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

  • O. A. Ladyzhenskaya
Part of the Seminars in Mathematics book series (SM, volume 11)


Let us consider the system
$$\left. \begin{gathered} {v_t} - v\Delta \upsilon + \sum\limits_{i = 1}^3 {{\upsilon _i}} {\upsilon _{{x_i}}} = - \nabla p + f(x,t) \hfill \\ div\upsilon = 0 \hfill \\ \end{gathered} \right\}$$
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
$$\upsilon {\left| {_{t = 0} = {\upsilon _0}(x),x \in {\Omega _{01}}_\upsilon } \right|_{s_{nbhd}^T}} = \psi (s,t)$$


Vector Function Finite Rate Solenoidal Vector Left Member Smooth Vector Function 
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Literature Cited

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    O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York (1968).Google Scholar
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    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
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    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).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1970

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  • O. A. Ladyzhenskaya

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