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Ignacio Guerra

Publications and source records attributed to Ignacio Guerra.

8 recordsLinked to original sources
arXiv2025

Nearly parallel helical vortex filaments in the three dimensional Euler equations

Klein, Majda, and Damodaran have previously developed a formalized asymptotic motion law describing the evolution of nearly parallel vortex filaments within the framework of the three-dimensional Euler equations for incompressible fluids. In this study, we rigorously justify this model for two configurations: the central configuration consisting of regular polygons of $N$ helical-filaments rotating with constant speed, and the central configurations of $N+1$ vortex filaments, where an $N$-polygonal central configuration surrounds a central straight filament.

math.AP↗
arXiv2014

Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane

As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis on the plane has global-in-time regular nonnegative solutions with total mass below the critical value $8π$. Solutions with mass above $8π$ blow up in a finite time. We show that the case of the parabolic-parabolic Keller-Segel is different: each mass may lead to a global-in-time-solution, even if the initial data is a finite signed measure. These solutions need not be unique, even if we limit ourselves to nonnegative solutions.

math.AP↗
arXiv2008

Perturbing singular solutions of the Gelfand problem

he equation $-Δu = λe^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $λ= 2(N-2)$. If $N\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,λ)$ close to $(U,2(N-2))$. In dimension $N\ge 11$ it corresponds to the extremal solution -- the one associated to the largest $λ$ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when $N\ge 10$, the extremal solution remains bounded in many cases.

math.AP↗
arXiv2008

Stable solutions for the bilaplacian with exponential nonlinearity

Let $λ^*>0$ denote the largest possible value of $λ$ such that \begin{align*} \left\{\begin{aligned} Δ^2 u & = \la e^u && \text{in $B $} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a solution, where $B$ is the unit ball in $\R^N$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^*$ this problem possesses a unique {\em weak} solution $u^*$. We prove that $u^*$ is smooth if $N\le 12$ and singular when $N\ge 13$, in which case $ u^*(r) = - 4 \log r + \log (8(N-2)(N-4) / λ^*) + o(1)$ as $r\to 0$. We also consider the problem with general constant Dirichlet boundary conditions.

math.AP↗
arXiv2006

On Regions of Existence and Nonexistence of solutions for a System of $p$-$q$-Laplacians

We give a new region of existence of solutions to the superhomogeneous Dirichlet problem $$ \quad \begin{array}{l} -Δ_{p} u= v^δ\quad v>0\quad {in}\quad B,\cr -Δ_{q} v = u^μ\quad u>0\quad {in}\quad B, \cr u=v=0 \quad {on}\quad \partial B, \end{array}\leqno{(S_R)} $$ where $B$ is the ball of radius $R>0$ centered at the origin in $\RR^N.$ Here $δ, μ>0$ and $ Δ_{m} u={\rm div}(|\nabla u|^{m-2}\nabla u) $ is the $m-$Laplacian operator for $m>1$.

math.AP↗
arXiv2006

Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -Δu &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad Ω, -Δv &=& u^{p},\:\:\:\quad u>0\quad {in}\quad Ω, u&=&v\:\:=\:\:0 \quad {on}\quad \partial Ω, \end{eqnarray*} where $Ω$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial Ω.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.

math.AP↗
arXiv2006

Solutions of an elliptic system with a nearly critical exponent

Consider the problem \begin{eqnarray*} -Δu_\e &=& v_\e^p \quad v_\e>0\quad {in}\quad Ω, -Δv_\e &=& u_\e^{q_\e}\quad u_\e>0\quad {in}\quad Ω, u_\e&=&v_\e\:\:=\:\:0 \quad {on}\quad \partial Ω, \end{eqnarray*} where $Ω$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial Ω.$ Here $p,q_\e>0,$ and \begin{equation*} ε:=\frac{N}{p+1}+\frac{N}{q_\e+1}-(N-2). \end{equation*} This problem has positive solutions for $\e>0$ (with $pq_\e>1$) and no non-trivial solution for $\e\leq 0.$ We study the asymptotic behaviour of \emph{least energy} solutions as $\e\to 0^+.$ These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given.

math.AP↗