On the constants in a basic inequality for the Euler and Navier-Stokes equations

Original · EN

We consider the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus Tᵈ; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w: Tᵈ -> Rᵈ into v. D w, and also involves the Leray projection L onto the space of divergence free vector fields. We derive upper and lower bounds for the constants in some inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants Kₙ d = Kₙ in the basic inequality || L(v. D w)||ₙ <= Kₙ || v ||ₙ || w ||ₙ₊₁, where n in (d/2, + infinity) and v, w are in the Sobolev spaces Hⁿ, Hⁿ⁺¹ of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. Some practical motivations are indicated for an accurate analysis of the constants Kₙ.