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

Chapter
Part of the Seminars in Mathematics book series (SM, volume 11)

## 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)
.

## Keywords

Vector Function Finite Rate Solenoidal Vector Left Member Smooth Vector Function

## Literature Cited

1. 1.
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York (1968).Google Scholar
2. 2.
O. A. Ladyzhenskaya, A Mixed Problem for Hyperbolic Equations [in Russian], Moscow (1953).Google Scholar
3. 3.
G. Prodi, Résultats récents et problèmes anciens dans la théorie des équations de Navier—Stokes, Institut de Mathématiques, Trieste (1962).Google Scholar
4. 4.
M. Shinbrot and S. Kaniel, “The initial value problem for the Navier—Stokes equations,” Archive for Rational Mechanics and Analysis, Vol. 21, No. 4, pp. 270–285 (1966).