José Lucas-Manchón | Searcharxiv

arXiv2026

Blow-up at finite time for the generalized SQG equations in the Sobolev well-posedness regime

We prove finite-time singularity formation for the forced generalized surface quasi-geostrophic equation in the singular velocity regime $γ\in(0,1)$, where $γ=0$ corresponds to SQG. For every such $γ$, we construct a smooth, compactly supported initial datum and a time-dependent force $F$ for which the corresponding solution $θ$ is classical on $[0,1)$ with finite energy for all times, but loses Sobolev regularity at $t=1$. More precisely, there exists \[ κ_0>2+γ+\frac{γ^2(1-γ)}{25(4+γ)}, \] such that the force satisfies $F\in L^1([0,1];H^κ(\mathbb{R}^2))$ for every $κ\in[2+γ,κ_0]$, whereas \[ \lim_{T\nearrow1}\int_0^T\|θ(\cdot,t)\|_{H^κ}\,dt=\infty \] for every exponent in the same interval. At the same time, the solution remains uniformly bounded in $H^{κ_1}$ throughout its lifespan for any \[κ_1\in\left[0,2+γ-\frac{γ(1-γ)}{2(4+γ)}\right].\] Then, the singularity occurs within the Sobolev well-posedness regime and cannot be attributed to insufficient regularity of the force or the initial conditions. To the best of our knowledge, this is the first finite-time blow-up result for classical finite-energy solutions of the generalized SQG equations in a well-posedness regime.

math.AP↗

arXiv2025

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 θ}{\partial t}+v^γ_1 \frac{\partial θ}{\partial x_1}+v^γ_2 \frac{\partial θ}{\partial x_2} =0 , \end{equation*} where the velocity comes defined by \begin{equation*} v^γ=\nabla^{\perp} ψ_γ=\left(\partial_{2} ψ_γ,-\partial_{1} ψ_γ\right), \quad ψ_γ=-Λ^{-1+γ} θ, \end{equation*} and $θ(\cdot,0)=θ_0(\cdot)$ is the initial condition. We consider the parameter $γ\in (-1,1)$ and the non-local operator $Λ^α=(-Δ)^{\fracα{2}}$ is defined on the Fourier side by $\widehat{Λ^α f}(ξ)=|ξ|^α \widehat{f}(ξ)$. The PDE is well-posed in the Sobolev spaces $H^s$ with $s>2+γ$. In this paper we prove strong ill-posedness in the super-critical regime $H^β$ with $β\in [1,2+γ)\cap(\frac{3}{2}+γ,2+γ)$. 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 $H^β$ and will leave it instantaneously. Nevertheless, this solution exists for a long time and remains the only classical solution in a high regularity class.

math.AP↗