The development of the cornea, a highly specialized transparent tissue located at the anterior of the eye, is coordinated by a variety of molecules and cells. Here, we report that mast cells (MCs), recently found to be involved in morphogenesis, played a potentially important role in corneal development in mice. We show that two different waves of MC migration occurred during corneal development. In the first wave, MCs migrated to the corneal stroma and became distributed throughout the cornea. This wave occurred by embryonic day 12.5, with MCs disappearing from the cornea at the time of eyelid opening. In the second wave, MCs migrated to the corneal limbus and became distributed around limbal blood vessels. The number of MCs in this region gradually increased after birth and peaked at the time of eyelid opening in mice, remaining stable after postnatal day 21. We also show that integrin α4β7 and CXCR2 were important for the migration of MC precursors to the corneal limbus and that c-Kit-dependent MCs appeared to be involved in the formation of limbal blood vessels and corneal nerve fibers. These data clearly revealed that MCs participate in the development of the murine cornea.

This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true, otherwise, there is a sharp critical viscosity, which distinguishes the nonlinear stability from instability.

This paper is concerned with the Rayleigh-Taylor instability for the nonhomogeneous incompressible Navier-Stokes equations with Navier-slip boundary conditions around a steady-state in an infinite slab, where the Navier-slip coefficients do not have defined sign and the slab is horizontally periodic. Motivated by [18], we extend the result from Dirichlet boundary condition to Navier-slip boundary conditions. Our results indicate the factor that "heavier density with increasing height" still plays a key role in the instability under Navier-slip boundary conditions.

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<p style='text-indent:20px;'>This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.</p>