In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities.In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity.Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4 ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.
We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E ∆ S 2m−1 ([0, R)) are uniformly bounded as operators fromWe also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions onSuppose that L is a positive definite, self-adjoint operator acting on L 2 (X, µ), where X is a measurable space with a measure µ. Such operator admits a spectral resolutionBy the spectral theorem, if F is a Borel bounded function on [0, ∞), then the operator F (L) given byis well-defined and bounded on L 2 (X, µ). One of the fundamental problems in the theory of spectral multipliers is to determine when F (L) is bounded on L p for some p = 2. An interesting example is the following family of functionsWe define the operator S δ R (L) using (1). S δ R (L) is called the Riesz mean or the Bochner-Riesz mean of order δ. The basic question in the theory
Notes on symplectic non-squeezing of the KdV flow J. É. D. P. (2005), Exposé AbstractWe prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle T. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.
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