The Navier–Stokes equations in the critical Morrey–Campanato space

We shall discuss various points on solutions of the 3D NavierStokes equations from the point of view of Morrey-Campanato spaces (global solutions, strong-weak uniqueness, the role of real interpolation, regularity). The classical Navier-Stokes equations describe the motion of a Newtonian fluid; we consider only the case when the fluid is viscous, homogeneous, incompressible and fills the entire space and is not submitted to external forces; then, the equations describing the evolution of the motion u(t, x) of the fluid element at time t and position x are given by: (1) { ρ ∂t u=μ ∆ u− ρ ( u. ∇) u− ∇p ∇. u= 0 ρ is the (constant) density of the fluid, and we may assume with no loss of generality that ρ = 1. μ is the viscosity coefficient, and we may assume as well that μ = 1. p is the (unknown) pressure, whose action is to maintain the divergence of u to be 0 (this divergence free condition expresses the incompressibility of the fluid). We shall use the scaling property of equations (1). When ( u, p) is a solution on (0, T ) × R of the Cauchy problem associated to equations (1) and initial value u0, then, for every λ > 0 and every x0 ∈ R, (λ u(λt, λ(x− x0)), λp(λt, λ(x− x0))) is a solution on (0, λ−2T ) × R of the Cauchy problem with initial value λ u0(λ(x−x0)). Therefore, we shall consider the Cauchy problem with initial 2000 Mathematics Subject Classification: 76D05, 76D03, 35Q30, 46E30, 46E35, 42C40.