arXiv subjects

Nikolaos Skouloudis

Publications and source records attributed to Nikolaos Skouloudis.

3 recordsLinked to original sources

arXiv2026

Strichartz Estimates and Small-Mass Global Well-Posedness for the Periodic Quintic NLS in 1D

We consider the periodic quintic nonlinear Schrödinger and prove small-mass global well-posedness in $H^s(\mathbb{T})$ for $s>0$. The proof relies on a new derivative-loss-free $L^6_{t,x}$ Strichartz estimate which is established using the high-low method, an asymmetric superlevel set estimate and a new refined broad-narrow argument. Although our $L^6_{t,x}$ Strichartz estimate is not sharp, being valid on slightly shorter time scales than the optimal logarithmic scale, combining it with the $I$-method enables the extension of local solutions to arbitrary times.

math.AP↗

arXiv2025

Global Well-posedness for the periodic fractional cubic NLS in 1D

We consider the defocusing periodic fractional nonlinear Schrödinger equation $$ i \partial_t u +\left(-Δ\right)^αu=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< α< 1$ and the operator $(-Δ)^α$ is the fractional Laplacian with symbol $\lvert k \rvert ^{2α}$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\geq \frac{1-α}{2}$ and we conjecture this threshold to be sharp as it corresponds to the pseudo-Galilean symmetry exponent. Our proof uses the $I$-method to control the $H^s(\mathbb{T})$-norm of solutions with infinite energy initial data. A key component of our approach is a set of improved long-time bilinear Strichartz estimates on the rescaled torus, which allow us to exploit the subcritical nature of the equation.

math.AP↗

arXiv2022

Mathematical certification of motion planning on uncertain terrain with limited perception: a case study

We design a controller for an agent whose mission is to reach a stationary target while avoiding a family of obstacles which are not known a-priori. The agent moves in the two dimensional plane with non-trivial double integrator dynamics and receives only local information from its surroundings. Under mild assumptions on the family of obstacles (smoothness, sufficient distance from each other, bounded curvature, etc), we prove that our control algorithm yields guaranteed obstacle avoidance and convergence to the target.

eess.SY↗