The I-method in its first version as developed by Colliander et al. in [2] is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H(R → R), provided s > − 1 42 .

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.

We prove that, the initial value problem associated to ∂tu+ iα∂ 2 x u+ β∂ x u+ iγ|u|u = 0, x, t ∈ R, is locally well-posed in Hs for s > −1/4.

This paper examines the properties of instrumental variables (IV) applied to models with essential heterogeneity, that is, models where responses to interventions are heterogeneous and agents adopt treatments (participate in programs) with at least partial knowledge of their idiosyncratic response. We present several empirical examples demonstrating the importance of unobserved heterogeneity in...

Bisectorial operators play an important role since exactly these operators lead to a well-posed equation u'(t) = Au(t) on the entire line. The simplest example of a bisectorial operator A is obtained by taking the direct sum of an invertible generator of a bounded holomorphic semigroup and the negative of such an operator. Our main result shows that each bisectorial operator A is of this form, ...

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in H−3/4(R) and the modified Korteweg-de Vries initial-value problem is globally well-posed in H1/4(R). The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at s = −3/4 by constructing some special resolution spaces in order to avoid some ’logarithmic d...

In this paper, we introduce two types of the Levitin-Polyak well-posedness for the system of weak generalized vector equilibrium problems. By using the gap function of the system of weak generalized vector equilibrium problems, we establish the equivalent relationship between the two types of Levitin-Polyak well-posedness of the system of weak generalized vector equilibrium problems and the cor...

In a recent work [12], Ionescu and Kenig proved that the Cauchy problem associated to the Benjamin-Ono equation is well-posed in L(R). In this paper we give a simpler proof of Ionescu and Kenig’s result, which moreover provides stronger uniqueness results. In particular, we prove unconditional well-posedness in Hs(R), for s > 1 4 .

We establish the local well-posedness in H(S) with any s > 72 for a modified Camassa-Holm equation derived as the EPDiff equation with respect to the H(S) metric, and obtain the global existence of the weak solution in H(S) under some sign assumption on the initial values and prove the convergence of the corresponding finite particle approximation method.

In this paper, we study Levitin-Polyak well-posedness for vector equilibrium problems with functional constraints. Two sufficient conditions of (generalized) Levitin-Polyak well-posedness are derived for vector equilibrium problems.