Low Regularity a Priori Bounds for the Modified Korteweg-de Vries Equation

We study the local well-posedness in the Sobolev space H(R) for the modified Korteweg-de Vries (mKdV) equation ∂tu+ ∂ 3 xu± ∂xu = 0 on R. KenigPonce-Vega [10] and Christ-Colliander-Tao [1] established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H(R) when s < 1 4 . In spite of this, we establish that for − 18 < s < 1 4 , the solution satisfies global in time H(R) bounds which depend only on the time and on the H(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-CollianderTao [2] and Koch-Tataru [11] employing a version of Bourgain’s Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.