Hard-sphere Boltzmann shock profiles at arbitrary Mach number
For a Mach number $\mathcal{M} > 1$ set $$\rho_- = T_- = 1, \quad u_- = \sqrt{5/3}\,\mathcal{M}, \quad r = \frac{4\mathcal{M}^2}{\mathcal{M}^2 + 3}, \quad \rho_+ = r, \quad u_+ = \frac{u_-}{r}, \quad T_+ = \frac{(5\mathcal{M}^2 - 1)(\mathcal{M}^2 + 3)}{16\mathcal{M}^2},$$ end states that obey the Rankine–Hugoniot relations of a monatomic ideal gas, and let $M_\pm(v) = \rho_\pm (2\pi T_\pm)^{-3/2} \exp\bigl(-|v - u_\pm e_1|^2 / (2T_\pm)\bigr)$. The stationary planar Boltzmann equation for hard spheres is $$v_1 \partial_z F = Q(F, F), \qquad z \in \mathbb{R}, \ v \in \mathbb{R}^3,$$ $$Q(F, F)(v) = \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} |(v - v_*) \cdot \omega| \bigl[F(v') F(v_*') - F(v) F(v_*)\bigr] \, d\omega \, dv_*,$$ where $v' = v - ((v - v_*) \cdot \omega)\omega$ and $v_*' = v_* + ((v - v_*) \cdot \omega)\omega$. Caflisch and Nicolaenko proved that shock profiles exist for weak shocks, with $\mathcal{M}$ close to $1$. Pogan and Zumbrun recovered this for hard spheres by a center-manifold construction, with the small-amplitude profiles localised in velocity at a near-Maxwellian rate.
Problem. For every $\mathcal{M} > 1$, does there exist a nonnegative profile $F(z, v)$, continuous in $z$ with values in $L^1((1 + |v|^2)\, dv)$, with locally finite entropy, solving the profile equation in the sense of distributions, such that $F(z, \cdot) \to M_\pm$ in $L^1((1 + |v|^2)\, dv)$ as $z \to \pm\infty$? The jump between the end states need not be small.
Source: The shock structure problem of kinetic theory, discussed by Truesdell, Boillat, Ruggeri and others; see Pogan and Zumbrun (2018), §1.3. Stated in this form, for hard spheres and every Mach number, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boltzmann equation, kinetic theory, shock profiles
References
@article{pogan2018,
author = {Pogan, Alin and Zumbrun, Kevin},
title = {Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks},
journal = {J. Differential Equations},
volume = {264},
number = {11},
year = {2018},
pages = {6752--6808},
doi = {10.1016/j.jde.2018.01.049},
eprint = {1612.05676},
archivePrefix = {arXiv},
}
@article{caflisch1982,
author = {Caflisch, Russel E. and Nicolaenko, Basil},
title = {Shock profile solutions of the {B}oltzmann equation},
journal = {Comm. Math. Phys.},
volume = {86},
number = {2},
year = {1982},
pages = {161--194},
doi = {10.1007/BF01206009},
}
@misc{aim-boltzmann-arbitrary-mach-shocks,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hard-sphere Boltzmann shock profiles at arbitrary Mach number}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/378-boltzmann-arbitrary-mach-shocks.md},
note = {Entry 378 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-00577, "Hard-sphere Boltzmann shock profiles at arbitrary Mach number", clopen.solutions.