Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems

By Tadahiro Oh. Differential and Integral Equations, 2009

71 citations recorded by the source corpus.

We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter \alpha \in (0, 4) \setminus 1, we show that the Gibbs measure is invariant under the flow and the system is globally well-posed almost surely on the statistical ensemble, provided that certain Diophantine conditions are satisfied.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2e10bd937acb17fde5bffb1d5843e6a86fc57a92

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

In Statistical and Nonlinear Physics

Citations in the corpus

Almost sure well-posedness of the cubic nonlinear Schr\ (2009, methodology): “es on a probability space (Ω,F,P). Lemma 3.4. Let ε,β>0 and δ≪ 1. Then, we have (3.17) |gn(ω)| ≤ Cδ− β 2 hniε for all n∈ Zfor ωoutside an exceptional set of measure <e−δ 1c . Proof. Recall from [26] that we have P(sup nhni−ε|g (ω)| >K) ≤ e−cK 2 for suﬃciently large K>0. Now, choose K∼ δ−”
Invariance of the Gibbs measure for the periodic quartic gKdV (2012, unknown): “Recall from [23] that P ( sup n∈Z\{0} 〈n〉|gn(ω)| > K) ≤ e−cK 2 for K sufficiently large.”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, background): “See [4, 5, 54, 49, 50, 17, 39, 40, 13] where this approach has been fruitful.”
Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\"odinger equation below the energy space (2013, background): “[28, 17]) For part (3) fix M ≫ 1 such that (2) holds for any |n| ≥M .”
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”
INVARIANT WEIGHTED WIENER MEASURES AND ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVE NLS. (2010, methodology): “Oh [33, 34, 35, 36] for the periodic KdV-type coupled systems, Schrödinger–Benjamin–Ono system and white noise for the KdV equation.”