Fluids Under Pressure by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9783030396398
Publisher: Springer International Publishing

where p k is a pressure, associated with the solution J ku of equation (4.47). Constants c 3 and c 4 are independent of k. Hence, passing to the limit for k →∞, we deduce that there exists a scalar function p with ∇p ∈ L r(0, T; L s( Ω)) such that u, p satisfy the Navier-Stokes equation (4.9) a.e. in Q T and

(4.48)

Function p can be chosen so that p ∈ L r(0, T; L 3s∕(3−s)( Ω)). (This basically follows from the inclusion ∇p ∈ L r(0, T; L s( Ω)) and the Poincaré-Sobolev inequality, see [36] for further references.)

Combining this procedure with the method of localization to a bounded domain ΩR :=  Ω ∩ B R(0) (described, e.g., in [61]) for sufficiently large R and considering the limit passage R →∞, we can also derive analogous results if or Ω is an exterior domain in . The case when Ω is a half-space can be reduced to the whole space by an appropriate symmetric extension of u and p. Thus, we obtain the theorem (see Theorem 3.1 in [36]):

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