Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations

The global regularity for the two- and three-dimensional Kuramoto-Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper and family of hyper-viscous Hamilton-Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton-Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p > 2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation

Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.Comment: 32 page

Role of a Compatibilizer in the Structure and Micromechanical Properties of Recycled Poly(ethylene terephthalate)/Polyolefin Blends with Clay

The comparison of the degree of crystallinity and the micromechanical properties in the blends of recycled amorphous poly(ethylene terephthalate) (PET)with isotactic polypropylene (iPP) and high-density polyethylene (HDPE) with a compatibilizer in different proportions is reported. The physical study of the composites of the compatibilized blends and clay is also discussed. The analysis, performed by means of wide-angle X-ray scattering and differential scanning calorimetry techniques, permits us to describe, at microscale level, the role of the compatibilizer on the structure and microhardness of the polymer blends that we studied. The results reveal that PET was incompatible with both iPP and HDPE. However, the presence of the compatibilizer, a styrene–ethylene/butylene–styrene block copolymer grafted with maleic anhydride, allowed the compatibilization of these polymers. In the PET/iPP blends, the clay seemed to have a nucleating effect on the iPP and also induced a hardness increase in the compatibilized blends. On the other hand, in case of PET/HDPE, the crystallinity of these samples (pure blends,blends with compatibilizer, and blends with compatibilizer plus clay) only depended on their composition. Similarly to the PET/iPP blends, the addition of clay induced an increase in the hardness of the ompatibilized blends.Peer reviewe

Uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system

We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system. Our proof extends the method used by Loeper in J. Math. Pures Appl. 86, 68-79 (2006) to obtain uniqueness results for the Vlasov-Poisson system.Comment: AMS-LaTeX, 21 page

On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations

We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius

Eternal solutions to a singular diffusion equation with critical gradient absorption

The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular diffusion equation with critical gradient absorption \begin{equation*} \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N \end{equation*} where $2N/(N+1) < p < 2$. Such solutions are shown to exist only if the parameter $\beta$ ranges in a bounded interval $(0,\beta_*]$ which is in sharp contrast with well-known singular diffusion equations such as $\partial_{t}\phi-\Delta_{p} \phi=0$ when $p=2N/(N+1)$ or the porous medium equation $\partial_{t}\phi-\Delta\phi^m=0$ when $m=(N-2)/N$. Moreover, the profile $f(r;\beta)$ decays to zero as $r\to\infty$ in a faster way for $\beta=\beta_*$ than for $\beta\in (0,\beta_*)$ but the algebraic leading order is the same in both cases. In fact, for large $r$, $f(r;\beta_*)$ decays as $r^{-p/(2-p)}$ while $f(r;\beta)$ behaves as $(\log r)^{2/(2-p)} r^{-p/(2-p)}$ when $\beta\in (0,\beta_*)$

DIAGNÓSTICO DA QUALIDADE DA ÁGUA AO LONGO DE UM CANAL DE CONCRETO: UM ESTUDO DE CASO DO CANAL DO SERTÃO ALAGOANO - BRASIL

Para atender a demanda, a transferência de água de rios por canal é uma prática comum no Nordeste brasileiro. O Canal do Sertão Alagoano capta água do rio São Francisco (no reservatório Apolônio Sales) para abastecer municípios do estado de Alagoas. O objetivo deste trabalho foi analisar a evolução de parâmetros físico-químicos da água (temperatura, pH, turbidez, condutividade, dureza, sulfatos, cloretos, nitrogênio total e fósforo total) ao longo dos 29 km iniciais. Foram realizadas duas coletas no período seco em 10 pontos. Por meio do teste não-paramétrico MannWhitney, evidenciou-se que temporalmente as duas coletas são significativamente diferentes para todos os parâmetros, mesmo sendo ambas realizadas no período seco. Longitudinalmente, nas duas coletas, temperatura, pH e condutividade, foram significativamente diferentes entre o início e o final dos 29 km, apresentando uma tendência crescente nos valores. Quanto à qualidade, conforme a Resolução 357/2005 do CONAMA, a água do Canal apresentou valores dentro da Classe 1, com exceção do fósforo total

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution