On the asymptotic stability of steady solutions of the Navier–Stokes equations in unbounded domains

2007 ◽  
Vol 30 (12) ◽  
pp. 1375-1401
Author(s):  
Francesca Crispo ◽  
Alfonsina Tartaglione
Keyword(s):  
Asymptotic Stability ◽  
Stokes Equations ◽  
Unbounded Domains ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Steady Solutions

scholarly journals Navier–Stokes equations with delays on unbounded domains

2006 ◽  
Vol 64 (5) ◽  
pp. 1100-1118 ◽  
Author(s):  
María José Garrido-Atienza ◽  
Pedro Marín-Rubio
Keyword(s):  
Stokes Equations ◽  
Unbounded Domains ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Wen-Juan Wang ◽  
Yan Jia
Keyword(s):  
Weak Solution ◽  
Asymptotic Stability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Perturbed System ◽  
Regular Class ◽  
Stability Issue ◽  
The Stability

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solutionuof the Navier-Stokes equations lies in the regular class∇u∈Lp(0,∞;Bq,∞0(ℝ3)),(2α/p)+(3/q)=2α,2<q<∞,0<α<1, then every weak solutionv(x,t)of the perturbed system converges asymptotically tou(x,t)asvt-utL2→0,t→∞.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

scholarly journals Random attractors for stochastic 2D-Navier–Stokes equations in some unbounded domains

2013 ◽  
Vol 255 (11) ◽  
pp. 3897-3919 ◽  
Author(s):  
Z. Brzeźniak ◽  
T. Caraballo ◽  
J.A. Langa ◽  
Y. Li ◽  
G. Łukaszewicz ◽  
...  
Keyword(s):  
Stokes Equations ◽  
Unbounded Domains ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Random Attractors
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