arXiv:2209.06969v1 [math.AP] 14 Sep 2022 Long-time solvability for the 2D inviscid Boussinesq equations with critical regularity and dispersive effects V. Angulo-Castillo 1 National University of Colombia, Campus Orinoquia, Department of Mathematics, Kilómetro 9 vía a Caño Limón, Arauca, Colombia L. C. F. Ferreira 2* and L. Kosloff 3 University of Campinas, IMECC-Department of Mathematics Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil Abstract We are concerned with the long-time solvability for 2D inviscid Boussinesq equations. First we show the local solvability in Besov spaces uniformly with respect to a physical parameter κ associated with the strength of gravity. After, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ and regardless of the size of initial data. Our results provide a larger class of initial data as well as cover borderline regularity for the system. AMS MSC: 35Q35; 76B03; 76U05; 35A01; 46E35 Key: Boussinesq equations; Convection problem; Long-time solvability; Dispersive effects; Besov spaces; Critical regularity 1 Introduction We consider the two-dimensional (2D) inviscid Boussinesq system with dispersive forcing              ∂ t u +(u ·∇)u + ∇p = κθe 2 , ∂ t θ +(u ·∇)θ =0, div u =0, u(x, 0) = u 0 (x),θ(x, 0) = θ 0 (x), (1.1) where (x,t) ∈ R 2 × (0, ∞), u is the fluid velocity, θ denotes the temperature (or the density in geophysical flows), p stands for the pressure, and the constant κ is a parameter associated with the strength of gravity. * Corresponding author. E-mail adresses: vlcastillo@unal.edu.co (V. Angulo-Castillo), lcff@ime.unicamp.br (L.C.F. Ferreira), kosloff@ime.unicamp.br (L. Kosloff). V. Angulo-Castillo was supported by CNPq, Brazil. LCF Ferreira was supported by FAPESP and CNPq, Brazil. L. Kosloff was supported by FAPESP grant 2016/15985-0, Brazil. 1