Global existence of weak solutions to dissipative transport equations with nonlocal velocity

We consider 1D dissipative transport equations with nonlocal velocity field: θt + uθx + δuxθ + Λγθ = 0, u = N (θ), where N is a nonlocal operator given by a Fourier multiplier. We especially consider two types of nonlocal operators: (1) N = H, the Hilbert transform, (2) N = (1 − ∂xx)−α. In this paper, we show several global existence of weak solutions depending on the range of γ, δ and α. When 0 <γ< 1, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when γ ∈ (0, 2).HB was supported by NRF-2015R1D1A1A01058892. RGB is funded by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘Investissements d’Avenir’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Both OL and RGB were partially supported by the Grant MTM2014-59488-P from the former Ministerio de Economía y Competitividad (MINECO, Spain). OL was partially supported by the Marie-Curie Grant, acronym: TRANSIC, from the FP7-IEF program