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On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion. (English) Zbl 1347.35076
Summary: We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $$H^k(w_{\lambda, \kappa}) \cap L^\infty$$, where $$k = \max (0, 3 / 2 - \alpha)$$ and $$w_{\lambda, \kappa}$$ is a given family of Muckenhoupt weights, we prove a global existence result in the subcritical case $$\alpha \in(1, 2)$$. We also prove a local existence theorem for large data in $$H^2(w_{\lambda, \kappa}) \cap L^\infty$$ in the supercritical case $$\alpha \in(0, 1)$$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.

##### MSC:
 35F10 Initial value problems for linear first-order PDEs 35R09 Integral partial differential equations
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