Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG

The general surface quasi-geostrophic equation is the scalar transport equation defined by \begin{equation*} \frac{\partial \theta}{\partial t}+v^\gamma_1 \frac{\partial \theta}{\partial x_1}+v^\gamma_2 \frac{\partial \theta}{\partial x_2} =0 , \end{equation*} where the velocity comes defined by \begin{equation*} v^\gamma=\nabla^{\perp} \psi_\gamma=\left(\partial_{2} \psi_\gamma,-\partial_{1} \psi_\gamma \right), \quad \psi_\gamma=-\Lambda^{-1+\gamma} \theta, \end{equation*} and $\theta(\cdot,0)=\theta_0(\cdot)$ is the initial condition. We consider the parameter $\gamma \in (-1,1)$ and the non-local operator $\Lambda^{\alpha}=(-\Delta)^{\frac{\alpha}{2}}$ is defined on the Fourier side by $\widehat{\Lambda^{\alpha} f}(\xi)=|\xi|^{\alpha} \widehat{f}(\xi)$. The PDE is well-posed in the Sobolev spaces $H^s$ with $s>2+\gamma$. In this paper we prove strong ill-posedness in the super-critical regime $H^\beta$ with $\beta\in [1,2+\gamma)\cap(\frac{3}{2}+\gamma,2+\gamma)$. To do this, we will derive an approximated PDE solvable by some family of functions that we will call pseudosolutions and that will allow us to control the norms of the real solutions. Using this result and a gluing argument we also prove non-existence of solutions in the same Sobolev spaces. Since the pseudosolution will control the real one, we can build a solution that will be initially in $H^{\beta}$ and will leave it instantaneously. Nevertheless, this solution exists for a long time and remains the only classical solution in a high regularity class.