Well-posedness in the generalized sense for the incompressible Navier-Stokes equation

Abstract

The theory of well-posedness in the generalized sense is developed for the linearized, time-dependent Navier-Stokes equation for incompressible flow, together with boundary conditions. This concept of well-posedness means existence and uniqueness of solutions together with an energy estimate where theL 2-norm is taken not only over the space variables, but also over the time. We prove that the existence of a unique solution of the Laplace-Fourier transformed problem, together with the corresponding energy estimate, implies well-posedness in the generalized sense. The Laplace-Fourier transform yields an ordinary differential equation for which the existence and uniqueness of solutions and the energy estimate is in general easy to show. This technique is therefore generally a considerable simplification of the investigation of well-posedness, and existence and uniqueness of solutions of the original problem can be proven for more general boundary conditions than if the classical concept of well-posedness is used. This is important when boundary conditions for open boundaries (inflow and outflow) are investigated, as one then typically wants to prescribe a high-order derivative of some of the dependent variables. We also show stability against perturbations with lower-order terms. This means that such terms can be omitted when investigating well-posedness in the generalized sense. All proofs are carried through in detail.