$\mathbf{L^q}$ estimates of weak solutions to the stationary Stokes equations around a rotating body

Abstract

We establish the existence, uniqueness and L q estimates of weak solutions to the stationary Stokes equations with rotation effect both in the whole space and in exterior domains. The equation arises from the study of viscous incompressible flows around a body that is rotating with a constant angular velocity, and it involves an important drift operator with unbounded variable coefficient that causes some difficulties.

Stationary Navier-Stokes Flow Around a Rotating Obstacle

“…This result is regarded as a generalization of [4], [22], [30], [31] in the usual case o ¼ 0, since the restriction on the exponent q is the same. Since the case q ¼ 3=2ð¼ n=2Þ needed to estimate the nonlinearity u Á 'u is missing in [28], an L q theory does not help to solve the nonlinear problem (1.2), (1.3). Note that L 3=2 is too restrictive at infinity to expect 'u A L 3=2 for (1.8) even if f ¼ div F with F A C y 0 .…”

“…We refer to [3], [9], [10], [12], [13], [14], [15], [16], [18], [19], [20], [21], [23], [24], [25], [26], [27], [28], [33], [34] and [36]. Nevertheless, our mathematical understanding is still far from complete.…”

“…In [28], based on an idea from [14], one of the present authors has established the existence, uniqueness and L q estimate…”

Time-Periodic Solutions to the Navier-Stokes Equations

“…Similar estimates for weak solutions in the Oseen case (λ = 0) were established in [37] and [38] by Kračmar, Nečasová, and Penel. The results in [31] and [37], [38] were obtained using the same very technical approach that was used in [9] and [6] based on the Littlewood-Paley decomposition. Simple proofs of these estimates were recently made available by Galdi and Kyed in [22].…”

The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

“…On the existence, uniqueness and regularity of solutions of (1), there are many papers, such as [1][2][3][4][5][6][7][8] and references therein. However, few results about the numerical analysis are developed.…”