𝐿^{𝑝}-𝐿^{𝑞} maximal regularity for some operators associated with linearized incompressible fluid-rigid body problems

Abstract

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is R sectorial in L q for every q ∈ (1, ∞), thus it has the maximal L p -L q regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the L q norm. Finally, we use the results to prove the global existence for small initial data, in an L p -L q setting, for the original nonlinear problem.

Energy Equality and Uniqueness of Weak Solutions of a “Viscous Incompressible Fluid + Rigid Body” System with Navier Slip-with-Friction Conditions in a 2D Bounded Domain

“…Finally C 1 and C 2 are linear and contiuous operators with finite dimention codomain. The proof is exactly the same of [15,Theorem 3.11], in fact the estimates are only based on the normal boundary condition and on the interior regularity (i.e. the fact that u ∈ W 2,q (F 0 ) or the fact that div u = 0).…”

“…To prove existence of strong solutions we use an idea introduced by Maity and Tucsnak in [15] where they view the "fluid+body system" as a perturbation of the system of a fluid alone. We start by recalling the result on L p − L q regularity from Shimada in [18].…”

“…Proof. The proof is contained in [15,Section 3.1], in fact the only boundary condition that they use is u · n = (l + rx ⊥ ) · n, and the second one is not relevant.…”

“…The most classical setting for this problem is to prescribe no-slip boundary condition on both ∂S(t) and ∂Ω. In this case a wide literature is available regarding the existence of both weak and strong solutions, see [11], [17], [7], [15]. Moreover in the 2D case weak solutions are also continuous in time with values in L 2 σ and unique [10].…”

“…Another interesting result is the work [1], where the authors study the imaginary power of the Stokes operator with some Navier slip-with-friction boundary condition. Such technics are useful to extend the theory of strong solutions from Hilbert setting, for which we refer to [19], to L p − L q setting, see [15] in the 3D case with no-slip conditions. Indeed the argument presented in section 7 can be implemented to prove similar existence results in both 2D and 3D for the Navier slip-with-friction boundary conditions (using a fixed point argument as in [7]).…”