Spatial asymptotics of mild solutions to the time-dependent Oseen system

Abstract

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in L q . They turn out to be also L q -weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on L p -integrability in time of the external force.This latter system is usually considered as a model for the flow of a viscous incompressible fluid around a rigid body moving with constant velocity and without rotation, with the set Ω corresponding to the rigid body. The functions u : Ω c × (0, ∞) → R 3 (velocity) and π : Ω c × (0, ∞) → R (pressure) are the unknowns of problem (1.1), (1.2), whereas the functions f : Ω c × (0, ∞) → R 3 (volume force) and U 0 : Ω c → R 3 (initial velocity), as well as the number τ ∈ (0, ∞) (Reynolds number), are given quantities. Mild solutions to (1.1), (1.2), which only involve the velocity among the two unknowns velocity and pressure, are introduced via the semigroup generated by the Oseen operator. We refer to (3.

Lq$L^q$‐weak solutions to the time‐dependent 3D Oseen system: Decay estimates

“…If U 0 has compact support, the decay rate of $$ | \partial _x^{\alpha } \mathfrak {I} ^{(\tau )} (U_0| \overline{ B_{S_0}}^c) (x,t) |$$ increases to $$-3/2-|\alpha |/2$$ ([6, Lemma 4.1]). In [7, Lemma 2.8], we identify a class of functions U 0 for which $$ | \partial _x^{\alpha } \mathfrak {I} ^{(\tau )} (U_0| \overline{ B_{S_0}}^c) (x,t) |$$ decreases even faster when $$|x|$$ tends to infinity.…”

“…The focus in this latter work is on improving the decay results in [32], [36] and [16] for nonlinear problems without requiring specific boundary conditions. Reference [7] deals with the spatial asymptotics of mild solutions to (1.1) under homogeneous Dirichlet boundary conditions. It turned out these solutions are $$L^q$$‐weak in the sense of the work at hand.…”