Well-posedness on a large Besov-weak-Morrey space and novel numerical insights for the inviscid SQG equation

Abreu E.; Pérez J.; Quispe-Cuba N.; Valencia-Guevara J.C. · Nonlinear Analysis: Real World Applications (2026) · DOI: 10.1016/j.nonrwa.2026.104645

Resumen

In this paper, we develop a mathematical framework for generating novel estimates of Bernstein type, commutator inequalities, and Logarithmic type inequality in the modified Besov-weak-Morrey spaces to allow us to get the local in time well-posedness for the inviscid Surface Quasi-Geostrophic (SQG) equation via the method of successive approximations. The purpose of this paper is two-fold. First, we prove local-in-time well-posedness theory for the inviscid SQG equation in a large class of Besov-weak-Morrey type, improving recent results as introduced in [22], covering now sharp results for critical and sub-critical regularity cases. In addition, we also present the criterion of blow-up for the SQG equation in this new framework. We also obtain a key estimate, which is new in our framework, as far as we know, which allows us to recover several well-known key results for the classic Besov spaces, such as Sobolev-like immersion and Log-Sobolev-type inequalities. Second, we show and present a qualitative discussion, based on numerical insights for the dynamics of the solution for the inviscid SQG. The novelty is a possible self-organization, as time evolves, of non-singular solutions into pairs of positive-negative vorticities, which continue evolving following a pair-trajectory. The novel approach has been developed in a non-periodic setting, leading solutions to travel beyond possible boundary periodic conditions, which allow us to capture the formation of strong fronts, also called frontogenesis, preserving L 1 and L 2 norms, thereby adding new improvements on similarities with the 3D incompressible Euler equations, as observed by Constantin, Majda and Tabak [12], but now on unbounded domains.

Instituciones: Universidad Nacional de San Agustín de Arequipa; Universidad Nacional Amazónica de Madre de Dios