arXiv subjects

Paulo R. Zingano

Publications and source records attributed to Paulo R. Zingano.

9 recordsLinked to original sources

arXiv2022

Upper and lower $\dot{H}^{m}$ estimates for solutions to parabolic equations

In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the $L^2$ norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the $L^2$ norm. We use these general results to prove new decay estimates for some equations and to recover some well known results.

math.AP↗

arXiv2021

Herd immunity for Covid-19 in homogeneous populations

In this note we estimate herd immunity levels for the Covid-19 epidemic using a standard SEIR system that models the disease dynamics in homogeneous populations. The results obtained are indicative of values between 80% and 90% for unprotected, fully susceptible populations. Basic protective measures such as hand hygiene and mask wearing may be effective to lower herd immunity levels down to values between 50% and 60% of the total population.

q-bio.PE↗

arXiv2020

A matlab code to compute reproduction numbers with applications to the Covid-19 outbreak

We discuss the generation of various reproduction ratios or numbers that are very useful to monitor an ongoing epidemic like Covid-19 and examine the effects of intervention measures. A detailed SEIR algorithm is described for their computation, with applications given to the current Covid-19 outbreaks in a number of countries (Argentina, Brazil, France, Italy, Mexico, Spain, UK and USA). The corresponding matlab script, complete and ready to use, is provided for free downloading.

q-bio.PE↗

arXiv2018

On the supnorm form of Leray's problem for the incompressible Navier-Stokes equations

We show that t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty for all (global) Leray solutions of the incompressible Navier-Stokes equations in R3. It is also shown that t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty, where v(.,t) is the Stokes approximation, as well as other fundamental results. In spite of the difficulty of these questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.

math.AP↗

arXiv2018

Two problems in Partial Differential Equations (in Portuguese)

In this work, we examine two important problems in the theory of nonlinear PDEs. In Part I, we propose and solve a more general and complete version of the celebrated Leray's problem for the incompressible Navier-Stokes equations in $ \mathbb{R}^{3} \!$, which in its simplest form was suggested by J.$\;$Leray in 1934 (and solved only in the 1980s by T.$\;$ Kato, K.$\;$Masuda and other authors). A number of related new results of clear interest to the theory of Leray's solutions are also given here. In Part II, which is independent of Part I and can be read separately, we introduce an important new collection of problems concerning global existence results and blow-up phenomena for solutions of conservative advection-diffusion equations in $ \mathbb{R}^{n} $ where heterogeneities in the lower order terms tend to destabilize the solution (everywhere or in certain regions), strongly competing with the viscous dissipation effects to determine the overall solution behavior. Here, we consider the case of superlinear advection (and arbitrary dimension), which may cause finite-time blow-up in several important spaces. We then point out a new kind of phenomena --- one that may be properly named "anti-Fujita" for its vivid contrast to the type of blow-up behavior discovered by Fujita in the 1960s, and which has been investigated ever since --- that has apparently been completely overlooked in the literature.

math.AP↗

arXiv2017

An inequality for solutions of the Navier-Stokes equations in Rn

We obtain a new inequality that holds for general Leray solutions of the incompressible Navier-Stokes equations in Rn (n <= 4). This recovers important results previously obtained by other authors regarding the time decay of solution derivatives (of arbitrary order).

math.AP↗

arXiv2017

Some remarks on the regularity time of Leray solutions to the Navier-Stokes equations

In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4} if n = 3, and t* <= K_{4} nu^{-3} || u(.,0) ||_{L2}^{2} if n = 4 (in particular, by reducing the current best known values for the constants K_{3}, K_{4}). Some related results of clear interest are also included (derived) in our discussion.

math.AP↗

arXiv2016

Sharp pointwise estimates for functions in the Sobolev spaces Hs(Rn)

We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u \,\|_{\dot{H}^{s}(\mathbb{R}^{n})}^{n/(2s)} $ where $ s > n/2 $.

math.FA↗

arXiv2015

The Navier-Stokes equations for Incompressible Flows: solution properties at potential blow-up times

In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can develop singularities in finite time. Assuming the maximum interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.

math.AP↗