GLOBAL EXISTENCE OF WEAK SOLUTIONS TO DISSIPATIVE TRANSPORT EQUATIONS WITH NONLOCAL VELOCITY
Résumé
We consider 1D dissipative transport equations with nonlocal velocity field: θt + uθx + δuxθ + Λ γ θ = 0, u = N (θ), where N is a nonlocal operator given by a Fourier multiplier. Especially we 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 and δ. 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 = 1.
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