On the Navier–Stokes Problem in Exterior Domains with Non Decaying Initial Data

2011 ◽  
Vol 14 (4) ◽  
pp. 633-652 ◽  
Author(s):  
Giovanni P. Galdi ◽  
Paolo Maremonti ◽  
Yong Zhou
Keyword(s):  
Initial Data ◽  
Stokes Problem ◽  
Navier Stokes ◽  
Exterior Domains

scholarly journals ON COUPLED PROBLEMS FOR VISCOUS FLOW IN EXTERIOR DOMAINS

1998 ◽  
Vol 08 (04) ◽  
pp. 657-684 ◽  
Author(s):  
M. FEISTAUER ◽  
C. SCHWAB
Keyword(s):  
Stokes Flow ◽  
Stokes System ◽  
Stokes Problem ◽  
Large Data ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Coupled Problems ◽  
Exterior Domains ◽  
Existence Of A Solution ◽  
Coupled Problem

The use of the complete Navier–Stokes system in an unbounded domain is not always convenient in computations and, therefore, the Navier–Stokes problem is often truncated to a bounded domain. In this paper we simulate the interaction between the flow in this domain and the exterior flow with the aid of a coupled problem. We propose in particular a linear approximation of the exterior flow (here the Stokes flow or potential flow) coupled with the interior Navier–Stokes problem via suitable transmission conditions on the artificial interface between the interior and exterior domains. Our choice of the transmission conditions ensures the existence of a solution of the coupled problem, also for large data.

On the stationary Navier–Stokes problem in 3D exterior domains

2018 ◽  
Vol 99 (9) ◽  
pp. 1485-1506
Author(s):  
Vincenzo Coscia ◽  
Remigio Russo ◽  
Alfonsina Tartaglione
Keyword(s):  
Stokes Problem ◽  
Navier Stokes ◽  
Exterior Domains

scholarly journals A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

2015 ◽  
Vol 8 (4) ◽  
pp. 549-581 ◽  
Author(s):  
Deepjyoti Goswami ◽  
Pedro D. Damázio
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Initial Data ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Time Dependent ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Smooth Initial Data ◽  
Element Method

AbstractWe analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h « H. This method gives optimal convergence for velocity in H1-norm and for pressure in L2-norm. The analysis mainly focuses on the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.

scholarly journals Comments on the Navier–Stokes Problem

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 95
Author(s):  
Alexander G. Ramm
Keyword(s):  
Initial Data ◽  
Fluid Mechanics ◽  
Stokes Problem ◽  
Exceptional Case ◽  
Navier Stokes ◽  
Correct Description ◽  
Broad Audience

The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)≢0, ∇·v(x,0)=0 and the solution to the NSP exists for all t≥0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t≥0 and is equal to zero, v(x,t)≡0. Thus, one of the millennium problems is solved.

scholarly journals A remark on the non-uniqueness in $$L^\infty $$ of the solutions to the two-dimensional Stokes problem in exterior domains

2020 ◽  
Author(s):  
Paolo Maremonti
Keyword(s):  
Initial Data ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Well Posedness ◽  
Two Dimensional ◽  
Exterior Domains ◽  
Divergence Free

AbstractThe paper is concerned with the IBVP in exterior domains of the two-dimensional Stokes equations. The goal was to investigate the well-posedness in the set of solutions assuming an initial data $$u_0\in L^\infty (\Omega )$$ u 0 ∈ L ∞ ( Ω ) , divergence free, and enjoying the property "Equation missing" for all $$t>0$$ t > 0 and c independent of u. For all $$u_0\in L^\infty $$ u 0 ∈ L ∞ , divergence-free one shows examples of non-uniqueness in the above set of solutions.

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

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