The general surface quasi-geostrophic equation is the scalar transport equation defined by    ∂θ ∂t + v γ 1 ∂θ ∂x1 + v γ 2 ∂θ ∂x2 = 0, vγ = ∇⊥ψγ = (∂2ψγ ,−∂1ψγ) , ψγ = −Λ−1+γθ, θ(·, 0) = θ0(·), for γ ∈ (−1, 1), where the non-local operator Λα = (−∆) α 2 is defined on the Fourier side by Λ̂αf(ξ) = |ξ|αf̂(ξ). The PDE is well-posed in the Sobolev spaces Hs with s > 2 + γ. In this paper we prove strong ill-posedness in the super-critical regimeHβ with β ∈ [1, 2+γ)∩( 3 2 +γ, 2+γ). 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 Hβ and will leave it instantaneously. Nevertheless, this solution exists for a long time and remains the only classical solution in a high regularity class.

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