Invariance of the Gibbs measure for the periodic quartic gKdV

By Geordie Richards. 2012

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We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on H^s for s<1/2, 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 and 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 H^{1/2}.

Subfields: Applied Mathematics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/ac2e96d3f0bb22ee351478a4feddba34b09b286b

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In Applied Mathematics

In Mathematical Physics

In Nuclear and High Energy Physics

Citations in the corpus

On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, result): “hane [32] and Ayache-Tzvetkov [2]. There are several works on Cauchy problems of evolution equations with random data that followed these results, including some on almost sure global well-posedness: [7, 58, 20, 45, 12, 21, 22, 15, 43, 23, 52, 13, 9, 10, 44, 51, 42]. 1.2. Randomization adapted to th”
Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$ (2014, unknown): “orous by exploiting the invariance of the ﬁnite dimensional Gibbs measures for the associated ﬁnite dimensional approximations of NLS. This approach has been used in subsequent works; see for example [8, 9, 65, 16, 18, 50, 45, 26, 14, 24, 57, 12]. In our context, however, there is no apparent (forma”