Global finite-energy compressible flow at the diatomic exponent
On the torus $\mathbb{T}^3$ consider the barotropic compressible Navier–Stokes system with constant viscosities and pressure $\rho^{7/5}$, $$\rho_t + \operatorname{div}(\rho u) = 0, \qquad (\rho u)_t + \operatorname{div}(\rho u \otimes u) + \nabla \rho^{7/5} = \Delta u + \nabla \operatorname{div} u,$$ with energy $$E(t) = \int_{\mathbb{T}^3} \Bigl( \frac{|m|^2}{2\rho} + \frac{5}{2}\rho^{7/5} \Bigr) dx, \qquad m = \rho u,$$ where the kinetic term is $0$ at $(\rho, m) = (0, 0)$ and $+\infty$ where $\rho = 0$ and $m \ne 0$. A global finite-energy weak solution with initial density $\rho_0 \ge 0$ and momentum $m_0$ is a pair $(\rho, u)$ with $\rho \ge 0$, $\rho \in L^\infty_{\mathrm{loc}}([0,\infty); L^{7/5})$ and $u \in L^2_{\mathrm{loc}}([0,\infty); H^1)$ that satisfies the two equations in the sense of distributions with initial traces $\rho_0$ and $m_0$, the renormalized continuity equation $$\partial_t b(\rho) + \operatorname{div}(b(\rho) u) + \bigl(\rho b'(\rho) - b(\rho)\bigr) \operatorname{div} u = 0$$ for every $b \in C^1$ with compactly supported derivative, and, for almost every $t$, the energy inequality $$E(t) + \int_0^t \int_{\mathbb{T}^3} \bigl( |\nabla u|^2 + |\operatorname{div} u|^2 \bigr)\, dx\, ds \le E_0,$$ where $E_0$ is the energy of the initial data, with $(\rho_0, m_0)$ in place of $(\rho, m)$. For the barotropic system with pressure $a\rho^\gamma$ in three dimensions, Lions proved the global existence of weak solutions for general initial data when $\gamma \ge 9/5$ (1998), and Feireisl, Novotný and Petzeltová extended this to $\gamma > 3/2$ (2001). Lukáčová-Medviďová and Schömer name the lack of suitable a priori estimates for the convective term $\rho u \otimes u$ as the main obstacle for $\gamma \le 3/2$.
Problem. Does every initial density $\rho_0 \ge 0$ and momentum $m_0$ with $E_0 < \infty$ admit a global finite-energy weak solution?
Source: After the existence theorems of Lions (1998, γ ≥ 9/5) and Feireisl, Novotný and Petzeltová (2001, γ > 3/2) for the three-dimensional barotropic system, which do not reach γ ≤ 3/2; see Lukáčová-Medviďová and Schömer (2022), §1. Stated by AIM — Open Applied Problems for γ = 7/5 on the three-dimensional torus.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, compressible Navier–Stokes equations, weak solutions
References
@book{lions1998,
author = {Lions, Pierre-Louis},
title = {Mathematical Topics in Fluid Mechanics. {V}ol. 2: Compressible Models},
series = {Oxford Lecture Series in Mathematics and its Applications},
volume = {10},
publisher = {The Clarendon Press, Oxford University Press},
address = {New York},
year = {1998},
}
@article{feireisl2001,
author = {Feireisl, Eduard and Novotn{\'y}, Anton{\'\i}n and Petzeltov{\'a}, Hana},
title = {On the existence of globally defined weak solutions to the {N}avier--{S}tokes equations},
journal = {J. Math. Fluid Mech.},
volume = {3},
number = {4},
year = {2001},
pages = {358--392},
doi = {10.1007/PL00000976},
}
@article{lukacova2022,
author = {Luk{\'a}{\v{c}}ov{\'a}-Medvi{\v{d}}ov{\'a}, M{\'a}ria and Sch{\"o}mer, Andreas},
title = {Existence of dissipative solutions to the compressible {N}avier--{S}tokes system with potential temperature transport},
journal = {J. Math. Fluid Mech.},
volume = {24},
number = {3},
year = {2022},
pages = {82},
doi = {10.1007/s00021-022-00713-3},
eprint = {2106.12435},
archivePrefix = {arXiv},
}
@misc{aim-compressible-ns-physical-exponent,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global finite-energy compressible flow at the diatomic exponent}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/230-compressible-ns-physical-exponent.md},
note = {Entry 230 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00582, "Global finite-energy compressible flow at the diatomic exponent", clopen.solutions.