Existence of weak solutions for a general porous medium equation with nonlocal pressure

We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $01$ by developing a new approximating method that allows to treat the range $m\ge 3$ that could not be covered by previous works. We also consider as initial data any non-negative measure $\mu$ with finite mass. In passing from bounded initial data to measure data we make strong use of an $L^1$-$L^\infty$ smoothing effect and other functional inequalities. Finite speed of propagation is established for all $m\ge 2$, which implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for $m<2$

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general symmetric diffusion operator of L ́evy type and φ is merely continuous and non-decreasing. We then use this theory to prove con- vergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion op- σ,μ α are the (fractional) Laplacians ∆ and −(−∆)2 for α ∈ (0,2), erators L discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L ́evy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and conver- gence of the methods under minimal assumptions – including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory

Modeling cardiac structural heterogeneity via space-fractional differential equations

We discuss here the use of non-local models in space and fractional order operators in the characterisation of structural complexity and the modeling of propagation in heterogeneous biological tissues. In the specific, we consider the application of space-fractional operators in the context of cardiac electrophysiology, where the lack of clear separation of scales of the highly heterogeneous myocardium triggers peculiar features such as the dispersion of action potential duration, that have been observed experimentally, but cannot be described by the standard monodomain or bidomain models. We describe the methodology and compare the results of a standard monodomain model with results of a model with a non-local component in space

Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions

In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretisations with order of convergence depending on the regularity of the domain and the function on which the fractional Laplacian is acting. Unlike other existing approaches in literature, our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.FdT acknowledges support of Toppforsk project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway. ERCIM ``Alain Benoussan" Fellowship programm

Chemiluminescence studies on comparison of antioxidant effectiveness on multiextruded polyethylenes

Several polyethylene resins; high density polyethylene (Ph-HDPE, Phillips metal oxide catalyst) and linear low density polyethytlenes (LLDPE, formed using Ziegler–Natta catalyst and metallocene catalyst technology, m-LLDPE) were used in order to acquire an insight in to the effect of different polymerization catalyst systems on the melt stabilization performance of single AO. Chemiluminescence (CL) and melt flow rate (MFR) were used to analyze the degradation as a function of the number of passes through a twin-screw extruder. A good correlation was obtained, and the additives resulting with the best melt flow stability in the polymer were the same as those that promote the best CL results

Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas

We introduce three representation formulas for the fractional p-Laplace operator in the whole range of parameters 0 A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional p-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional p-Laplacian on manifolds, as well as alternative characterizations of the Ws, p(ℝn) seminorms.</p

A new approach for sensory characterization of grape: relationship with chemical composition

AIM: Characterize taste and mouthfeel properties of grapes elicited by the phenolic fraction (PF) of grape berries and establish relationships with chemical variables. METHODS: As many as 31 diverse grape lots of Tempranillo Tinto and Garnacha Tinta from three different regions were harvested. Grapes were destemmed and macerated in 15% of ethanol for one week and extracts were submitted to solid phase extraction. The recovered polyphenolic fraction was reconstituted in wine model and characterized by a panel of 21 wine experts employing a list of 23 taste and mouthfeel-related attributes following a rate-k-attributes methodology. RESULTS: Six significant attributes among the 31 samples differed based on ANOVA results: “dry”, “coarse”, “bitter”, “dry on tongue”, “sticky” and “watery”. PCA with VARIMAX algorithm was calculated. Three main independent dimensions defining the sensory space of PFs were identified: D1, “dry on the tongue”; D2, “bitter/ sticky”; and D3: “coarse/dry”. Two out of the three dimensions could be satisfactory modeled by PLS-regression from chemical parameters. Tannin activity and tannin concentration along with mDP of tannins proved to be good predictors of perceived dryness. Flavonols have a good prediction power for “bitter” attribute and the “sticky/bitter” dimension. In addition, the low molecular weight anthocyanins seem to be involved in the formation of the “dry” attribute, whereas large polymeric pigments in the “sticky” attribute and the “sticky/bitter” dimension. CONCLUSIONS: This study has increased our knowledge about some of the chemical drivers of grape sensory properties and presents a powerful tool for the wine industry to assess grape quality

Carbon Dioxide Embolism Associated with Transanal Total Mesorectal Excision Surgery: A Report From the International Registries

BACKGROUND: Carbon dioxide embolus has been reported as a rare but clinically important risk associated with transanal total mesorectal excision surgery. To date, there exists limited data describing the incidence, risk factors, and management of carbon dioxide embolus in transanal total mesorectal excision. OBJECTIVE: This study aimed to obtain data from the transanal total mesorectal excision registries to identify trends and potential risk factors for carbon dioxide embolus specific to this surgical technique. DESIGN: Contributors to both the LOREC and OSTRiCh transanal total mesorectal excision registries were invited to report their incidence of carbon dioxide embolus. Case report forms were collected detailing the patient-specific and technical factors of each event. SETTINGS: The study was conducted at the collaborating centers from the international transanal total mesorectal excision registries. MAIN OUTCOME MEASURES: Characteristics and outcomes of patients with carbon dioxide embolus associated with transanal mesorectal excision were measured. RESULTS: Twenty-five cases were reported. The incidence of carbon dioxide embolus during transanal total mesorectal excision is estimated to be 480.4% (25/6375 cases). A fall in end tidal carbon dioxide was noted as the initial feature in 22 cases, with 13 (52%) developing signs of hemodynamic compromise. All of the events occurred in the transanal component of dissection, with mean (range) insufflation pressures of 15mm Hg (12\u201320mm Hg). Patients were predominantly (68%) in a Trendelenburg position, between 30\ub0 and 45\ub0. Venous bleeding was reported in 20 cases at the time of carbon dioxide embolus, with periprostatic veins documented as the most common site (40%). After carbon dioxide embolus, 84% of cases were completed after hemodynamic stabilization. Two patients required cardiopulmonary resuscitation because of cardiovascular collapse. There were no deaths. LIMITATIONS: This is a retrospective study surveying reported outcomes by surgeons and anesthetists. CONCLUSIONS: Surgeons undertaking transanal total mesorectal excision must be aware of the possibility of carbon dioxide embolus and its potential risk factors, including venous bleeding (wrong plane surgery), high insufflation pressures, and patient positioning. Prompt recognition and management can limit the clinical impact of such events. See Video Abstract at http://links. lww.com/DCR/A961