Low regularity bounds for mKdV
Skip to main content
Open Access Publications from the University of California

Low regularity bounds for mKdV

  • Author(s): Christ, M
  • Holmer, J
  • Tataru, D
  • et al.

We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H^s when s<1/4. In spite of this, we establish that for -1/8 < s < 1/4, the solution satisfies global in time H^s(R) bounds which depend only on the time and on the H^s(R) norm of the initial data. This result is weaker than global well-posedness, as we have no control on differences of solutions. Our proof is modeled on recent work by Christ-Colliander-Tao and Koch-Tataru employing a version of Bourgain's Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.

Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you.

Main Content
Current View