Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.

NUCLEAR ENERGY AND ITS IMPORTANCE IN ROMANIA'S ENERGY IDENTITY

Uncertainties caused by energy dependencies affect the energy security of the states in the region, and implicitly their national security. Romania is in a relatively unique positionbetween the states in the region and those in the EU, given the energy potential. In the next period, taking into account the commitments assumed at the European level, but also the need to revive the National Energy System, Romania will define its energy identity. In this equation, a topic of interest at the moment is the question of nuclear energy, and whether the current historical logic still brings relevance to this way of producing energy, considering also the appropriate security implications. This paper deals with the issue first in terms of defining the place of the nuclear sector in the energy identity of Romania, then pointing out some aspects related to security. Finally, the conclusions are presented in the form of development perspectives.Keywords: Romania; EU; nuclear energy; energy identity; security.

Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces

We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $$\alpha $$ α -(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.

The relaxation limit of bipolar fluid models

<p style='text-indent:20px;'>This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid models, and it is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.</p>