Se invita a la comunidad de DMAT, el día de hoy jueves 30 de junio de 2022, a las 12:15 horas, a participar de una nueva sesión del Seminario de Investigación del DMAT, donde expondrá Argenis Mendez, académico del Instituto de Matemáticas de la Pontificia Universidad Católica de Valparaíso con la charla se denomina “On the fractional Zakharov-Kuznetsov equation”
Lugar: Sala de Seminarios DMAT, Casa Central, Valparaíso.
Resumen: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation ∂tu−∂x1 (−∆) α/2u+u∂x1 u = 0, 0 < α < 2, u(x,0) = u0(x), x = (x1, x2,…, xn) ∈ R n ,n ≥ 2, t ∈ R, (1) where (−∆) α/2 denotes the n−dimensional fractional Laplacian. In the case that α = 2, the equation is known as the Zakharov-Kuznetsov-(ZK) equation, Zakharov and Kuznetsov proposed it as a model to describe the propagation of ion-sound waves in magnetic fields in dimension n = 3. A property that enjoys the solutions of the ZK equation is Kato’s smoothing effect. Roughly speaking, the solution to the the initial value problem is, locally, one derivative smoother (in all directions) in comparison to the initial data. The goal of this talk is to show that despite the non-local character of the operator (−∆) α 2 , the solution of the equation (1) is locally smoother. It becomes α 2− smoother in all directions. As a byproduct, we show the applicability of this result in establishing the propagation of localized regularity of the solutions of (1) in a suitable Sobolev space.
