Contributions to Current Challenges in Mathematical Fluid by Giovanni P. Galdi, John G. Heywood, Rolf Rannacher

By Giovanni P. Galdi, John G. Heywood, Rolf Rannacher

The mathematical idea of the Navier-Stokes equations offers nonetheless basic open questions that symbolize as many demanding situations for the mathematicians. This quantity collects a chain of articles whose goal is to provide new contributions and concepts to those questions, with specific regard to turbulence modelling, regularity of options to the initial-value challenge, stream in zone with an unbounded boundary and compressible flow.

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1 dt . 1 O J0 (iou'ii. 1 Multiplying with bU"+' on both sides of (Nt) and integrating over 1R3, we have 2 dt IIbU,1+1 II2 2 + IIA"bU' IIVU" 11 L- II oU" IILl 'U II6U"+1 2 11 L2 + Ilou' IIL'° IISU"+1 IIL2II5Un+1 111, IIbU"IIL2II6U"+'11L2 +C15IIulII

2°)nC((0, oo); Bz 1), - 6 2, if 0 < a < 2 for any 62 > 0. We provide only a where a > 0 and -y = l 2,if2 0 if 0 < a < 2. If a > 21 u satisfies atu = -(u 0)u - A2au - Op. 1) n L' (a, oo; B21 u belongs to C([a, oo); B2,1). This completes the proof of Theorem 1. Proof of Theorem 2.

Data, extending the previous results for the case a = 1. for small initial Mathematics Subject Classification (2000). 35Q30, 76D03. Keywords. Navier-Stokes equations, global well-posedness, stability. 1. Introduction In this paper, we are concerned with the sub-dissipative or hyper-dissipative Navier-Stokes equations. (SNS) div u = 0, u(0,x) = ue(x), where it represents the velocity vector field, p is the scalar pressure. For simplicity, we assume that the external force vanishes, but it is easy to extend our results to the case of nonzero external force.

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