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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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