Vorticity blow-up for the 2D incompressible non-homogeneous Euler equations with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}$ force

Abstract

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In order to achieve this, we adapt the Boussinesq blow-up we set up in arXiv:2505.20988 to the non-homogeneous Euler setting. Furthermore, we bring the potential existence of two different types of singularities of the forced system to light.