About

Gustavo Ponce is a professor at the Department of Mathematics at the University of California, Santa Barbara. His research interests lie at the intersection of mathematical physics, harmonic analysis, and nonlinear partial differential equations. His work focuses on describing the qualitative behavior of solutions to model physical situations by developing techniques derived from harmonic analysis.

Research topics

• Computer Science
• Chemistry
• Mathematical analysis
• Mathematics
• Waste management
• Organic chemistry
• Environmental science
• Engineering
• Applied mathematics
• Pulp and paper industry
• Process engineering

Selected publications

• On decay and regularly of solutions of the Benjamin-Ono equation

  ArXiv.org · 2025-09-08

  preprintOpen accessSenior author

  We study persistence properties of solutions of the Benjamin-Ono equation in weighted Sobolev spaces. Roughly, we show that for $β<7/2$, the solution $u(x,t)$ of the BO remains in the space $L^2(|x|^{2β} dx)$ if and only if its data $u(x,0)$ belongs to this space and it is regular enough, i.e. $u_0\in H^β(\mathbb R)$.

  PublisherOA PDFDOI

• On the fractional Schrödinger equation with variable coefficients

  Journal of Functional Analysis · 2025-10-08

  articleOpen access
  PublisherDOI

• On the fractional Schödinger equation with variable coefficients

  arXiv (Cornell University) · 2024-11-02

  preprintOpen access

  We study the initial value problem (IVP) associated to the semi-linear fractional Schödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

  PublisherOA PDFDOI

• On special properties of solutions to Camassa-Holm equation and related models

  arXiv (Cornell University) · 2024-05-14

  preprintOpen accessSenior author

  We study unique continuation properties of solutions to the b-family of equations. This includes the Camassa-Holm and the Degasperi-Procesi models. We prove that for both, the initial value problem and the periodic boundary value problem, the unique continuation results found in \cite{LiPo} are optimal. More precisely, the result established there for the constant $c_0=0$ fails for any constant $c_0\neq 0$.

  PublisherOA PDFDOI

• Comparative Study of Active Power Filters with Least Mean Square algorithm and conventional strategy in discrete-time domain

  2024-11-28

  article

  This paper compares the performance of an active power filter implemented using the Least Mean Square (LMS) algorithm, with that of the traditional filter. An active power filter is designed to improve power quality in an electrical system. This entails reducing harmonics and reactive power, as well as eliminating unbalance. Since the LMS algorithm is among the fastest and simplest filters, which tends to make it easy to implement, the performance is evaluated using the Fast Fourier Transfer. Performance indicators for this application considered are harmonic removal and the stability period. The simulations are implemented in Matlab software, and the results allow a comparison of the LMS-based APF and reflect when the method is better and when it is necessary to have to stick to the traditional one. The LMS algorithm demonstrates results due to its ability to adapt to changes in real-time operation even under varying load operation frequency. This paper presents the mathematical model of two strategies of power compensation in continuous and discrete-time domain.

  PublisherDOI

• On decay and asymptotic properties of solutions to the Intermediate Long Wave equation

  arXiv (Cornell University) · 2024-06-27

  preprintOpen accessSenior author

  We consider solutions to the initial value problem associated to the intermediate long wave (ILW) equation. We establish persistence properties of the solution flow in weighted Sobolev spaces, and show that they are sharp. We also deal with the long time dynamics of large solutions to the ILW equation. Using virial techniques, we describe regions of space where the energy of the solution must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.

  PublisherOA PDFDOI

• On special properties of solutions to the Benjamin-Bona-Mahony equation

  Journal of Differential Equations · 2024-02-27 · 4 citations

  articleOpen accessSenior author

  This work is concerned with the Benjamin-Bona-Mahony equation. This model was deduced as an approximation to the Korteweg-de Vries equation in the description of the unidirectional propagation of long waves. Our goal here is to study unique continuation and regularity properties on solutions to the associated initial value problem and initial periodic boundary value problems.

  PublisherDOI

• A Master in Harmony and Differential Equations

  Vietnam Journal of Mathematics · 2023-10-01

  articleOpen access
  PublisherOA PDFDOI

• On local energy decay for solutions of the Benjamin–Ono equation

  Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2023-03-09 · 2 citations

  articleOpen accessSenior author

  We consider the long time dynamics of large solutions to the Benjamin–Ono equation. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.

  PublisherOA PDFDOI

• On Special Properties of Solutions to the Benjamin-Bona-Mahony Equation

  arXiv (Cornell University) · 2023-11-02

  preprintOpen accessSenior authorCorresponding

  This work is concerned with the Benjamin-Bona-Mahony equation. This model was deduced as an approximation to the Korteweg-de Vries equation in the description of unidirectional propagation of long waves. Our goal here is to study unique continuation and regularity properties on solutions to the associated initial value problem and initial periodic boundary value problems.

  PublisherOA PDFDOI

Recent grants

• Nonlinear Evolution Equations

  NSF · $115k · 2008–2012

• Nonlinear Evolution Equations

  NSF · $126k · 2011–2015

• Nonlinear Evolution Equations

  NSF · $133k · 2005–2008

Frequent coauthors

• Luis Vega

  Basque Center for Applied Mathematics

  89 shared

• Carlos E. Kenig

  University of Chicago

  85 shared

• Felipe Linares

  Instituto Nacional de Matemática Pura e Aplicada

  81 shared

• Luis Escauriaza

  University of the Basque Country

  19 shared

• Pedro Isaza

  Universidad Nacional de Colombia

  12 shared

• Jean–Claude Saut

  Laboratoire de Mathématiques d'Orsay

  10 shared

• Claudio Muñoz

  University of Chile

  9 shared

• Rubens Maciel Filho

  Universidade Estadual de Campinas (UNICAMP)

  9 shared