Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations

We consider a family of dispersion generalized Benjamin-Ono equations (dgBO) which are critical with respect to the L2 norm and interpolate between the critical modified (BO) equation and the critical generalized Korteweg-de Vries equation (gKdV). First, we prove local well-posedness in the energy space for these equations, extending results by Kenig, Ponce and Vega concerning the (gKdV) equations. Second, we address the blow up problem in the spirit of works of Martel and Merle on the critical (gKdV) equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of solitons. We prove that when the model is close to critical (gKdV), solutions of negative energy close to solitons blow up in finite or infinite time in the energy space. The blow up proof requires in particular extensions to (dgBO) of monotonicity results for localized versions of L2 norms by pseudo-differential operator tools.Comment: Submitte

Well-posedness in energy space for the periodic modified Benjamin-Ono equation

We prove that the periodic modified Benjamin-Ono equation is locally well-posed in the energy space $H^{1/2}$. This ensures the global well-posedness in the defocusing case. The proof is based on an $X^{s,b}$ analysis of the system after gauge transform.Comment: 26 pages, 0 figure. Title changed, introduction modifie

Integrity Constraints Revisited: From Exact to Approximate Implication

Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked by considering only databases containing a single transaction

Low regularity solutions of two fifth-order KdV type equations

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in $H^s({\mathbf R})$ with $s>-\frac74$ and the local well-posedness for the modified Kawahara equation in $H^s({\mathbf R})$ with $s\ge-\frac14$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $[k; Z]$ multiplier norm method of Tao \cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.Comment: 17page

Uniform Lipschitz Estimates in Bumpy Half-Spaces

This paper is devoted to the proof of uniform H\"older and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscillating coefficients. Our main point is that no structure is assumed on the oscillations of the boundary. In particular, those are neither periodic, nor quasiperiodic, nor stationary ergodic. We investigate the consequences of our estimates on the large scales of Green and Poisson kernels. Our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries, which is an area of active research.Comment: 54 page