Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

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Nonlinear dispersive equations : local and global analysis

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrodinger (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 -¾ , 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 -¾ to a distributional solution u ∈ C 0 ([- T, T ]; H -¾ ) 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.

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Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

On the ill-posedness of some canonical dispersive equations

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Propagation of ultra-short optical pulses in cubic nonlinear media

We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrodinger equation in Hs(R) when n = 2, 3 and s > 4 7 , 5 6 , respectively.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2002/0009/0005/MRL-2002-0009-0005-a009.pdf

Almost Conservation Laws and Global Rough Solutions to a Nonlinear Schrödinger Equation

The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.

Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation

On remarque que l'on peut construire un exemple tel que le probleme de Cauchy pour l'equation de KdV soit mal pose dans H s (R), s < -3/4, si on suppose que l'application qui a une donnee initiale associe la solution a l'instant t soit de classe C 2 (dans [2], une regularite C 3 est supposee).

Remark on the local ill-posedness for KdV equation

On the local regularity of the Kadomtsev-Petviashvili-II equation

We prove the global well-posedness of the KP-I equation in ${\Bbb R}^2$
 for data of arbitrary size in a suitable Sobolev class. We use a compactness method to get local solutions. We extend the solutions thanks to some conservation laws, an almost conserved quantity and Strichartz estimates.

Global well-posedness for the KP-I equation

We analyze bilinear estimates related to periodic KP-I type equations. We prove that the Cauchy problem for the fifth order KP-I equation, posed on ?×ℝ, is globally well-posed in the energy space. Despite the loss of a smoothing relation (which is essential to recuperate the derivative loss), used in the context of the KdV and KP-II type equations, we are able to gain a moothing thanks to some specific properties of the set of Fourier modes where the smoothing relation fails. Counterexamples in the context of KP-I type equations posed on ?×? are given. These examples indicate that contrary to the KP-II case it would be difficult to obtain the local well-posedness for the periodic KP-I type equations by an iteration scheme using Bourgain spaces.

N. Tzvetkov

Papers

Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation

Remark on the local ill-posedness for KdV equation

On the local regularity of the Kadomtsev-Petviashvili-II equation

Global well-posedness for the KP-I equation

On Periodic KP-I Type Equations