2025/02/10 by Diego Córdoba, Córdoba, Diego, José Lucas-Manchón +3 · 1 citation
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2502.06357
openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The general surface quasi-geostrophic equation is the scalar transport equation defined by (∂ θ)/(∂ t)+vγ1 (∂ θ)/(∂ x1)+vγ2 (∂ θ)/(∂ x2) =0 , where the velocity comes defined by vγ=∇⊥ ψγ=(∂2 ψγ,-∂1 ψγ), ψγ=-Λ-1+γ θ, and θ(⋅,0)=θ0(⋅) is the initial condition. We consider the parameter γ∈ (-1,1) and the non-local operator Λα=(-Δ)^\fracα2 is defined on the Fourier side by \widehatΛα f(ξ)=|ξ|α \widehatf(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 regime Hβ 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.