Boltzmann equations with dilatation and shear: global well-posedness and stability
Abstract
We study global well-posedness and stability for Boltzmann equations with linear velocity deformation, allowing for arbitrarily large deformation strength. For diagonal dilatation with −2<γ≤0 and cylindrical dilatation with −3/2<γ≤0, with or without shear, we construct global homogeneous solutions for small data and prove their stability under small spatially inhomogeneous perturbations. In scaled variables, the perturbations remain uniformly controlled in deformation-adapted spaces and decay in mixed spatial--velocity norms, algebraically in the diagonal regime and exponentially in the cylindrical regime. For homogeneous simple shear with −2≤γ≤0, we prove global existence and uniqueness for nonnegative data with finite mass and Fisher information, without smallness or a finite-energy assumption. These solutions conserve mass and propagate additional Sobolev regularity. For spatially inhomogeneous simple shear with −2<γ≤0, we establish global well-posedness for small data. The analysis combines commuting vector fields, dispersive and bilinear collision estimates, and integrability of the rescaled collision coefficients. For homogeneous shear, a positivity-preserving splitting transfers Fisher-information control to rough data, while a Lorentz-space estimate includes the endpoint γ=−2 in the global continuation argument.
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BibTeX
@misc{liu2026boltzmann,
title = {{Boltzmann equations with dilatation and shear: global well-posedness and stability}},
author = {Shuangqian Liu and Haoran Peng and Jiatong Zou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07340},
eprint = {2611.07340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07340},
}
RIS
TY - UNPB
TI - Boltzmann equations with dilatation and shear: global well-posedness and stability
AU - Shuangqian Liu
AU - Haoran Peng
AU - Jiatong Zou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07340
UR - https://arxiv.org/abs/2611.07340
AB - We study global well-posedness and stability for Boltzmann equations with linear velocity deformation, allowing for arbitrarily large deformation strength. For diagonal dilatation with −2<γ≤0 and cylindrical dilatation with −3/2<γ≤0, with or without shear, we construct global homogeneous solutions for small data and prove their stability under small spatially inhomogeneous perturbations. In scaled variables, the perturbations remain uniformly controlled in deformation-adapted spaces and decay in mixed spatial--velocity norms, algebraically in the diagonal regime and exponentially in the cylindrical regime. For homogeneous simple shear with −2≤γ≤0, we prove global existence and uniqueness for nonnegative data with finite mass and Fisher information, without smallness or a finite-energy assumption. These solutions conserve mass and propagate additional Sobolev regularity. For spatially inhomogeneous simple shear with −2<γ≤0, we establish global well-posedness for small data. The analysis combines commuting vector fields, dispersive and bilinear collision estimates, and integrability of the rescaled collision coefficients. For homogeneous shear, a positivity-preserving splitting transfers Fisher-information control to rough data, while a Lorentz-space estimate includes the endpoint γ=−2 in the global continuation argument.
ER -