Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space. (English) Zbl 0754.35110

Maremonti, Paolo

Ric. Mat. 40, No. 1, 81-135 (1991).

The author studies the Navier-Stokes-equations with a slip boundary condition in \(\mathbb{R}_ +^ 3\), i.e.: \[ v_ t-\Delta v+(v\cdot\nabla)v+\nabla\pi=F,\quad\text{div} v=0 \quad\text{in }(0,T)\times\mathbb{R}_ +^ 3;\quad v(x,t)\to 0,\quad | x|\to\infty,\;t>0, \]
\[ v\cdot n\mid_{x_ 3=0}=0,\quad [D(v)- (nD(v)n)n]\mid_{x_ 3=0}=0;\qquad v(0)=v_ 0\quad\text{in } \mathbb{R}_ +^ 3, \] where \(D(v)_{ij}={1\over 2}(v_{x_ j}^ i+v_{x_ i}^ j)\) (\(i,j=1,2,3\)) denotes the deformation tensor of the velocity \(v\). Under suitable conditions on \(v_ 0\) and \(F\) (which is not required to be summable in time) the existence and uniqueness of global (in time) solutions is shown. The main part of the proof consists of the study of a linearized problem for which several a-priori estimates are established. Finally the theorem on global solvability is used in order to prove the existence of time periodic and stationary solutions.

Reviewer: K.Deckelnick (Freiburg i.Br.)

MSC:

35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

time periodic solutions; stationary solutions; existence; uniqueness; a- priori estimates