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On the well-posedness of various one-dimensional model equations for fluid motion. (English) Zbl 06756450
Summary: We consider 1D equations with nonlocal velocity of the form $w_t + u w_x + \delta u_x w = - \nu \varLambda^\gamma w$ where the nonlocal velocity $$u$$ is given by (1) $$u = (1 - \partial_{x x})^{- \beta} w$$, $$\beta > 0$$ or (2) $$u = \mathcal{H} w$$ ($$\mathcal{H}$$ is the Hilbert transform). In this paper, we address several local well-posedness results with blow-up criteria for smooth initial data. We then establish the global well-posedness by using the blow-up criteria.

##### MSC:
 76D99 Incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics
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