arXiv subjects

Miguel M. G. Pascual-Caballo

Publications and source records attributed to Miguel M. G. Pascual-Caballo.

3 recordsLinked to original sources

arXiv2026

On the instability of De Gregorio excited states

We prove that all excited states $-\sin(nx)$, $n\geq2$, of the De Gregorio equation on the torus are nonlinearly unstable in the sense of Hadamard. More precisely, perturbations arbitrarily small in $H^2$ leave a fixed $H^1$ neighborhood of the state. For every $n\geq2$, the linearized operator has a real eigenvalue $λ_n\in(0.56,0.6)$. To prove this, we transform the eigenvalue problem into a Fuchsian equation on the complex disk and obtain a matching condition for the eigenvalue. This condition is verified rigorously using interval arithmetic. The nonlinear instability follows from a bootstrap argument in the spirit of Guo, Hallstrom, and Spirn.

math.AP↗

arXiv2026

Stationary homogeneous solutions for the inviscid SQG equation

In this paper, we prove the existence of nontrivial stationary homogeneous solutions with infinite energy (unbounded at infinity) for the SQG equation. Our analysis also covers the existence of stationary solutions for the generalized De Gregorio equation $\partial_t w+αu\,\partial_x w= w\,\partial_x u,\ \partial_x u=Hw$ with $α>\tfrac{1}{2}$, where $H$ denotes the Hilbert transform.

math.AP↗

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.

math.AP↗