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On a transport equation with nonlocal drift. (English) Zbl 1334.35254
Summary: In [Ann. Math. (2) 162, No. 3, 1377–1389 (2005; Zbl 1101.35052)], A. Córdoba et al. proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial _t \theta +u \; \partial _x \theta = 0, \quad u = H \theta , \] where \( H\) is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial _t \theta + u \; \partial _x \theta + \Lambda ^\gamma \theta = 0, \quad u = H \theta, \] where \( \Lambda = (-\Delta )^{1/2}\), and \( 1/2 \leq \gamma <1\). Our results also apply to the model with velocity field \( u = \Lambda ^s H \theta \), where \( s \in (-1,1)\). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in \( C^{(s+1)/2}\), for all positive time.

MSC:
35Q35 PDEs in connection with fluid mechanics
35L60 First-order nonlinear hyperbolic equations
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