Navier-Stokes equations in a rotating frame in R3 with initial data nondecreasing at infinity

Yoshikazu Giga, Katsuya Inui, Alex Mahalov, Shin’ya Matsui

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30 Scopus citations

Abstract

Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter Ω and initial data nondecreasing at infinity. In contrast to the non-rotating case (Ω = 0), it is shown for the problem with rotation (Ω ≠ 0) that Green’s function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to L1(R3). Moreover, the corresponding integral operator is unbounded in the space [formula] of solenoidal vector fields in R3 and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in [formula]. Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space [formula] which consists of L solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space [formula], 1 which is smaller than L but still contains various periodic and almost periodic functions. This restriction of initial data to [formula] which is a subspace of [formula] is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time solvability of the classical 3D Navier-Stokes equations in R3 with initial velocity and vorticity of the form V(0) = Ṽ0(y) + (Ω/2)e3 × y, curl V(0) = curl Ṽ0(y) + Ωe3 where V0(y) ∈ [formula].

Original languageEnglish (US)
Pages (from-to)321-364
Number of pages44
JournalHokkaido Mathematical Journal
Volume35
Issue number2
DOIs
StatePublished - 2006

Keywords

  • Homogeneous besov spaces
  • Nondecreasing initial data
  • Riesz operators
  • Rotating Navier-Stokes equations

ASJC Scopus subject areas

  • Mathematics(all)

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