Low regularity bounds for mKdV

2012/07/29 by Michael Christ, Christ, Michael, Justin Holmer +3
Mathematics · #35Q53 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1207.6738

openalex publication_date 2012/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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 \citeKPV2 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.

Citations

• On the ill-posedness of some canonical dispersive equations
• A-priori bounds for KdV equation below H^-3/4
• Interpolation of linear operators

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