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

Abstract

We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space H k (w λ,κ )∩L ∞ , where k = max(0, 3/2− α) and w λ,κ is a given family of Muckenhoupt weights, we prove a global existence result in the subcritical case α ∈ (1, 2). We also prove a local existence theorem for large data in H 2 (w λ,κ ) ∩ L ∞ in the supercritical case α ∈ (0, 1). The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.

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

“…These weights also satisfy the following properties. For the proofs, see [18] (for the first two points), and [17] (for the last two points).…”

“…In the two next subsections we shall prove a global existence theorem and a local existence theorem, for respectively the subcritical case and the supercritical case), we shall just focus on the a priori estimates since the construction by compactness is classical (see [17]).…”

“…A priori estimate. Since the case δ = 0 has been already done in [17], one observes that it suffices to estimate the L 2 (wdx) part T 1 = δ w β,k θ 2 Λθ dx, and the weighted homogeneous Sobolev part, that is…”

On the dynamics of the roots of polynomials under differentiation

“…Last but not least, it is worth mentioning that equation (1) shares many similarity with 1D transport equations with a nonlocal velocity given by the Hilbert transform (see e.g. [18,20,33,23,30,29,7].…”

On the local and global existence of solutions to 1d transport equations with nonlocal velocity

“…where the velocity u is recovered from the vorticity ω through u = ∇ ⊥ (−∆) −1 ω or equivalently u(ξ) = iξ ⊥ |ξ| 2 ω(ξ). Other nonlocal and quadratically nonlinear equations, such as the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magneto-geostrophic equation in multi-dimensions, have been studied intensively as one can see in [1,2,5,6,7,8,9,12,15,16,18,19,21] and references therein.…”