On nonlinear problems of parabolic type with implicit constitutive equations involving flux

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone [Formula: see text]-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

From discrete to continuous monotone C*-algebras via quantum central limit theorems

We prove that all finite joint distributions of creation and annihilation operators in monotone and anti-monotone Fock spaces can be realised as Quantum Central Limit of certain operators in a [Formula: see text]-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone [Formula: see text]-algebras, the weights being related to the above cited test functions.

Conformal Geometry Solitons

In this paper, we deal with a generalization of the Yamabe flow named conformal geometry flow. Firstly we derive a monotone formula of the Einstein-Hilbert functional under the conformal geometry flow. Then we prove the properties that the conformal geometry solitons and conformal geometry breather both have constant scalar curvature at each time by using the modified Einstein-Hilbert function. Finally we present some properties of Yamabe solitons in compact manifold and noncompact manifolds through the equation of Yamabe soliton.