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

Abstract

We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C_t^0{C}_x^{1,\sqrt{\frac{4}{3}}-1-\epsilon }\cap C_t^0{L}_x^2$ 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.

49 pages