New results of general n-dimensional incompressible Navier–Stokes equations

Let u=u(x,t,u0)u=u(x,t,u0) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier–Stokes equationsut+εΔ2nut−αΔu+(u⋅∇)u+∇p+(β⋅∇)u=0,∇⋅u=0,u(x,0)=u0(x),∇⋅u0=0, where the spatial dimension n⩾2n⩾2, 0⩽ε⩽10⩽ε⩽1 is a constant and β=T(β1,β2,…,βn)∈Rnβ=(β1,β2,…,βn)T∈Rn is a constant vector. Note that if ε=0ε=0 and β=0β=0, then the problem reduces to the traditional Navier–Stokes equations. Let the scalar functions ϕij∈C2(Rn)∩L1(Rn)ϕij∈C2(Rn)∩L1(Rn), ∂ϕij∂xj∈L1(Rn)∩H2n(Rn), i,j∈{1,2,…,n}i,j∈{1,2,…,n}. Define the real vector-valued functions Φi=T(ϕi1,ϕi2,…,ϕin)Φi=(ϕi1,ϕi2,…,ϕin)T. Let the initial datau0(x)=(∑j=1n∂ϕ1j∂xj(x),∑j=1n∂ϕ2j∂xj(x),…,∑j=1n∂ϕnj∂xj(x))T satisfy∑i=1n∑j=1n∂2ϕij∂xi∂xj(x)=0. Thenlimt→∞{(1+t)1+n/2∫Rn[|u(x,t)|2+ε|Δnu(x,t)|2]dx}=1(2π)n(π2α)n/214α∑k=1n[∫RnΦk(x)dx]2. For any integer m⩾1m⩾1, we will establish the following limitlimt→∞{(1+t)2m+1+n/2∫Rn[|Δmu(x,t)|2+ε|Δm+nu(x,t)|2]dx}=1(2π)n(π2α)n/2(14α)2m+1[∏l=12m(2l+n)]∑k=1n[∫RnΦk(x)dx]2. This kind of exact limit will have great influence on the Hausdorff dimension of the global attractor of the model equations.