We prove that the Cauchy problem for Muskat equation is well-posed locally in time any initial data critical space of Lipschitz functions with three-half derivative $$L^2$$ . Moreover, we solution exists globally under a smallness assumption.

The most plausible see-saw explanation of the smallness of the neutrino masses is based on the assumption that total lepton number is violated at a large scale and neutrinos with definite masses are Majorana particles. In this review we consider in details difference between Dirac and Majorana neutrino mixing and possibilities to reveal Majorana nature of neutrinos with definite masses.

The simplest unified extension of the Minimal Supersymmetric Standard Model with bi-linear R–Parity violation provides a predictive scheme for neutrino masses which can account for the observed atmospheric and solar neutrino anomalies. Despite the smallness of neutrino masses Rparity violation is observable at present and future high-energy colliders, providing an unambiguous cross-check of the...

Let f be a locally homeomorphic mapping of finite distortion in dimension larger than two. We show that when the distortion of f satisfies a certain subexponential integrability condition, small sets are removable. The smallness is measured by a weighted modulus.

A new mechanism is proposed to explain neutrino masses and their mixing via SU(1, 1) horizontal symmetry breaking. Based on this mechanism, the smallness of neutrino masses is realized and a hierarchical spectrum with large mixing is given. ∗) E-mail: [email protected] ∗∗) E-mail: [email protected]

The property of smallness for Π1 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π1 class depends only on the Muchnik degree of a Π1 class. A comparison is made with the idea of thinness for Π1 classes

Euler's equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymptotic expansion of Euler's equations is considered (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modelling the motion of shallow water waves are reviewed in ...

Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes.

We show that a model, recently used to describe all the dynamical regimes of the magnetic field generated by the dynamo effect in the von Kármán sodium experiment, also provides a simple explanation of the reversals of Earth's magnetic field, despite strong differences between both systems. The validity of the model relies on the smallness of the magnetic Prandtl number.

We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.