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Ángel Castro

Publications and source records attributed to Ángel Castro.

18 recordsLinked to original sources
arXiv2026

Time-periodic non-radial solutions near monotone vortices in linearized 2D Euler

We study the linearized 2D Euler equations around radial vortex profiles. Previous works have shown that the strict monotonicity of the vorticity profile leads to axisymmetrization and inviscid damping of non-radial perturbations. Focusing on a representative strictly decreasing radial vortex profile, we construct arbitrarily close (in low Hölder norms $C^α$, with $0<α<1$) radial profiles that are merely non-increasing and for which non-radial, time-periodic solutions to the linearized 2D Euler equations exist. This shows that axisymmetrization and inviscid damping are not robust under small, low-regularity perturbations of the background profile that violate strict monotonicity.

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arXiv2026

Smooth nonradial stationary solutions to SQG via the half-Yamabe equation

We prove the existence of infinitely many smooth nonradial stationary solutions to the surface quasi-geostrophic (SQG) equation with finite kinetic energy. Our construction is based on a family of nonradial sign-changing solutions to the two-dimensional half-Yamabe equation, obtained via a Lyapunov--Schmidt reduction and concentrated at the vertices of a regular polygon. As the number of vertices tends to infinity, the associated stationary SQG solutions converge to a radial stationary profile centered at the origin, together with a lower-order vortex sheet correction.

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arXiv2026

Linear instability of a Burgers--Hilbert traveling wave

We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $ω$ and large wave speed $c$. For $ω= 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states.

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arXiv2026

Global-in-time estimates for the 2D one-phase Muskat problem with contact points

In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. A similar contact scenario was recently analyzed for more regular viscous flows, modeled by the Stokes [GuoTice2018] and Navier-Stokes [GuoTice2024] equations. Here, we adopt the same framework but use the more singular Darcy's law for modeling the flow. We prove global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted $L^2$-based Sobolev spaces and without imposing restrictions on the contact angles.

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arXiv2025

Three-phase Muskat problem: uniform lifespan with respect to the width of the strip between interfaces

We consider the three-phase Muskat problem with different densities and the same viscosities. The lifespan of the solutions with respect to the width of the strip between interfaces is studied. Indeed, the interfaces are parameterized by the graph of two functions $f(x,t)$ and $g(x,t)$ and we impose that $||f(\cdot,0)-g(\cdot,0)||_{L^\infty}\leq Cσ$ and $\inf_x |f(\cdot,0)-g(\cdot,0)|\geq cσ.$ It is shown, under stronger assumption on $f(x,0)$ and $g(x,0)$, local existence independent of the parameter $σ$ (with $σ$ small enough). In order to prove such a result, we need to work in analytic spaces.

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arXiv2025

Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

We prove non-uniqueness of weak solutions to the forced $α$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < α+ \frac{2}{p}$, covering the 2D Euler equation ($α= 0$), the Surface Quasi-Geostrophic equation ($α= 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $α$-SQG equation that are neither rotating nor traveling.

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arXiv2024

Entropy solutions to macroscopic IPM

We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarse grained solutions) in the setting of the unstable Muskat problem. We show that both, imposing this criterion on the level of convex integration subsolutions, and the strategy of Otto based on a relaxation via minimizing movements, lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface.

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arXiv2024

A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.

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arXiv2023

Time periodic solutions for the 2D Euler equation near Taylor-Couette flow

In this paper we consider the incompressible 2D Euler equation in an annular domain with non-penetration boundary condition. In this setting, we prove the existence of a family of non-trivially smooth time-periodic solutions at an arbitrarily small distance from the stationary Taylor-Couette flow in $H^s$, with $s<3/2$, at the vorticity level.

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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 $ε>\frac{2}{e}$.

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arXiv2021

Traveling waves near Couette flow for the 2D Euler equation

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $H^s$, with $s<3/2$, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see arXiv:1306.5028), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $H^{\frac{3}{2}+}$ neighborhoods of Couette flow (see arXiv:1004.5149).

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arXiv2021

Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.

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arXiv2021

Mixing solutions for the Muskat problem

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

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arXiv2021

Localized mixing zone for Muskat bubbles and turned interfaces

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.

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arXiv2020

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $ε$ from a radial stationary solution $θ=|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}ε$ to a time scale of $\frac{1}{ε^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.

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arXiv2019

Singularity formation for the fractional Euler-Alignment system in 1D

We study the formation of singularities for the Euler-Alignment system with influence function $ψ=\frac{k_α}{|x|^α}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $α$ in the range $0<α<2$ in both the real line and the periodic case.

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arXiv2018

Degraded mixing solutions for the Muskat problem

We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme presented in [DS10, Sze12] applied to the subsolution in [CCF16]. More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance $\mathbf{P}$ (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations.

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