Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions

Paul Deuring (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We consider the incompressible time-dependent Navier-Stokes system with Oseen term, in a 3D exterior domain, with the option of adding to the system another term arising in the study of stability of stationary incompressible Navier-Stokes flows. We do not impose any boundary conditions. The solutions we consider are supposed to possess properties of L^2-strong solutions: The velocity u is an L^∞-function in time and L^κ-integrable in space for some κ ∈ [1, 3) and some κ ∈ (3, ∞), the spatial gradient ∇_x u is L 2-integrable in space and in time, and the nonlinearity (u · ∇_x)u is L^2-integrable in time and L^3/2-integrable in space. We show that if the right-hand side of the equation and the initial data decay pointwise in space sufficiently fast, then u and ∇_x u also decay pointwise in space, with rates which are higher than those exhibited in previous articles.

Mots clés: Navier-Stokes system; Oseen term; strong solutions; spatial decay; Navier-Stokes system

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Paul Deuring. Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions. 2021. ⟨hal-02508815v3⟩