Condensation and Non-condensation in the Quasilinear 4-wave Kinetic Equation
Abstract
We study finite-time condensation at zero frequency for the isotropic quasilinear 4-wave kinetic equation in three dimensions. With Schrodinger dispersion and a product interaction coefficient with exponent β∈(0,3/4], we construct global measure solutions (gt)t≥0 for every nonnegative initial measure g0 with finite wave action and energy. We then establish the sharp threshold β=1/4 for finite-time condensation of the constructed solutions. For β∈[1/4,3/4], the constructed solutions satisfy gt({0})=g0({0}) for all t≥0. For β∈(0,1/4), every constructed solution with (g0)ac≠0 satisfies |gt−𝒩0δ0|TV≤ C𝒩0(1+𝒩02t)−(1−4β)/40 for all t≥0, where 𝒩0=g0([0,∞)) and C=C(β,g0/𝒩0)>0. In particular, for every ζ∈(0,1), there exists Tζ<∞ such that gt({0})≥(1−ζ)𝒩0 for all t≥ Tζ.
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BibTeX
@misc{luo2026condensation,
title = {{Condensation and Non-condensation in the Quasilinear 4-wave Kinetic Equation}},
author = {Siwei Luo and Jian-Guo Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25207},
eprint = {2610.25207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25207},
}
RIS
TY - UNPB
TI - Condensation and Non-condensation in the Quasilinear 4-wave Kinetic Equation
AU - Siwei Luo
AU - Jian-Guo Liu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25207
UR - https://arxiv.org/abs/2610.25207
AB - We study finite-time condensation at zero frequency for the isotropic quasilinear 4-wave kinetic equation in three dimensions. With Schrodinger dispersion and a product interaction coefficient with exponent β∈(0,3/4], we construct global measure solutions (g_t)_t≥0 for every nonnegative initial measure g_0 with finite wave action and energy. We then establish the sharp threshold β=1/4 for finite-time condensation of the constructed solutions. For β∈[1/4,3/4], the constructed solutions satisfy g_t({0})=g_0({0}) for all t≥0. For β∈(0,1/4), every constructed solution with (g_0)_ac≠0 satisfies |g_t−𝒩_0δ_0|_TV≤ C𝒩_0(1+𝒩_0^2t)^−(1−4β)/40 for all t≥0, where 𝒩_0=g_0([0,∞)) and C=C(β,g_0/𝒩_0)>0. In particular, for every ζ∈(0,1), there exists T_ζ<∞ such that g_t({0})≥(1−ζ)𝒩_0 for all t≥ T_ζ.
ER -