Abstract.: Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge are established in an L²-setting of monomially weighted spaces. A mathematical framework is developed that allows us to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.

In this paper, we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the limit when the volume of the rigid bodies converges to zero while their densities are a fixed constant.

In this paper, we highlight a set of ad hoc test functions to study the homogenization of viscous compressible fluids in domains with very tiny holes. This set of functions allows to improve previous results in dimensions two and three. As an application, we show that the presence of a small obstacle does not influence the dynamics of a viscous compressible fluid in dimension two.

In this paper, we perform the fast rotation limit ε→ 0 ⁺ of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip Ωε:=R2×]-ℓε,ℓε[ , where ε∈]0,1] is the size of the Rossby number and ℓ_ε> 0 for any ε> 0 . By letting ℓ_ε⟶ 0 ⁺ for ε→ 0 ⁺ and considering Navier-slip boundary conditions at the boundary of Ω _ε , we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a 3-D setting. In this respect, we remark that the case ℓ_ε⩾ ℓ> 0 for all ε> 0 remains largely open at present.

In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.