The Oseen–Navier–Stokes flow in the exterior of a rotating obstacle: The non-autonomous case

Abstract

Consider the incompressible Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to a non-autonomous problem with unbounded drift terms on a fixed exterior domain R d . It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system ¹T .t; s/º t s 0 on L p . / for 1 < p < 1. Moreover, L p -L q smoothing properties and gradient estimates of T .t; s/, 0 Ä s Ä t, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for initial value in L p . /, p d .

‐Theory for Schrödinger systems

“…Besides being interesting in their own right, such systems appear naturally in the study of backward‐forward stochastic differential systems, in the study of Nash equilibria to stochastic differential games, in the analysis of the weighted ‐problem in , in the time‐dependent Born–Openheimer theory and also in the study of Navier–Stokes equations. We refer the reader to [, Section 6], for further details. In these applications unbounded drift and diffusion coefficients as well as unbounded potential terms appear.…”

Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle

“…For the autonomous case (in which ω ∈ R 3 \ {0} is a constant vector), this was observed first by the present author [31], [32] (see also [33] even for the specific non-autonomous case) within the framework of L 2 and, later on, by Geissert, Heck and Hieber [24] within the one of L q . Thus the result of Hansel and Rhandi [30] may be regarded as a desired generalization of [24] to the non-autonomous case. What is remarkable is that they constructed the evolution operator in their own way without relying on any theory of abstract evolution equations although the idea of iteration is somewhat similar to the one in the Tanabe-Sobolevskii theory mentioned above.…”

“…Global existence of a unique solution in the non-autonomous setting has still remained open even if the data are small enough, while weak solutions (in the sense of Leray-Hopf) were constructed globally in time by Borchers [2]. The essential contribution of [30] is not only to construct the evolution operator {T (t, s)} t≥s≥0 on L q σ (D), the space of solenoidal L q -vector fields (1 < q < ∞) with vanishing normal trace at ∂D, which provides a solution operator to the initial value problem for the linearized system ∂ t u = ∆u + (η + ω × x) · ∇u − ω × u − ∇p, div u = 0, u| ∂D = 0, u → 0 as |x| → ∞, u(·, s) = f, (1.5) in D × [s, ∞), where s ≥ 0 is the given initial time, but also to show the L q -L r smoothing action (1 < q ≤ r < ∞) near the initial time, namely,…”

On coupled systems of PDEs with unbounded coefficients

“…Among them we quote the study of backward-forward stochastic differential systems, the study of Nash equilibria to stochastic differential games, the analysis of the weighted ∂-problem in C d , in the time-dependent Born-Openheimer theory and also in the study of Navier-Stokes equations. We refer the reader to [2,Section 6] and [7,9,12,13,15,16].…”