Classical Solutions

In this chapter, we study 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/eq308.tif"/> and kinematic viscosity v > 0): Navier–Stokes equations

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/eq309.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/eq310.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/eq311.tif"/> and p on [0, T] × ℝ³ solutions to (4.1) ∂ t u → = v 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/eq312.tif"/>