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

We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

/pdf/sharp-global-well-posedness-for-kdv-and-modified-kdv-on-and-35jzrmuq91.pdf

Sharp global well-posedness for KdV and modified KdV on ℝ and

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.

/pdf/asymptotics-frequency-modulation-and-low-regularity-ill-3ve0ij81n1.pdf

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

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrodinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

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Endpoint Strichartz estimates

We obtain global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain (4) and Grillakis (20), which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain (4), but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in (12), (13)). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of L 2 mass in fre- quency space, rules out the possibility of energy concentration.

http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n3-p03.pdf

Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3

We consider the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes This behavior is quantified by the growth of higher Sobolev norms: given any δ � 1 ,K � 1 ,s >1, we construct smooth initial data u0 withu0� H s K at some time T This growth occurs despite the Hamiltonian's bound onu(t)� ˙ H 1 and despite the conservation of the quantityu(t)� L2

https://link.springer.com/content/pdf/10.1007%2Fs00222-010-0242-2.pdf

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

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

Markus Keel

Papers

Endpoint Strichartz estimates

Sharp global well-posedness for KdV and modified KdV on ℝ and

Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

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