A Priori Bounds for Time-Dependent Models

Diogo A. Gomes*, Edgard A. Pimentel, Vardan Voskanyan

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


We continue our study of the regularity of MFGs by considering the time-dependent problem (formula presented) acts as a perturbation of the heat equation and the main regularity tool is the Gagliardo–Nirenberg inequality. In the second instance, the Hopf–Cole transformation gives an explicit way to study (8.1). However, this transformation cannot be used to superquadratic problems. As a consequence, here, we use a technique that extends for superquadratic problems, 2, based on the nonlinear adjoint method. In the next chapter, we investigate two time-dependent problems with singularities—the logarithmic nonlinearity and the congestion problem—for which different methods are required.

Original languageEnglish (US)
Title of host publicationSpringerBriefs in Mathematics
PublisherSpringer Science and Business Media B.V.
Number of pages5
StatePublished - 2016

Publication series

NameSpringerBriefs in Mathematics
ISSN (Print)2191-8198
ISSN (Electronic)2191-8201

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing Switzerland.


  • Gagliardo-Nirenberg Inequality
  • Logarithmic Nonlinearity
  • Priori Bounds
  • Subquadratic Case
  • Time-dependent Problems

ASJC Scopus subject areas

  • General Mathematics


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