I-method: Difference between revisions
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The '''I-method''' is a method for constructing global solutions to nonlinear dispersive and wave equations in situations when the relevant conserved quantity (such as the energy E(u)) is subcritical but infinite. One applies a mollifying operator ''I'' to the solution (dependent on a large frequency truncation parameter ''N'') to make the conserved quantity E(Iu) finite. The catch is that this quantity is no longer conserved, but one hopes to show an ''almost conservation'' law for this quantity which makes it stable over long periods of time (going to infinity as <math>N \to \infty</math>). Letting ''N'' go to infinity one obtains global well-posedness. | The '''I-method''' is a method for constructing global solutions to nonlinear dispersive and wave equations in situations when the relevant conserved quantity (such as the energy E(u)) is subcritical but infinite. One applies a mollifying operator ''I'' to the solution (dependent on a large frequency truncation parameter ''N'') to make the conserved quantity E(Iu) finite. The catch is that this quantity is no longer conserved, but one hopes to show an ''almost conservation'' law for this quantity which makes it stable over long periods of time (going to infinity as <math>N \to \infty</math>). Letting ''N'' go to infinity one obtains global well-posedness. | ||
The I-method was inspired by the earlier [[Fourier truncation method]] of Bourgain, which achieves similar effects by approximating the evolution of the rough solution by the evolution if its low and high frequency components. | The I-method was inspired by the earlier [[Fourier truncation method]] of Bourgain, which achieves similar effects by approximating the evolution of the rough solution by the evolution if its low and high frequency components. | ||
== Explanation for the name ''I'' == | |||
The name ''I'' was chosen for the mollification operator, because this operator acts like the '''I'''dentity on low freqeuencies (less than N), and like an '''I'''ntegration operator on high frequencies. The ''I-team'' (([[User:Colliand|James Colliander]], Mark Keel, [[User:Gigliola|Gigliola Staffilani]], Hideo Takaoka, and [[User:tao|Terry Tao]]) used this operator extensively in their joint papers, hence the name (though the ''I'' was also interpreted to indicate the '''I'''nternational nature of the collaboration). | |||
[[Category:Methods]] | [[Category:Methods]] | ||
Revision as of 22:18, 5 August 2006
The I-method is a method for constructing global solutions to nonlinear dispersive and wave equations in situations when the relevant conserved quantity (such as the energy E(u)) is subcritical but infinite. One applies a mollifying operator I to the solution (dependent on a large frequency truncation parameter N) to make the conserved quantity E(Iu) finite. The catch is that this quantity is no longer conserved, but one hopes to show an almost conservation law for this quantity which makes it stable over long periods of time (going to infinity as ). Letting N go to infinity one obtains global well-posedness.
The I-method was inspired by the earlier Fourier truncation method of Bourgain, which achieves similar effects by approximating the evolution of the rough solution by the evolution if its low and high frequency components.
Explanation for the name I
The name I was chosen for the mollification operator, because this operator acts like the Identity on low freqeuencies (less than N), and like an Integration operator on high frequencies. The I-team ((James Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and Terry Tao) used this operator extensively in their joint papers, hence the name (though the I was also interpreted to indicate the International nature of the collaboration).
