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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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