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On a generalized maximum principle for a transport-diffusion model with $$\log$$-modulated fractional dissipation. (English) Zbl 1304.35538
Summary: 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 $$\nu\geq 0$$, $$\beta\geq 0$$, $$0\leq \gamma \leq 2$$, $$\lambda>1$$. We introduce a novel nonlocal decomposition of the operator $$\mathcal{A}$$ in terms of a weighted integral of the usual fractional operators $$|\nabla|^{s}$$, $$0\leq s \leq \gamma$$ 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 $$\| \theta(t)\|_\infty \leq e^{Ct} \| \theta_0 \|_{\infty}$$ 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 $$1\leq p \leq \infty$$. Under the additional assumption that $$\theta_0\in L^2$$, we show that $$\|\theta(t)\|_p$$ is uniformly bounded for $$2\leq p\leq \infty$$. At the cost of a possible exponential factor, this extends a recent result of T. Hmidi [Anal. PDE 4, No. 2, 247–284 (2011; Zbl 1264.35173)].

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B50 Maximum principles in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35R11 Fractional partial differential equations
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