Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow

We prove that solutions of a mildly regularized Perona-Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.Comment: 19 page

A novel mechanism underlying the innate immune response induction upon viral-dependent replication of host cell mRNA: A mistake of +sRNA viruses' replicases

Viruses are lifeless particles designed for setting virus-host interactome assuring a new generation of virions for dissemination. This interactome generates a pressure on host organisms evolving mechanisms to neutralize viral infection, which places the pressure back onto virus, a process known as virus-host cell co-evolution. Positive-single stranded RNA (+sRNA) viruses are an important group of viral agents illustrating this interesting phenomenon. During replication, their genomic +sRNA is employed as template for translation of viral proteins; among them the RNA-dependent RNA polymerase (RdRp) is responsible of viral genome replication originating double-strand RNA molecules (dsRNA) as intermediates, which accumulate representing a potent threat for cellular dsRNA receptors to initiate an antiviral response. A common feature shared by these viruses is their ability to rearrange cellular membranes to serve as platforms for genome replication and assembly of new virions, supporting replication efficiency increase by concentrating critical factors and protecting the viral genome from host anti-viral systems. This review summarizes current knowledge regarding cellular dsRNA receptors and describes prototype viruses developing replication niches inside rearranged membranes. However, for several viral agents it's been observed both, a complex rearrangement of cellular membranes and a strong innate immune antiviral response induction. So, we have included recent data explaining the mechanism by, even though viruses have evolved elegant hideouts, host cells are still able to develop dsRNA receptors-dependent antiviral response.Fil: Delgui, Laura Ruth. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos. Universidad Nacional de Cuyo. Facultad de Cienicas Médicas. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos; ArgentinaFil: Colombo, Maria Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos. Universidad Nacional de Cuyo. Facultad de Cienicas Médicas. Instituto de Histología y Embriología de Mendoza Dr. Mario H. Burgos; Argentin

Obstructions to regularity in the classical Monge problem

We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counterexample in \cite{LSW} to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains

On the singular local limit for conservation laws with nonlocal fluxes

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.Comment: 26 page