We study Gibbs measures with log-correlated base Gaussian fields on the ddimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When d = 2, our argument provides an alternative proof of the non-normalizability result for the focusing Φ 4 2 -measure by Brydges and Slade (1996). We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

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We consider the Cauchy problem for the fourth order cubic nonlinear Schrödinger equation (4NLS)The main goal of this paper is to prove the low regularity well-posedness and ill-posedness problem for the fourth order NLS. We prove three results. One is the local well-posedness in H s (R) , s ≥ −1/2 via the contraction principle on the X s.b space, also known as Bourgain space. Another is the global wellposedness in H s (R) , s ≥ −1/2. To prove this, we use the I-method with a higher order correction term presented in Colliander-Keel-Staffilani-Takaoka-Tao [8]. The other is the ill-posedness in the sense that the flow map fails to be uniformly continuous for s < − 1 2 . Therefore, we show that s = −1/2 is the sharp regularity threshold for which the well-posedness problem can be dealt with the iteration argument.

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