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].

Finite time singularities for a class of generalized surface quasi-geostrophic equations

“…The proof of Theorem 1.2 is completely similar to that of a corresponding result in [15]. Therefore we devote the rest of this paper to the proof of Theorem 1.1.…”

Global smoothness for a 1D supercritical transport model with nonlocal velocity

“…Then, by Theorem 3.1, θ ∈ C ∞ (T×(T * , T max ]) and, in particular, θ(T max ) ∈ H 3 2 (T). So, by using standard arguments and the local-existence of [11], we can extend θ in the class (3.1) to a time-interval [0, T 2 ) with T max < T 2 , which is a contradiction. It follows that T max = ∞ and θ is a global H…”

On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

“…In this article, we study equation (T α ) with subcritical or supercritical diffusion. We extended the class of initial data of some recent existence results (essentially some of those in [17], and [1]) to the weighted setting. By considering data in weighted Lebesgue or Sobolev spaces with a weight which has the property to be sufficently decaying at infinity say like |x| −λ , with λ > 0, one is allowed to consider the problem (T α ) with an initial data that behaves for instance like θ 0 (x) ∼ |x| −1/2 at infinity, and therefore does not necessarily belong to L 2 .…”