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

Abstract

Let u = u ( x , t , u 0 ) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier–Stokes equations u t + ε Δ 2 n u t − α Δ u + ( u ⋅ ∇ ) u + ∇ p + ( β ⋅ ∇ ) u = 0 , ∇ ⋅ u = 0 , u ( x , 0 ) = u 0 ( x ) , ∇ ⋅ u 0 = 0 , where the spatial dimension n ⩾ 2 , 0 ⩽ ε ⩽ 1 is a constant and β = ( β 1 , β 2 , … , β n ) T ∈ R n is a constant vector. Note that if ε = 0 and β = 0 , then the problem reduces to the traditional Navier–Stokes equations. Let the scalar functions ϕ i j ∈ C 2 ( R n ) ∩ L 1 ( R n ) , ∂ ϕ i j ∂ x j ∈ L 1 ( R n ) ∩ H 2 n ( R n ) , i , j ∈ { 1 , 2 , … , n } . Define the real vector-valued functions Φ i = ( ϕ i 1 , ϕ i 2 , … , ϕ i n ) T . Let the initial data u 0 ( x ) = ( ∑ j = 1 n ∂ ϕ 1 j ∂ x j ( x ) , ∑ j = 1 n ∂ ϕ 2 j ∂ x j ( x ) , … , ∑ j = 1 n ∂ ϕ n j ∂ x j ( x ) ) T satisfy ∑ i = 1 n ∑ j = 1 n ∂ 2 ϕ i j ∂ x i ∂ x j ( x ) = 0 . Then lim t → ∞ { ( 1 + t ) 1 + n / 2 ∫ R n [ | u ( x , t ) | 2 + ε | Δ n u ( x , t ) | 2 ] d x } = 1 ( 2 π ) n ( π 2 α ) n / 2 1 4 α ∑ k = 1 n [ ∫ R n Φ k ( x ) d x ] 2 . For any integer m ⩾ 1 , we will establish the following limit lim t → ∞ { ( 1 + t ) 2 m + 1 + n / 2 ∫ R n [ | Δ m u ( x , t ) | 2 + ε | Δ m + n u ( x , t ) | 2 ] d x } = 1 ( 2 π ) n ( π 2 α ) n / 2 ( 1 4 α ) 2 m + 1 [ ∏ l = 1 2 m ( 2 l + n ) ] ∑ k = 1 n [ ∫ R n Φ k ( x ) d x ] 2 . This kind of exact limit will have great influence on the Hausdorff dimension of the global attractor of the model equations.