Fourier truncation method: Difference between revisions
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The '''Fourier truncation method''' of Bourgain aims to establish long-time control (and in particular global [[well-posedness]]) of an equation with rough initial data by splitting the data into low and high frequency components. The low component is evolved nonlinearly and controlled using a conserved quantity such as the energy. The high frequency component is evolved linearly. Superimposing the two evolutions yields an ''approximate'' solution to the original equation; perturbation theory (and additional smoothing effects of the equation) is then used to convert the approximate solution to an exact one. The perturbation theory is only sufficient to control the evolution for short times, so at periodic intervals the decomposition of solution into low frequency, high frequency, and error terms must be updated to avoid an exponential loss of control of the error term. | The '''Fourier truncation method''' of [http://en.wikipedia.org/wiki/Bourgain Bourgain] aims to establish long-time control (and in particular global [[well-posedness]]) of an equation with rough initial data by splitting the data into low and high frequency components. The low component is evolved nonlinearly and controlled using a conserved quantity such as the energy. The high frequency component is evolved linearly. Superimposing the two evolutions yields an ''approximate'' solution to the original equation; perturbation theory (and additional smoothing effects of the equation) is then used to convert the approximate solution to an exact one. The perturbation theory is only sufficient to control the evolution for short times, so at periodic intervals the decomposition of solution into low frequency, high frequency, and error terms must be updated to avoid an exponential loss of control of the error term. | ||
The [[I-method]] achieves similar goals by the slightly different method of mollifying the high frequency components of the equation and analyzing the energy of the resulting field, rather than | The [[I-method]] achieves similar goals by the slightly different method of mollifying the high frequency components of the equation and analyzing the energy of the resulting field, rather than | ||
Latest revision as of 06:40, 16 January 2007
The Fourier truncation method of Bourgain aims to establish long-time control (and in particular global well-posedness) of an equation with rough initial data by splitting the data into low and high frequency components. The low component is evolved nonlinearly and controlled using a conserved quantity such as the energy. The high frequency component is evolved linearly. Superimposing the two evolutions yields an approximate solution to the original equation; perturbation theory (and additional smoothing effects of the equation) is then used to convert the approximate solution to an exact one. The perturbation theory is only sufficient to control the evolution for short times, so at periodic intervals the decomposition of solution into low frequency, high frequency, and error terms must be updated to avoid an exponential loss of control of the error term.
The I-method achieves similar goals by the slightly different method of mollifying the high frequency components of the equation and analyzing the energy of the resulting field, rather than splitting the solution into high and low frequency components and evolving them separately.
