arXiv subjects

Francisco Mengual

Publications and source records attributed to Francisco Mengual.

10 recordsLinked to original sources

arXiv2026

Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation

For any smooth bounded domain $Ω\subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(Ω)$ for all $p < \infty$, and magnetic fields $B^ε\in L_t^p C^{m}(Ω)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $ε\to 0$. The construction is based on the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations.

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arXiv2026

Sharp nonuniqueness for the forced 2D Navier-Stokes and dissipative SQG equations

We prove a sharp nonuniqueness result for the forced generalized SQG equation. First, this yields nonunique $\dot{H}^s$- energy solutions below the Miura-Ju class. In particular, this shows that the solutions constructed by Resnick and Marchand for the dissipative SQG equation are not necessarily unique. Second, this establishes nonuniqueness below the Ladyzhenskaya-Prodi-Serrin class for the 2D Navier-Stokes equation, as well as below the Constantin-Wu and Dong-Chen-Zhao-Liu classes for the dissipative SQG equation.

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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

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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arXiv2024

Dissipative Euler flows originating from circular vortex filaments

In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in $C([0,T],L^{2^-})$. The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.

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arXiv2023

Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data

For any $2<p<\infty$ we prove that there exists an initial velocity field $v^\circ\in L^2$ with vorticity $ω^\circ\in L^1\cap L^p$ for which there are infinitely many bounded admissible solutions $v\in C_tL^2$ to the 2D Euler equation. This shows sharpness of the weak-strong uniqueness principle, as well as sharpness of Yudovich's proof of uniqueness in the class of bounded admissible solutions. The initial data are truncated power-law vortices. The construction is based on finding a suitable self-similar subsolution and then applying the convex integration method. In addition, we extend it for $1<p<\infty$ and show that the energy dissipation rate of the subsolution vanishes at $t=0$ if and only if $p\geq 3/2$, which is the Onsager critical exponent in terms of $L^p$ control on vorticity in 2D.

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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

Dissipative Euler flows for vortex sheet initial data without distinguished sign

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a byproduct, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength.

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arXiv2020

H-principle for the 2D incompressible porous media equation with viscosity jump

In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_μ=0$) to the case of viscosity jump ($|A_μ|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi for $A_μ=0$, between the subsolution and the Lagrangian relaxed solution of Otto, holds for $|A_μ|<1$ too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh-Taylor nor vorticity at the interface.

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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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