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

Abstract

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).