# Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs

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We improve the regularity criterion for the incompressible Navierâ€“Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see â€œRegularity criteria for the three dimensional...

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We give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier-Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial...

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Based on two-grid discretizations, in this paper, some new local and parallel finite element algorithms are proposed and analyzed for the stationary incompressible Navier-Stokes problem. These algorithms are motivated by the observation that for a solution to the Navier-Stokes problem, low...

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We consider the stationary Navierï¿½Stokes equations in a bounded domain O in R n with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in L n ( O) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we...

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We show that if a Lerayï¿½Hopf solution u of the three-dimensional Navierï¿½Stokes equation belongs to $${C((0,T]; B^{-1}_{\infty,\infty})}$$ or its jumps in the $${B^{-1}_{\infty,\infty}}$$-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses...

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We consider inverse extremal problems for the stationary Navier-Stokes equations. In these problems, one seeks an unknown vector function occurring in the Dirichlet boundary condition for the velocity and the solution of the considered boundary value problem on the basis of the minimization of...

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We consider the problem of partial controllability for an evolution equation with a quadratic nonlinearity, in which one should provide, at a given time, a given projection of the solution onto some finite-dimensional subspace by using the action of external forces that belong to one and the...

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In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in (Rï¿½ckner and Zhang in Probab. Theory Relat. Fields 145(1ï¿½2),...

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The flow around the gap of a semi-spade rudder was simulated as part of an effort to minimize gap cavitation. Simulations were performed for various devices aimed at controlling the gap flow in two-dimensional and three-dimensional domains using the Reynolds-averaged Navier-Stokes equations. A...