scholarly journals Existence and stability of time periodic solution to the compressible Navier–Stokes equation for time periodic external force with symmetry

2015 ◽  
Vol 258 (2) ◽  
pp. 399-444 ◽  
Author(s):  
Yoshiyuki Kagei ◽  
Kazuyuki Tsuda
Keyword(s):  
Periodic Solution ◽  
Stokes Equation ◽  
External Force ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Existence And Stability ◽  
Time Periodic ◽  
Time Periodic Solution

scholarly journals The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

2021 ◽  
Author(s):  
Riccardo Montalto
Keyword(s):  
Periodic Solutions ◽  
Stokes Equation ◽  
External Force ◽  
Invariant Torus ◽  
Invariant Tori ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Existence And Stability ◽  
Exponential Rate Of Convergence ◽  
Incompressible Navier Stokes Equation

AbstractWe prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus $$\mathbb T^d$$ T d , with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $$H^s$$ H s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $$t \rightarrow + \infty $$ t → + ∞ , with an exponential rate of convergence $$O( e^{- \alpha t })$$ O ( e - α t ) for any arbitrary $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) .

scholarly journals DECAY PROPERTIES OF SOLUTIONS TO THE LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATION AROUND TIME-PERIODIC PARALLEL FLOW

2012 ◽  
Vol 22 (07) ◽  
pp. 1250007 ◽  
Author(s):  
JAN BŘEZINA ◽  
YOSHIYUKI KAGEI
Keyword(s):  
Periodic Function ◽  
Stokes Equation ◽  
Parallel Flow ◽  
Navier Stokes ◽  
Decay Estimates ◽  
Convective Term ◽  
Navier Stokes Equation ◽  
Decay Properties ◽  
Linearized Problem ◽  
Time Periodic

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

Sign in / Sign up

Export Citation Format

Share Document