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Open Access Publications from the University of California

Scholarly Works (4 results)

  • Article
  • Peer Reviewed
UC Berkeley Previously Published Works (2017)
We show that for small, localized initial data in a Galilean-invariant space, there exists a global solution to the Kadomtsev-Petviashvili equation, using the method of testing by wave packets.
    Cover page: The Lifespan of Small Data Solutions to the KP-I
    • Article
    • Peer Reviewed
    UC Berkeley Previously Published Works (2014)
    We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.
      Cover page: The lifespan of small data solutions to the KP-I
      • Article
      • Peer Reviewed
      UC Berkeley Previously Published Works (2017)
      In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth result in Hunter et al. (Commun Math Phys, 2014).
        Cover page: Finite Depth Gravity Water Waves in Holomorphic Coordinates
        • Thesis
        • Peer Reviewed
        UC Berkeley Electronic Theses and Dissertations (2015)

        We consider the behavior of nonlinear KdV-type equations that admit quasilinear dynamics in the sense that the nonlinear flow cannot be simply treated as a perturbation of the linear flow, even for small initial data.

        We treat two problems in particular. First we study the local dynamics of KdV-type equations with nonlinearities involving two spatial derivatives. A key obstruction to well-posedness arises from to the Mizohata condition. This leads to an additional integrability requirement for the solution in the absence of a suitable null structure. In this case we prove local well-posedness for large, low-regularity data in translation-invariant spaces.

        Second we explore the global dynamics of the modified Korteweg de-Vries equation. We establish modified asymptotic behavior without relying on the integrable structure of the equation. This approach has the advantage that it can be used for a wide class of short-range perturbations of the mKdV. To give a thorough description of the asymptotic behavior we prove an asymptotic completeness result that relates mKdV solutions to the 1-parameter family of solutions to the Painlevé II equation.

          Cover page: Quasilinear dynamics of KdV-type equations
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