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

We consider a transport-diffusion equation of the form $\partial_t\theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is agiven 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 theparameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$,$\lambda>1$. We introduce a novel nonlocal decomposition of theoperator $\mathcal{A}$ in terms of a weighted integral of the usualfractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus asmooth remainder term which corresponds to an $L^1$ kernel. For ageneral vector field $v$ (possibly non-divergence-free) we prove ageneralized $L^\infty$ maximum principle of the form $ \|\theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where theconstant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the sameinequality 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 ofHmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le2$ and removes the incompressibility assumption in the $L^\infty$case.

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