scholarly journals On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

2015 ◽  
Vol 219 (2) ◽  
pp. 637-678 ◽  
Author(s):  
Kazuyuki Tsuda
Keyword(s):  
Periodic Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Existence And Stability ◽  
Whole Space ◽  
Time Periodic ◽  
Time Periodic Solution

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.

scholarly journals Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

2020 ◽  
Vol 22 (2) ◽  
Author(s):  
Zdzisław Brzeźniak ◽  
Gaurav Dhariwal
Keyword(s):  
Invariant Measure ◽  
Stokes Equation ◽  
Diffusion Processes ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Unique Strong Solution ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Whole Space

Abstract Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

Time-Periodic Solution to the Compressible Navier-Stokes/Allen-Cahn System

2020 ◽  
Vol 36 (4) ◽  
pp. 419-442
Author(s):  
Chang Ming Song ◽  
Jian Lin Zhang ◽  
Yuan Yuan Wang
Keyword(s):  
Periodic Solution ◽  
Navier Stokes ◽  
Time Periodic ◽  
Time Periodic Solution
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