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Global existence of weak solutions to dissipative transport equations with nonlocal velocity. (English) Zbl 1402.35077
The authors consider non-local transport equations. Non-locality concerns velocity, which is defined by a non-local operator of the scalar unknown field \(\theta\). The considered equations are spatially one-dimensional, linear with respect to the unknown scalar, velocity is obtained either as the Hilbert transform of the scalar \(\theta\) or expressed as \((1-\partial_{xx})^{-\alpha}\theta\). The authors prove global existence of weak solutions for some cases, for fewer cases also uniqueness and/or regularity are established.

MSC:
35D30 Weak solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
35Q31 Euler equations
35R11 Fractional partial differential equations
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