| Abstract: |
| In this talk, we will discuss the existence of statistically stationary solutions to the linearly damped stochastic compressible Euler equations in one spatial dimension. The system of isentropic compressible Euler equations, describing the evolution of the density and momentum of a compressible fluid, is driven by a multiplicative noise. Additionally, the momentum equation consists of a small linear damping term. The power law for the pressure is given in terms of the density. We show the existence of statistically stationary weak martingale entropy solutions to these equations for any adiabatic constant and any damping coefficient, namely stochastic solutions for which the law of the solution is constant in time. To establish this result, we consider an approximate system for which we show the existence of an invariant measure, and then pass to the limit in the approximation parameters in order to obtain a limiting statistically stationary solution. We develop new techniques for obtaining uniform-in-time estimates for entropies of all orders, and study invariant regions of the approximate system. Such a result is a significant step in understanding the long-time statistics of stochastically perturbed compressible inviscid fluids. |
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