JournalspmVol. 65, No. 3pp. 387–429

Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation

  • Paulo Amorim

    Universidade de Lisboa, Portugal
  • Philippe G. LeFloch

    Université Pierre et Marie Curie - Paris 6, France
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Abstract

We establish sharp estimates for distributional solutions to the Euler–Poisson–Darboux equation posed in a periodic domain. These equations are highly singular, and setting the Cauchy problem requires a precise understanding of the nature of the singularities that may arise in weak solutions. We consider initial data in a space of functions with fractional derivatives such that weak solutions are solely integrable, and we derive sharp continuous dependence estimates for solutions to the initial-value problem. Our results strongly depend on a key parameter arising in the Euler–Poisson–Darboux equation.

Cite this article

Paulo Amorim, Philippe G. LeFloch, Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation. Port. Math. 65 (2008), no. 3, pp. 387–429

DOI 10.4171/PM/1819