On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation

Hongjie Dong; Dong Li

doi:10.3934/dcds.2014.34.3437

Article Contents

2014, Volume 34, Issue 9: 3437-3454. Doi: 10.3934/dcds.2014.34.3437

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On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation

Hongjie Dong^1, and
Dong Li^2,

1.
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912
2.
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

Received: August 31, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

We consider a transport-diffusion equation of the form $\partial_t \theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\mathcal{A}$ represents log-modulated fractional dissipation: $\mathcal{A}=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator $\mathcal{A}$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $ \| \theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $\|\theta(t)\|_p$ with $1\le p \le \infty$. Under the additional assumption that $\theta_0\in L^2$, we show that $\|\theta(t)\|_p$ is uniformly bounded for $2\le p\le \infty$. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.

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Mathematics Subject Classification: Primary: 35Q35, 35B50; Secondary: 35B65.

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