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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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