Invariant Measures for the Stochastic One-Dimensional Compressible Navier-Stokes Equations

Area of Science:

• Fluid Dynamics
• Stochastic Analysis
• Partial Differential Equations

Background:

• The study focuses on the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system.
• This system models the motion of a compressible viscous fluid with a linear pressure law.

Purpose of the Study:

• To investigate the long-time behavior of solutions for the specified Navier-Stokes system.
• To prove the existence of an invariant measure for the associated Markov process.

Main Methods:

• Generalization of the classical Krylov-Bogoliubov method to handle non-Feller Markov semigroups on non-complete metric spaces.
• Derivation of polynomial and exponential moment bounds for the solutions.
• Utilization of pathwise estimates to analyze the system's behavior.

Main Results:

• Existence of an invariant measure for the Markov process generated by strong solutions has been proven.
• The study successfully overcomes analytical challenges associated with non-complete metric spaces and non-Feller semigroups.

Conclusions:

• The findings provide a fundamental understanding of the long-term statistical properties of the stochastically forced fluid system.
• This research contributes to the rigorous mathematical analysis of complex fluid dynamics models.