arXiv subjects

Giacomo Maria Leccese

Publications and source records attributed to Giacomo Maria Leccese.

4 recordsLinked to original sources

arXiv2026

Existence and Blow-Up for Non-linear Fokker-Planck with Controlled Drift

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = Δu \end{equation*} for $b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$ non-autonomous fields, in the case \begin{equation*} b\in L^\infty_{\mathrm{loc}} \bigl((0,\infty),L^{p,\infty}(\mathbb{R}^d)\bigr), \qquad p>d\ge 2. \end{equation*} For the critical case, we study the difference between two ordered solutions whose conserved small mass absorbs the drift. Finitely many increments then reach an arbitrary solution.

math.AP↗

arXiv2026

Existence and Blow-Up for Nonlinear Fokker-Planck with Controlled Drift Divergence

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = Δu \end{equation*} for $b:\mathbb R\times{\mathbb R}^d\to{\mathbb R}^d$ non-autonomous fields, in the case \begin{equation*} (\text{div } b)_-\in L^\infty_{\mathrm{loc}} \big([0,\infty);L^{p,\infty}({\mathbb R}^d)\big), \qquad p>d. \end{equation*} The proof of global existence follows the parabolic symmetrization approach of [Bandle, J. Analyse Math. (1976)] and is based on the study of the function \begin{equation*} m(t,s)=\int_0^s u^*(t,r)\,dr. \end{equation*}

math.AP↗

arXiv2023

Existence and Blow-up for Non-autonomous Scalar Conservation Laws With Viscosity

We consider a question posed in [GSZZ22], namely the blow-up of the PDE $$u_t + (b(t,x) u^{1+k})_x = u_{xx}$$ when $b$ is uniformly bounded, Lipschitz and $k = 2$. We give a complete answer to the behavior of solutions when $b$ belongs to the Lorentz spaces $b \in L^{p,\infty}$, $p \in (2,\infty]$, or $b_x \in L^{p,\infty}$, $p \in (1,\infty]$.

math.AP↗

arXiv2022

On the Hausdorff Measure of $\R^n$ with the Euclidean Topology

In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb{R}^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb{R}^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

math.MG↗