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

Abstract

We consider a transport-diffusion equation of the form ∂tΦ + v·δ Φ + vAΦ = 0, where v is a given time-dependent vector field on Rd. The operator A represents log-modulated fractional dissipation: A = /δ/γ/logβ(λ /δ/) and the parameters v ≥ 0, β 0, 0 ≤ γ 2, λ > 1. We introduce a novel nonlocal decomposition of the operator A in terms of a weighted integral of the usual fractional operators /delta;/s, 0 ≤ s ≤ γ plus a smooth remainder term which corresponds to an L1 kernel. For a general vector field v (possibly non-divergence-free) we prove a generalized L∞ maximum principle of the form ∥ 0(t) ∥ ∞ ≤ eCt∥ Φ0∥ ∞ where the constant C = C(v, β, γ) > Φ. In the case div(u) = 0 the same inequality holds for ∥0 (t) ∥p with 1≤ p≤ ∞. Under the additional assumption that Φ0 2 L2, we show that ∥ Φ (t) ∥p is uniformly bounded for 2 ≤ p ≤ ∞. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime d ≥ 1, 0 ≤ γ ≤ 2 and removes the incompressibility assumption in the L∞ case.