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On the global well-posedness of one-dimensional fluid models with nonlocal velocity. (English) Zbl 1435.35317
A one-dimensional fluid flow model \(w_t+(1-\partial_{xx})^{-\beta}w w_x+\delta ((1-\partial_{xx})^{-\beta}w)_x +\nu (-\partial_{xx})^\gamma w=0\) includes as particular examples the (fractional) Burgers equation and the Camassa-Holm equations. The author studies local- and global-in-time solvability of this equation as well as blow-up criteria. Similar results are also obtained for a related equation involving the Hilbert transform as is in the dissipative surface quasigeostrophic model.
MSC:
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B60 Continuation and prolongation of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B44 Blow-up in context of PDEs
44A15 Special integral transforms (Legendre, Hilbert, etc.)
35Q86 PDEs in connection with geophysics
76U60 Geophysical flows
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