Initial-Boundary Problem for Navier-Stokes Equations in Domains with Time-Varying Boundaries

Abstract

Let us consider the system

$$\left. \begin{gathered} {v_t} - v\Delta \upsilon + \sum\limits_{i = 1}^3 {{\upsilon _i}} {\upsilon _{{x_i}}} = - \nabla p + f(x,t) \hfill \\ div\upsilon = 0 \hfill \\ \end{gathered} \right\}$$

(1)

in the bounded domain ${Q^T} = \left\{ {(x,t):t \in (o,T),x \in {\Omega _t}} \right\}$ of the space ${E_u}\left\{ {(x,t):t \in ( - \infty ,\infty ),x = ({x_1},{x_2},{x_3}) \in {E_3}} \right\}$ and let us assume that the boundary S _t of the domain Ω _t belongs to C ² for all t ∈ [0,T] (where the “norms” of S _t in C ² are uniformly bounded) and changes with time at a finite rate. With the system (1) we shall associate the initial and boundary conditions

$$\upsilon {\left| {_{t = 0} = {\upsilon _0}(x),x \in {\Omega _{01}}_\upsilon } \right|_{s_{nbhd}^T}} = \psi (s,t)$$

(2)

.