Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman-Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry-\'Emery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.Comment: 41 page

Nonlinear System Identification of Laboratory Heat Exchanger Using Artificial Neural Network Model

This paper addresses the nonlinear identification of liquid saturated steam heat exchanger (LSSHE) using artificial neural network model. Heat exchanger is a highly nonlinear and non-minimum phase process and often its working conditions are variable. Experimental data obtained from fluid outlet temperature measurement in laboratory environment is used as the output variable and the rate of change of fluid flow into the system as input too. The results of identification using neural network and conventional nonlinear models are compared together. The simulation results show that neural network model is more accurate and faster in comparison with conventional nonlinear models for a time series data because of the independence of the model assignment.DOI:http://dx.doi.org/10.11591/ijece.v3i1.195

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

This article presents new elliptic gradient estimates for positive solutions to nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-\'Emery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed

On Multiple Solutions to a Family of Nonlinear Elliptic Systems in Divergence Form Coupled with an Incompressibility Constraint

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll} \dive\{\A(|x|,|u|^2,|\nabla u|^2) \nabla u\} + \B(|x|,|u|^2,|\nabla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,\nabla u] \} \quad &\text{ in} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det \nabla u \equiv 1$ and \A = \A(|x|,|u|^2,|\nabla u|^2), \B = \B(|x|,|u|^2,|\nabla u|^2) are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draws upon intimate links with the Lie group ${\bf SO}(n)$, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity \Delta=\Delta(\A,\B), prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.Comment: 24 page

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman-Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry-\'Emery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed

Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman-Ricci flow and implications

In this paper we derive new differential Harnack estimates of Li-Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form Laϕ[w]:=[∂∂t−a(x,t)−Δϕ]w(x,t)=G(t,x,w(x,t)),t>0,on smooth metric measure spaces where the metric and potential are time dependent and evolve under a $({\mathsf k}, m)$-super Perelman-Ricci flow. A number of consequences, most notably, a parabolic Harnack inequality, a class of Hamilton type global curvature-free estimates and a general Liouville type theorem together with some consequences are established. Some special cases are presented to illustrate the strength of the results.</p

On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll} \dive\{\A(|x|,|u|^2,|\nabla u|^2) \nabla u\} + \B(|x|,|u|^2,|\nabla u|^2) u = \dive \{ \mcP(x) [{\rm cof}\,\nabla u] \} \quad &\text{ in} \ \Omega , \\ \text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\ u =\varphi \ &\text{ on} \ \partial \Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det \nabla u \equiv 1$ and \A = \A(|x|,|u|^2,|\nabla u|^2), \B = \B(|x|,|u|^2,|\nabla u|^2) are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draws upon intimate links with the Lie group ${\bf SO}(n)$, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity \Delta=\Delta(\A,\B), prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-\'Emery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.</p

Effect of the simultaneous curing and foaming kinetics on the morphology development of polyisoprene closed cell foams

Pianki o zamkniętych komórkach oparte na kauczuku poliizoprenowym (IR) wytworzono przez formowanie tłoczne z zastosowaniem azodikarbonamidu (ADC), jako chemicznego poroforu. Zbadano wpływ temperatury przetwarzania na rozkład ADC oraz na wulkanizację IR z użyciem ADC i bez ADC, w celu określenia wpływu tego parametru na końcową morfologię pianki i właściwości mechaniczne. Badanie kinetyczne wykazało, że do interpretacji danych eksperymentalnych odpowiedni jest model autokatalityczny. Stwierdzono, że energia aktywacji rozkładu ADC (Ea = 181,8 kJ/mol) jest znacznie wyższa niż wulkanizacji IR bez ADC (Ea = 79,6 kJ/mol) lub z ADC (Ea = 72,3 kJ/mol) Wynika z tego, że wraz ze wzrostem temperatury, szybkość rozkładu ADC wzrasta bardziej niż szybkość wulkanizacji kauczuku, więc należy przeprowadzić optymalizację procesu. Zwiększenie temperatury ze 140 do 150°C zmniejszyło średni rozmiar komórek z 355 do 290 μm, zwiększając jednocześnie gęstość komórek z 73 do 118 komórek/mm3. Dalszy wzrost temperatury doprowadził jednak, ze względu na równowagę pomiędzy koalescencją komórek a sieciowaniem, do zwiększenia rozmiaru komórek oraz niższej gęstości komórek. Dla zoptymalizowanej temperatury (150°C) pianki miały najwyższy moduł sprężystości przy ściskaniu oraz twardość.Closed cell foams based on polyisoprene rubber (IR) were produced via compression molding using azodicarbonamide (ADC) as a chemical blowing agent. The effect of processing temperature on ADC decomposition, as well as IR curing with and without ADC were studied to determine the effect of this parameter on the final foam morphology and mechanical properties. The kinetic study showed that the autocatalytic model is appropriate to represent the experimental data. The activation energy for ADC decomposition (Ea = 181.8 kJ/mol) was found to be much higher than for IR curing without (Ea = 79.6 kJ/mol) or with (Ea = 72.3 kJ/mol) ADC. This indicates that with increasing temperature the rate of ADC decomposition accelerates faster than rubber vulcanization, so an optimization must be performed. For example, increasing the temperature from 140 to 150°C decreased the average cell size from 355 to 290 μm while increased the cell density from 73 to 118 cell/mm3. But further temperature increase led to larger cell size and lower cell density because of a balance between cell coalescence and crosslinking. For the optimized temperature (150°C), the foams had the highest modulus of elasticity and hardness