Well-posedness for a transport equation with nonlocal velocity

Abstract

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Córdoba, Córdoba and Fontelos [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377–1389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in H max { 3 / 2 − γ , 0 } . While in the supercritical case, the problem is locally well-posed with initial data in H 3 / 2 − γ , and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377–1389].