Invariance of the Gibbs measure for the periodic quartic gKdV
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- 作者：Geordie Richards1 ; grichar5@z.rochester.edu" class="auth_mail" title="E-mail the corresponding author
- 关键词：gKdV ; Gibbs measure ; Nonlinear smoothing ; A.s. global well-posedness
- 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
- 出版年：2016
- 出版时间：May-June 2016
- 年：2016
- 卷：33
- 期：3
- 页码：699-766
- 全文大小：1045 K
文摘
We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on Hs(T) for 
, and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below .
, and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below .