Global Solutions for 3D Quadratic Schrödinger Equations

By Pierre Germain, Nader Masmoudi, Jalal Shatah. 2008

178 citations recorded by the source corpus.

This is the firstof the three papers where we present a new method based on the concept of space-time resonance to prove global existence of small solutions to nonlinear dispersive equations. The idea is that time resonances (dynamical systems resonances) correspond to interactions between plane waves; but since for dispersive equations we deal with localized solutions, it is crucial to take also into account the traveling speeds of the different wave packets. Here we show how this idea, and the analytical method that this naturally suggests, leads to a simple proof of global existence and scattering for quadratic nonlinear Schrodinger equations in three dimensions.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fc8274bb9dfd2c3eac35300552cb1790e9796924

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. F. Pusateri
2. Sijue Wu
3. Yan Guo
4. Nader Masmoudi
5. Pierre Germain
6. Benoit Pausader
7. A. Ionescu
8. J. Kato

In Statistical and Nonlinear Physics

1. Sijue Wu
2. F. Pusateri
3. Yan Guo
4. Nader Masmoudi
5. Benoit Pausader
6. Pierre Germain
7. J. Kato
8. Jalal Shatah

Citations in the corpus

• Global solutions for the gravity water waves equation in dimension 3 (2009, methodology): “Our proof of global existence is based on the method of space-time resonances introduced in [GMS09].”
• The second iterate for the Navier–Stokes equation (2008, background): “We would like to mention here the article of Germain, Masmoudi and Shatah [ 9 ].”
• MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “We remark that ‖Fηκm̃‖L1(R2) = ‖I(x1, x2)‖L1x1,x2 , 10In the terminology of Germain-Masmoudi-Shatah [43], O1 corresponds to space nonresonant interactions and O2 to time nonresonant interactions.”
• Almost global wellposedness of the 2-D full water wave problem (2009, methodology): “While their method is algorithmic, and applicable to some other problems [12], we note that their assumptions on the initial data implies some quite strong decay properties of the velocity potential ψ .”
• Global wellposedness of the 3-D full water wave problem (2011, methodology): “While their method is algorithmic, and applicable to some other problems [12], we note that their assumptions on the initial data implies some quite strong decay properties of the velocity potential ψ .”
• A new proof of long-range scattering for critical nonlinear Schrödinger equations (2010, methodology): “Although the present example is quite elementary, this typeof analysis of “generalized null-structures” constitutes an important part of the methods developed by Germain, Masmoudi and Shatah.”
• Global Smooth Ion Dynamics in the Euler-Poisson System (2010, methodology): “[22] Shatah, J., Normal forms and quadratic nonlinear Klein-Gordon equations.”
• Global existence for the Euler-Maxwell system (2011, methodology): “The former is achieved by an energy estimate; and the latter by the method of space-time resonances, which was introduced in [12].”