Uniqueness of the invariant law for a stochastic one-dimensional compressible fluid
For $A > 0$, on $I = (0,1)$ consider the Itô system $$\begin{aligned} d\rho + (\rho u)_x\,dt &= 0,\\ d(\rho u) + \bigl(\rho u^2 + A^2\rho\bigr)_x\,dt &= u_{xx}\,dt + \rho \sum_{k \ge 1} \sigma_k(x)\,dW^k_t, \end{aligned}$$ with $u(t,0) = u(t,1) = 0$ and total mass $\int_0^1 \rho\,dx = 1$, where the $W^k$ are independent standard real Brownian motions and the deterministic coefficients satisfy $$\sigma_k \in H^2(I) \cap H^1_0(I), \qquad \sum_{k \ge 1} \|\partial_{xx}\sigma_k\|_{L^2(I)}^2 < \infty.$$ Let $(P_t)_{t \ge 0}$ be the Markov semigroup generated by its strong solutions on $$\mathcal X = \Bigl\{(\rho,u) \in H^1(I) \times H^1_0(I) : \rho(x) > 0 \text{ for every } x \in [0,1],\ \int_0^1 \rho\,dx = 1\Bigr\},$$ with the $L^2(I) \times L^2(I)$ metric and its Borel $\sigma$-algebra; positivity refers to the continuous representative of $\rho$. A Borel probability measure $\mu$ on $\mathcal X$ is invariant if $$\int_{\mathcal X} P_t\varphi\,d\mu = \int_{\mathcal X} \varphi\,d\mu \qquad (t \ge 0)$$ for every bounded Borel function $\varphi$ on $\mathcal X$. Strong solutions exist globally and are pathwise unique (Coti Zelati, Glatt-Holtz and Trivisa, Theorem 2.1, from the deterministic theory), and Coti Zelati, Glatt-Holtz and Trivisa proved that $(P_t)$ has an invariant probability measure for every $A > 0$. They conjectured that it is unique, noting that the noise acts on only one component of the phase space and that the unforced system tends to the steady state $(\rho, u) = (1, 0)$.
Problem. Is it true that for every $A > 0$ and every choice of the $\sigma_k$ as above, $(P_t)$ has exactly one invariant Borel probability measure on $\mathcal X$?
Source: Coti Zelati, Glatt-Holtz and Trivisa (2021), conjectured in Section 1, "Extensions and further developments". Stated by AIM — Open Applied Problems for every admissible noise.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, compressible Navier–Stokes equations, ergodicity, invariant measures
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{cotizelati2021,
author = {Coti Zelati, Michele and Glatt-Holtz, Nathan and Trivisa, Konstantina},
title = {Invariant measures for the stochastic one-dimensional compressible {N}avier--{S}tokes equations},
journal = {Appl. Math. Optim.},
volume = {83},
number = {3},
year = {2021},
pages = {1487--1522},
doi = {10.1007/s00245-019-09594-x},
eprint = {1802.04000},
archivePrefix = {arXiv},
}
@misc{aim-uniqueness-of-the-invariant-law-for-a-stochastic-one-dimensional-compressible-fluid,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the invariant law for a stochastic one-dimensional compressible fluid}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/1001-uniqueness-of-the-invariant-law-for-a-stochastic-one-dimensional-compressible-fluid.md},
note = {Entry 1001 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01168, "Uniqueness of the invariant law for a stochastic one-dimensional compressible fluid", clopen.solutions.