scholarly journals On Unique Continuation for Navier-Stokes Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
Zhiwen Duan ◽  
Shuxia Han ◽  
Peipei Sun
Keyword(s):  
Stokes Equations ◽  
Sufficient Conditions ◽  
Unique Continuation ◽  
Navier Stokes ◽  
Carleman Inequality ◽  
Navier Stokes Equations ◽  
Logarithmic Convexity ◽  
Weighted Estimates ◽  
Rotation Transformation

We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different timest0=0andt1=1which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations.

scholarly journals The inf-sup stability of the lowest order Taylor–Hood pair on affine anisotropic meshes

2019 ◽  
Vol 40 (4) ◽  
pp. 2377-2398
Author(s):  
Gabriel R Barrenechea ◽  
Andreas Wachtel
Keyword(s):  
Finite Element ◽  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Element Solution ◽  
Navier Stokes Equations ◽  
Anisotropic Meshes ◽  
Theoretical Results

Abstract Uniform inf-sup conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier–Stokes equations. In this work we prove a uniform inf-sup condition for the lowest-order Taylor–Hood pairs $\mathbb{Q}_2\times \mathbb{Q}_1$ and $\mathbb{P}_2\times \mathbb{P}_1$ on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalize Verfürth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.

On the identification of a single body immersed in a Navier-Stokes fluid

2007 ◽  
Vol 18 (1) ◽  
pp. 57-80 ◽  
Author(s):  
A. DOUBOVA ◽  
E. FERNÁNDEZ-CARA ◽  
J. H. ORTEGA
Keyword(s):  
Inverse Problem ◽  
Rigid Body ◽  
Stokes Equations ◽  
Outer Boundary ◽  
Unique Continuation ◽  
Uniqueness Result ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Friction Forces ◽  
Regularity Results

In this work we consider the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Navier-Stokes equations. It is assumed that friction forces are known on a part of the outer boundary. We first prove a uniqueness result. Then, we establish a formula for the observed friction forces, at first order, in terms of the deformation of the rigid body. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs we use unique continuation and regularity results for the Navier-Stokes equations and domain variation techniques.

Transport-stabilized Semidiscretizations of the Incompressible Navier—Stokes Equations

2006 ◽  
Vol 6 (3) ◽  
pp. 239-263 ◽  
Author(s):  
L. Angermann
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Approximation Technique ◽  
Convective Term ◽  
Navier Stokes Equations ◽  
Velocity Error ◽  
Sufficient Conditions For Convergence ◽  
Upwind Method

AbstractWithin the framework of finite element methods, the paper investigates a general approximation technique for the nonlinear convective term of Navier — Stokes equations. The approach is based on an upwind method of the finite volume type. It has been proved that the discrete convective term satisfies the well-known collection of sufficient conditions for convergence of the finite element solution. For a particular nonconforming scheme, the assumptions have been verified in detail and the estimate of the semidiscrete velocity error has been proved.

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