ON OPTIMAL DECAY RATES FOR WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS IN n
نویسندگان
چکیده
0 ∇ · e−(t−s)AP (u⊗ u)(s) ds, for prescribed initial velocity a(x) = (a1(x), . . . , an(x)), x = (x1, . . . , xn) ∈ n , and unknown velocity u(x, t) = (u1(x, t), . . . , un(x, t)). Here, A = −∆ is the Laplacian on n ; {e−tA}t 0 is the heat semigroup; P = (Pjk) is the bounded projection onto divergence-free vector fields; u ⊗ v is the matrix with entries (u ⊗ v)jk = ujvk; ∇ = (∂1, . . . , ∂n) with ∂j = ∂/∂xj; and (∇ · e−tAP (u⊗ u))j = n ∑
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