In this paper,we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d 3.Moreover,u has a unique trajectory which is Hlder continuous with respect to the space variables.
In this paper, we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d ≥ 3. Moreover, u has a unique trajectory which is HSlder continuous with respect to the space variables.
Given initial data(ρ0, u0) satisfying 0 < m ρ0≤ M, ρ0- 1 ∈ L2∩˙W1,r(R3) and u0 ∈˙H-2δ∩ H1(R3) for δ∈ ]1/4, 1/2[ and r ∈ ]6, 3/1- 2δ[, we prove that: there exists a small positive constant ε1,which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution(ρ, u) whenever‖ u0‖ L2 ‖▽u0 ‖L2 ≤ε1 and ‖μ(ρ0)- 1‖ L∞≤ε0 for some uniform small constant ε0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.
