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A family of dissipative active scalar equations with singular velocity and measure initial data. (English) Zbl 1268.35123

Summary: We consider a family of generalized active scalar equations, with fractional dissipation, whose velocity fields are more singular than Riesz transform. We prove global well-posedness results for small initial data belonging to Besov-Morrey spaces, which contain strongly singular functions and measures concentrated at points (Diracs) and on smooth curves. Self-similar solutions are obtained for initial data and coupling-velocity operator with correct homogeneities. We also show an asymptotic behavior result and obtain a class of asymptotically self-similar solutions.

MSC:

35R11 Fractional partial differential equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
35R06 PDEs with measure
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