arXiv subjects
Publications and source records attributed to Felice Iandoli.
We prove local second-order regularity near the critical set for widely degenerate elliptic systems, and show that this degenerate region has zero Lebesgue measure provided the source term is non-vanishing.
We study higher regularity for weak solutions of the $p$-Laplace equation $-Δ_p u = f$ in a domain $Ω\subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and Hölder regularity conditions, we prove that the nonlinear quantity $|\nabla u|^{m-2}\nabla u$ belongs to $W^{m-1,q}_{loc}(Ω)$, and that $|\nabla u|^{m-2} D^2u$ belongs to $W^{m-2,q}_{loc}(Ω)$, for any $q\ge 2$. Furthermore, we obtain uniform $L^\infty$ bounds for the weighted $(m-1)$-th derivatives of $|\nabla u|^{m-2}\nabla u$ and the weighted $(m-2)$-th derivatives of $|\nabla u|^{m-2} D^2u$, providing quantitative control even near critical points of $\nabla u$.
We consider the equation $- Δ_{p} u = f(x)$ in $Ω,$ where $Δ_{p}$ is the $p$-Laplace operator. We provide $L^{\infty}$-type estimates for the second derivatives of solutions when $p$ approaches to $2$.
We prove the strong ill-posedness in the sense of Hadamard of the two-dimensional Boussinesq equations in $W^{1, \infty}(\mathbb{R}^2)$ without boundary, extending to the case of systems the method that Shikh Khalil \& Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large class of initial data with velocity and density of small $W^{1, \infty}(\mathbb{R}^2)$ size, for which the horizontal density gradient has a strong $L^{\infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi's decomposition of the Biot-Savart law, we apply the method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with velocity field uniformly bounded in $W^{1, \infty}(\mathbb{R}^2)$ provides a solution whose swirl component has a strong $W^{1, \infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the potential vorticity remains bounded at least for small times. Finally, the $W^{1,\infty}$-norm inflation of the swirl (and the $L^{\infty}$-norm inflation of the vorticity field) is quantified from below by an explicit lower bound which depends on time, the size of the data and it is valid for small times.
We improve the result by Feola and Iandoli [J. de Math. Pures et App., 157:243-281, 2022], showing that quasilinear Hamiltonian Schrödinger type equations are well posed on $H^s(\mathbb{T}^d)$ if $s>d/2+3$. We exploit the sharp paradifferential calculus on $\mathbb{T}^d$ introduced by Berti, Maspero and Murgante [J. Dynam. and Differential Equations, 33 (3): 1475-1513, 2021].
In this paper we provide a local well posedness result for a quasilinear beam-wave system of equations on a one-dimensional spatial domain under periodic and Dirichlet boundary conditions. This kind of systems provides a refined model for the time-evolution of suspension bridges, where the beam and wave equations describe respectively the longitudinal and torsional motion of the deck. The quasilinearity arises when one takes into account the nonlinear restoring action of deformable cables and hangers. To obtain the a priori estimates for the solutions of the linearized equation we build a modified energy by means of paradifferential changes of variables. Then we construct the solutions of the nonlinear problem by using a quasilinear iterative scheme à la Kato.
We prove exact controllability for quasi-linear Hamiltonian Schrödinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity assumptions, provided that the region of control satisfies the geometric control condition.
We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R 3 and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary, Strichartz estimates, all matching the R 3 case.
We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous result in \cite{Mietka}, generalising the considered class of equations and improving the regularity assumption on the initial data.
We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time $\varepsilon^{-2}$. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$, in the Klein-Gordon case, we prove that the solutions exist at least for a time of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order $O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and $d\geq4$, the general results in [17,24].
We prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on $\mathbb{T}^d$ for any $d\geq 1$. For any initial condition in the Sobolev space $H^s$, with $s$ large, we prove the existence and unicity of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map.
We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from $\varepsilon$-small initial conditions, remain bounded in $H^s$ for a time scale of order $O(\varepsilon^{-1-1/(d-1)+})$, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising from three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes.This is due to the irrationality of the torus which prevent to have "good separation" properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable \emph{high/low} frequencies analysis, which gives an \emph{a priori} estimate on the solutions.
We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $ε$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$ε$ of solutions is O($ε$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$ε$ = O($ε$ --6 --), much better than O($ε$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $Δ$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.
In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation. In the case of Hamiltonian equations we also put the system in Poincaré-Birkhoff normal forms. We apply this transformation to quasi-linear perturbations of the Schrödinger and Beam equations, obtaining a long time existence result without requiring any symmetry on the initial data. We also provide the local in time well-posedness for quasi-linear perturbations of the Benjamin-Ono equation.
In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\varepsilon$ sufficiently small, the lifespan of the solution is of order $\varepsilon^{-N}$ for any $N\in\mathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.
We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.
The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p-L^q$ estimates of the free Laplacian in the smaller regime $q\in[2,3)$. These estimates are implied by a recent result concerning the $L^p$ boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q\geq 3$.