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

Abstract

This article deals with 𝐿 𝑞 -weak solutions to the 3D time-dependent Oseen system. This type of solution is defined in terms of the velocity only. It is shown that the velocity may be represented by a sum of integrals none of which involves the pressure and without a surface integral of the spatial gradient of the velocity. On the basis of this representation formula, an estimate of the spatial decay of the velocity and its spatial gradient is derived. No boundary conditions have to be imposed for these results.

Anisotropically weighted Lq$L^q$‐Lr$L^r$ estimates of the Oseen semigroup in exterior domains, with applications to the Navier–Stokes flow past a rigid body

“…Deuring [11] used a representation formula for the solution to the Oseen system, see [9, 10, 12], to deduce $$$$\begin{align*} \nabla ^iu(x,t)=O({(1+|x|)}^{-1-\frac{i}{2}} {(1+|x|-x_1)}^{-1-\frac{i}{2}}) \end{align*}$$$$for $$i=0,1$$ uniformly in t under some assumptions on the initial perturbation from the stationary solution and on the solution u . In [15], by employing another integral representation, see [13, 14], he also established the estimate $$$$\begin{align*} \nabla ^i{\left(u(x,t)-u_s\right)}=O({(1+|x|)}^{-\frac{5}{4}-\frac{i}{2}} {(1+|x|-x_1)}^{-\frac{5}{4}-\frac{i}{2}}) \end{align*}$$$$for $$t>0$$ without the boundary condition except the zero‐flux condition.…”

“…for 𝑖 = 0, 1 uniformly in 𝑡 under some assumptions on the initial perturbation from the stationary solution and on the solution 𝑢. In [15], by employing another integral representation, see [13,14], he also established the estimate for 𝑡 > 0 without the boundary condition except the zero-flux condition.…”

“…Note that the exponents 𝑞 𝑖 (𝑖 = 1, 2, 3, 4) may not coincide with each other and this remark plays an important role when we treat the terms 𝑓 1 and 𝑓 2 (see (10)- (11)) to analyze the starting problem, which is described in Section 8. The estimate (13) is not employed in the nonlinear problems, but it is seen that the rate in (13) is better than the one in (14) with 𝑞 𝑖 = 𝑞 (𝑖 = 1, 2, 3, 4), that is, 𝑡 −3(1∕𝑞−1∕𝑟)∕2−𝑗∕2+𝛼+𝛽∕2 . We thus use (13) in the study of the Oseen semigroup 𝑒 −𝑡𝐴 𝑎 in 𝐷 for a better understanding of 𝑒 −𝑡𝐴 𝑎 .…”