APDE seminar@UPV/EHU: Low regularity well-posedness of nonlocal dispersive perturbations of Burger's equation

Abstract

We show that the Cauchy problem associated with a class of dispersive perturbations of Burgers’ equations is locally well-posed. This class includes the low-dispersion Benjamin–Ono equation, also known as the low-dispersion fractional KdV equation:

∂ₜu − Dₓ^α ∂ₓu = ∂ₓ²(u).

We prove local well-posedness in the Sobolev space H^s(K), where K = ℝ or 𝕋, for s > s_α = 1 − (3α)/4, when 2/3 ≤ α ≤ 1. Moreover, we obtain a priori estimates for solutions at a lower regularity threshold, namely s > s̃_α > 1/2 − α/4. The result also extends to other values of s_α when 0 < α < 2/3. As a consequence of these results, and using the Hamiltonian structure of the equation, we obtain global well-posedness in H^s(K) for s > s_α when α > 2/3, and in the energy space H^{α/2}(K) when α > 4/5.

In the first part of the talk, I introduce the equations, explain their connection to fluid mechanics, and review several existing mathematical results as well as open problems.

In the second part, I give an overview of the proof. The argument combines: an energy method for strongly non-resonant dispersive equations, introduced by Molinet and Vento, refined Strichartz estimates, and modified energy methods.

In addition, we use a full symmetrization of the modified energy both for the a priori estimates and for estimating the difference between two solutions. This symmetrization yields crucial cancellations in the associated symbols, which are essential to close the estimates.