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Diego Córdoba

Publications and source records attributed to Diego Córdoba.

At least 19 recordsLinked to original sources
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↗
arXiv2026

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

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

math.AP↗
arXiv2025

Splash-squeeze singularities and analytic breakdown in ideal incompressible MHD

We construct splash-squeeze singularities for the free boundary ideal incompressible plasma-vacuum system, in which two arcs of the plasma boundary come together to form a smooth, glancing self-intersection. As the interface self-intersects, Sobolev norms remain bounded, although analyticity is necessarily lost. This contrasts classical splash singularities, in which solutions remain analytic up to the time of self-intersection. The narrowing gap bounded by these arcs is not occupied by plasma, as squeezing the plasma itself would cause blow-up in Sobolev norms. Instead, the gap represents the region outside the plasma, a vacuum carrying a nontrivial magnetic field. The plasma on either side pinches the field as the gap closes, and, in response, the field flattens to infinite order at the intersection point (and nowhere else), thereby forming an analytic singularity. This gives the first example of analytic breakdown without Sobolev blow-up in a locally well-posed free-boundary incompressible fluid system, and can be viewed as the first rigorous construction of a squeeze-type singularity, in which we study and quantify the precise behavior of an active, incompressible vector field as it is completely pinched off by a free-boundary in finite time. The proof combines a magnetically-aligned Lagrangian formulation of ideal MHD together with weighted elliptic estimates in the vacuum that remain uniform as the width of the gap tends to zero. Our framework may provide a starting point for the analysis of squeeze-type singularities in other incompressible fluid models.

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arXiv2025

Instantaneous continuous loss of regularity for the SQG equation

Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $θ_0$ such that $\| θ_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $θ$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$, and, for each $t\in [0,T]$, $ θ(\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ θ(\cdot ,t ) \not \in {H^β}$ for any $β> s/(1+ct)$. Moreover, $θ$ is unique among all solutions with initial condition $θ_0$ which belong to $C([0,T];H^{1+α})$ for any $α>0$ and is continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$.

math.AP↗
arXiv2025

Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-ε}\cap L^2$ force

We establish the existence of compactly supported solutions of the inviscid incompressible 2D Boussinesq equation with $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}\cap L^{2}$ force that develop a singularity in finite time. Importantly, the force preserves this regularity at the blow-up time. Moreover, the forces in the vorticity and density equations have compact support. The mechanism behind the blow-up is an accumulated hysteresis effect on the vorticity caused by an infinite chain of "degenerate" pendula and flickering density.

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.

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arXiv2024

Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation

We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of Córdoba and Mart\'ınez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis.

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arXiv2024

Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$

In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla|^α$ for any $α\in [0, α_0)$ ($α_0 = \frac{22-8\sqrt7}{9} > 0$). We construct solutions in $\mathbb{R}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L^1_t([0, T]) C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$, such that on the time interval $0 \le t < T$, the velocity $u$ is in the space $C^\infty\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity.

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arXiv2023

Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,α}\cap L^2$

We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in $\mathbb{R}^3\times [-T,0]$ such that the velocity is in the space $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,α}\cap L^2$ for times $t\in (-T,0)$ and is not $C^1$ at time 0.

math.AP↗
arXiv2023

Blow-up for the incompressible 3D-Euler equations with uniform $C^{1,\frac{1}{2}-ε}\cap L^2$ force

This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space $C^{3,\frac12}\cap L^2$ for the time interval $[0, T)$, where $T > 0$ is finite, subject to a uniform force in $C^{1,\frac12 -ε}\cap L^2$. Remarkably, our methodology results in a blow-up: as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity, all while preserving the solution's smoothness throughout, except at the origin. In the process of our blow-up construction, self-similar coordinates are not utilized and we are able to treat solutions beyond the $C^{1,\frac13+}$ threshold regularity of axy-symmetric solutions without swirl.

math.AP↗
arXiv2023

Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

We construct solutions of the 2D incompressible Euler equations in $\mathds{R}^2\times [0,\infty)$ such that initially the velocity is in the super-critical Sobolev space $H^β$ for $1<β<2$, but are not in $H^{β'}$ for $β'>1+\frac{(3-β)(β-1)}{2 - (β-1)^2}$ for $0<t<\infty$. These solutions are not in the Yudovich class, but they exists globally in time and they are unique in a determined family of classical solutions.

math.AP↗
arXiv2022

Non-existence and strong ill-posedness in $C^{k,β}$ for the generalized Surface Quasi-geostrophic equation

We consider solutions to the generalized Surface Quasi-geostrophic equation ($γ$-SQG) when the velocity is more singular than the active scalar function (i.e. $γ\in(0,1)$). In this paper we establish strong ill-posedness in $C^{k,β}$ ($k\geq 1$, $β\in(0,1]$ and $k+β>1+γ$) and we also construct solutions in $\mathbb{R}^2$ that initially are in $C^{k,β}\cap L^2$ but are not in $C^{k,β}$ for $t>0$. Furthermore these solutions stay in $H^{k+β+1-2δ}$ for some small $δ$ and an arbitrarily long time.

math.AP↗
arXiv2022

Stability of traveling waves for the Burgers-Hilbert equation

We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $δ$ from a global periodic traveling wave with small amplitude $ε$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $\frac{1}{εδ}$ with $0<δ\llε\ll1$ and on a time scale of $\fracε{δ^2}$ with $0<δ\llε^2\ll1$. Moreover, we show that the traveling wave exists for an amplitude $ε$ in the range $(0,ε^*)$ with $ε^*\sim 0.23$ and fails to exist for $ε>\frac{2}{e}$.

math.AP↗
arXiv2022

Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG

We construct solutions in $\mathbb{R}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\geq 2$) but that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$. Moreover, we prove strong ill-posedness in the critical space $H^{2}$.

math.AP↗
arXiv2021

Existence of gravity-capillary Crapper waves with concentrated vorticity

The aim of this paper is to prove the existence of gravity-capillary Crapper waves with the presence of vorticity. In particular, we consider a concentrated vorticity: point vortex and vortex patch. We show that for small gravity and small vorticity it is possible to demonstrate that the waves are overhanging.

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