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We consider the Navier–Stokes equations in R 3 subject to the initial condition with initial velocity field in L loc 2 (R 3) such that lim sup R → + ∞ R − 1 ‖ u 0 ‖ L 2 (B (R)) < + ∞. Our aim is to show the local existence of a weak solution, global existence of weak solution if C = 0 and the partial regularity in the sense of Caffarelli–Kohn–Nirenberg. [ABSTRACT FROM AUTHOR]

We provide a local compactness lemma, that can be used for various models of incompressible fluids in a general domain, such as generalized Newtonian fluids. In particular, it helps for the proof of the existence of weak solution to those systems, when it comes to carrying out the passage to limit for sequences of weak solution to a corresponding approximate system. The key argument of our approach is the use of the local pressure decomposition in order to avoid the construction of a global pressure. [ABSTRACT FROM AUTHOR]

We address the existence of strong solutions to a system of equations of motion of an incompressible non-Newtonian fluid. Our aim is to prove the short-time existence of strong solutions for the case of shear thickening viscosity, which corresponds to the power law ν ( D ) = ∣ D ∣ q − 2 ( 2 < q < + ∞ ) . In particular, we find that global strong solutions exist whenever q > 2.23 … . The results are obtained by flattening the boundary and by using the difference quotient method. Near the boundary, we use weighted estimates in the normal direction. [ABSTRACT FROM AUTHOR]

Abstract: We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on physical parameters and which appear on the right hand side. [Copyright &y& Elsevier]