Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies

2016 ◽  
pp. 25-41
Author(s):  
Helmut Abels ◽  
Josef Weber
Keyword(s):  
Stationary Solutions ◽  
Navier Stokes ◽  
Free Energies

scholarly journals Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

2021 ◽  
Vol 2090 (1) ◽  
pp. 012046
Author(s):  
Nikolay M. Evstigneev
Keyword(s):  
Flow Problem ◽  
Stokes Equations ◽  
Continuation Method ◽  
Fourier Method ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
System Of Nonlinear Equations ◽  
Arc Length ◽  
Navier Stokes Equations ◽  
Numerical Bifurcation

Abstract The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

scholarly journals Lr-LpStability of the Incompressible Flows with Nonzero Far-Field Velocity

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Jaiok Roh
Keyword(s):  
Constant Velocity ◽  
Incompressible Flows ◽  
Temporal Stability ◽  
Stationary Solutions ◽  
Decay Rates ◽  
Navier Stokes ◽  
Far Field ◽  
Spatial Direction ◽  
Field Velocity ◽  
The Stability

We consider the stability of stationary solutionswfor the exterior Navier-Stokes flows with a nonzero constant velocityu∞at infinity. Foru∞=0with nonzero stationary solutionw, Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability inLpspaces for1<pand obtained good stability decay rates. For the spatial direction, we recently obtained some results. Foru∞≠0, Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability inL2space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability inLpspaces forp≥3. Then, Bae and Roh recently improved Enomoto and Shibata's results in some sense. In this paper, we improve Bae and Roh's result in the spacesLpforp>1and obtainLr-Lpstability as Kozono and Ogawa and Borchers and Miyakawa obtained foru∞=0.

STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
Author(s):  
YURI BAKHTIN ◽  
JONATHAN C. MATTINGLY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Solution ◽  
Landau Equation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
With Memory

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

Sign in / Sign up

Export Citation Format

Share Document