Enhanced Dissipation by Transport Noise for Stochastic p(x)-Laplace Equations
Abstract
We are concerned with dissipation enhancement by divergence-free transport noise for stochastic p(x)-Laplace equations on 𝕋d, d≥2. Spatially variable growth removes homogeneous scaling and produces two flux growth regimes. For periodic log-Hölder continuous exponents satisfying 2<p-≤ p+<∞, we first establish global variational well-posedness and the exact energy identity in the natural dual spaces. Under the additional condition p+<2d/(d−2) when d≥3, we then prove averaged algebraic decay with any prescribed coefficient, uniformly on bounded zero-mean L2 balls, by estimating the two flux powers separately. A deterministic lower bound yields strict averaged comparison with the unperturbed flow at prescribed positive times on shells with bounded gradient modular. We also prove optimality of the modular coercivity exponent p+ near zero. Finally, combining these estimates with intervalwise noise acceleration from the constant-exponent theory gives exponential decay in mean square and almost surely, with moments of the random prefactors bounded uniformly on the initial-data ball. For the exponential estimates, the noise depends on the prescribed rate and moment order; its intensity is fixed in time and its frequency range increases.
Related
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BibTeX
@misc{zhang2026enhanced,
title = {{Enhanced Dissipation by Transport Noise for Stochastic $p(x)$-Laplace Equations}},
author = {Jiangwei Zhang and Juntao Wu and Shang Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21242},
eprint = {2610.21242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21242},
}
RIS
TY - UNPB
TI - Enhanced Dissipation by Transport Noise for Stochastic p(x)-Laplace Equations
AU - Jiangwei Zhang
AU - Juntao Wu
AU - Shang Wu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.21242
UR - https://arxiv.org/abs/2610.21242
AB - We are concerned with dissipation enhancement by divergence-free transport noise for stochastic p(x)-Laplace equations on 𝕋^d, d≥2. Spatially variable growth removes homogeneous scaling and produces two flux growth regimes. For periodic log-Hölder continuous exponents satisfying 2<p^-≤ p^+<∞, we first establish global variational well-posedness and the exact energy identity in the natural dual spaces. Under the additional condition p^+<2d/(d−2) when d≥3, we then prove averaged algebraic decay with any prescribed coefficient, uniformly on bounded zero-mean L^2 balls, by estimating the two flux powers separately. A deterministic lower bound yields strict averaged comparison with the unperturbed flow at prescribed positive times on shells with bounded gradient modular. We also prove optimality of the modular coercivity exponent p^+ near zero. Finally, combining these estimates with intervalwise noise acceleration from the constant-exponent theory gives exponential decay in mean square and almost surely, with moments of the random prefactors bounded uniformly on the initial-data ball. For the exponential estimates, the noise depends on the prescribed rate and moment order; its intensity is fixed in time and its frequency range increases.
ER -