A capacitary approach of the Navier–Stokes integral equations

In Chapter 4, we have studied classical solutions of the Cauchy initial value problem for the Navier–Stokes equations (with reduced (unknown) pressure p, reduced force density f → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373393/e1c1ff9e-5083-409e-83f8-9c892d4b410a/content/eq686.tif"/> and kinematic viscosity v > 0): given a divergence-free vector field u → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373393/e1c1ff9e-5083-409e-83f8-9c892d4b410a/content/eq687.tif"/> on ℝ³ and a force f → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373393/e1c1ff9e-5083-409e-83f8-9c892d4b410a/content/eq688.tif"/> on (0, +∞) × ℝ³, find a positive T and regular functions u → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373393/e1c1ff9e-5083-409e-83f8-9c892d4b410a/content/eq689.tif"/> and p on [0, T] × ℝ³ solutions to (5.1) ∂ t u → = ν Δ u → − ( u → . ∇ → ) u → + f → − ∇ → p div   u →   =     0 u → | t = 0     =   u → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315373393/e1c1ff9e-5083-409e-83f8-9c892d4b410a/content/eq690.tif"/>