ˆ ε and integrating over the volume. Here ˆ ε = a $^{3}$d ¯ x ∧ d ¯ y ∧ d ¯ z is the volume element which satisfies ∂ $_{∂t}$ˆ ε = 3 H ˆ ε . Considering Eqs.(1), the LHS of Eq.(12) can be multiplied by aV$_{m}$ and integrated to yield where T$_{m}$ = 1 2 ∫ V 2 $_{m}$ρ$_{m}$ ˆ ε is the kinetic energy of DM associated with peculiar motions of DM particles. The RHS of Eqs.( 12) can be treated by the same token. Using partial integration, the potential part can be changed into Taking account of the first equation in (7), it can turn into where U$_{mm}$ = 1 2 ∫ σ$_{m}$ψ$_{m}$ ˆ ε , U$_{dm}$ = ∫ σ$_{d}$ψ$_{m}$ ˆ ε , U$_{md}$ = ∫ σ$_{m}$ψ$_{d}$ ˆ ε , and U$_{dd}$ = 1 2 ∫ σ$_{d}$ψ$_{d}$ ˆ ε . The second term in the RHS of Eq.(12) can be changed into, Combining Eqs.(13,14,15), we obtain 6 ˙ T$_{m}$ + ˙ U$_{mm}$ + H (2 T$_{m}$ + U$_{mm}$ ) = − ∫ ψ$_{d}$ ∂ ∂t ( σ$_{m}$ ˆ ε ) − 3 H ( ξ$_{1}$ + ξ$_{2}$/r ) T$_{m}$ +3 H { ξ$_{1}$U$_{md}$ + ξ$_{2}$U$_{dm}$ + 2 ξ$_{1}$U$_{mm}$ + 2 ξ$_{2}$U$_{dd}$ } . (16) − ∫ ( aV$_{m}$ ) $^{2}$3 H ( ξ$_{1}$ + ξ$_{2}$/r ) ρ$_{m}$ ˆ ε = − a $^{2}$6 H ( ξ$_{1}$ + ξ$_{2}$/r ) T$_{m}$ (15) − ∫ aV$_{m}$ ∇$_{x}$ ( aψ$_{m}$ + aψ$_{d}$ ) ρ$_{m}$ ˆ ε = − a $^{2}$( ˙ U$_{mm}$ + HU$_{mm}$ ) − a 2 ∫ ψ$_{d}$ ∂ ∂t ( σ$_{m}$ ˆ ε ) + 3 a $^{2}$H { ξ$_{1}$U$_{md}$ + ξ$_{2}$U$_{dm}$ + 2 ξ$_{1}$U$_{mm}$ + 2 ξ$_{2}$U$_{dd}$ } (14) − ∫ aV$_{m}$ ∇$_{x}$ ( aψ$_{m}$ + aψ$_{d}$ ) ρ$_{m}$ ˆ ε = a 2 ∫ ∇$_{x}$ ( ρ$_{m}$V$_{m}$ ) ψ$_{m}$ ˆ ε + a 2 ∫ ∇$_{x}$ ( ρ$_{m}$V$_{m}$ ) ψ$_{d}$ ˆ ε. ∂ ∂t ( a $^{2}$T$_{m}$ ) − a $^{2}$3 H ( ξ$_{1}$ + ξ$_{2}$/r ) T$_{m}$. (13) ∂ ∂t ( aV$_{m}$ ) = −∇$_{x}$ ( aψ$_{m}$ + aψ$_{d}$ ) − 3 H ( ξ$_{1}$ + ξ$_{2}$/r )( aV$_{m}$ ) . (12)