Constant Exponents Research Articles

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems

Abstract We establish existence results for a class of mixed anisotropic and nonlocal p-Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.

Intermittency across Reynolds numbers – the influence of large-scale shear layers on the scaling of the enstrophy and dissipation in homogenous isotropic turbulence

Direct numerical simulations up to Reλ = 1445 show that the scaling exponents for the enstrophy and the dissipation rate extrema are different and depend on the Reynolds number. A similar Reynolds number dependence of the scaling exponents is observed for the moments of the dissipation rate, but not for the moments of the enstrophy. Significant changes in the exponents occur at approximately Reλ ≈ 250, where Reλ is the Taylor based Reynolds number, which coincides with structural changes in the flow, in particular the development of large-scale shear layers. A model for the probability density functions (PDFs) of the enstrophy and dissipate rate is presented, which is an extension of our existing model (Proc. R. Soc. A, vol. 476, 2020, p. 20200591) and is based on the mentioned development of large-scale layer regions within the flow. This model is able to capture the observed Reynolds number dependencies of the scaling exponents, in contrast to the existing theories which yield constant exponents. Moreover, the model reconciles the scaling at finite Reynolds number with the theoretical limit, where the enstrophy and dissipation rate scale identically at infinite Reynolds number. It suggests that the large-scale shear layers are vital for understanding the scaling of the extrema. Furthermore, to reach the theoretical limit, the scaling exponents must remain Reynolds number dependent beyond the present Reλ range.

Exploring a family of Bernoulli-like shift chaotic maps and its amplitude control

Chaos theory is a branch of mathematics that studies systems highly sensitive to initial conditions. Various methods for analyzing and controlling these systems, such as bifurcation theory and amplitude control, have been developed and applied in diverse fields, including cryptography, physics, and engineering.In this study, we introduce a new class of Bernoulli-Like Shift (BLS) maps. These maps are generated by using a specific class of Piecewise Linear (PWL) maps combined with a modulo operation. We conduct a fixed point analysis by establishing propositions and corollaries, which outline the conditions that guarantee the presence or absence of fixed points. Additionally, we provide numerical examples to validate these theoretical results.Utilizing bifurcation diagrams and Lyapunov exponent calculations, we demonstrate that the range of chaotic behavior in the BLS maps is broader than those of the logistic and Bernoulli Shift maps. Moreover, we show that the BLS maps possess higher values of Lyapunov exponents and a better distribution than both the Bernoulli shift and logistic maps.Lastly, we introduce a transformation that generates a modified BLS map, allowing for amplitude control. This modification enables the regulation of amplitude while maintaining constant Lyapunov exponents.

Multiplicity results for double phase problems involving a new type of critical growth

Using variational methods, we obtain several multiplicity results for double phase problems that involve variable exponents and a new type of critical growth. This new critical growth is better suited for double phase problems when compared to previous works on the subject. In order to overcome the lack of compactness caused by the critical exponents, we establish a concentration-compactness principle of Lions type for spaces associated with double phase operators, which is of independent interest to us. Our results are new, even in the case of constant exponents.

Versions of the drainage area ratio method for transfer of daily or monthly stream flows

Using the drainage area ratio (DAR) method, daily or monthly flows gauged at a nearby station are transferred to an ungauged section where a dam is to be built by multiplying them by a real number resulting from raising the ratio of the drainage areas of the dam section and of the station to an exponent n. Commonly, n is taken as 1. Using the daily flows of two stations nearby a dam section on the same stream or a neighbouring stream, having a common N-year-long gauging record, it is proposed that 365 linear regressions be fitted to N pairs of 365 daily flows and relate them to the DARs. The daily flows at the dam section can then be computed using 365 values of n' each belonging to 1 day. Next, using the same analytical model applied to monthly flows, 12 monthly flows are transferred to the dam section. Five versions of the DAR method – three of which have constant exponents – were investigated. The method with n = 1 was found to be the worst performer.

A new approach to the correlation of SPT-CPT depending on the soil behavior type index

In this study, a model based on soil behavior type index (Ic) has been proposed for the conversion of field cone penetration test (CPT) data to the standard penetration test (SPT) blow counts (N). The field data used in the study were obtained from the fluvial soils formed by the Sakarya river in Adapazarı (Turkey). According to the proposed model, it is understood that there are exponential relationships between SPT-N and CPT data with varying coefficients (constant multipliers and exponents) for different Ic intervals. While the coefficients of the equations were kept constant in the previous formulations developed based on Ic, the current study revealed that the mentioned coefficients should be changed for different Ic intervals. The normalized cone resistance (qt/ Pa) or corrected SPT-N values (N60), found with the model created with the final coefficients, were evaluated. The results obtained by using the new model proposed in this paper were compared with other models generally accepted in the literature. It has been revealed that the results achieved by this new model are the closest ones to the measured values compared to the other methods. To verify the proposed model, CPT-SPT data pairs that were not used in the model generation were analyzed. Verification analyses confirmed that the developed method has given more realistic results than the other methods suggested in the literature. It was understood that the error rate in the model, in which conversion between CPT and SPT data can be made by considering the variability in the soil behavior type index, is lower than the others. As a result, the paper gives a new perspective by introducing a simple and newly updated form of estimating N60 values.

Fixed-time observer-based homogeneous controller with state-dependent exponent for fault tolerant control of an underwater vehicle

This paper is concerned with designing a fault tolerant control system in the presence of unknown vehicle dynamics and saturation and rate limits of the thrusters for an underwater remotely operated vehicle. For this, a novel variable-exponent finite-time controller based on homogeneity is proposed for the underwater vehicle. The error-dependent exponent is introduced to provide better performance in transient and obtain faster converge to the desired states. Stability analyses are carried out for the controller and it is proven that system states will converge to the origin in finite time. In addition, phase plane analyses are performed and it is shown that system trajectories converge to the origin faster under proposed controller compared to controller with constant exponents. A fixed-time extended state observer with a new fault detection and identification unit is introduced which provides faster estimation of failures in transients despite the presence of uncertainties and disturbances. State estimation error dynamics is proven to be fixed-time convergent and fault estimation error is globally uniformly ultimately bounded. Simulations are carried out and comparisons are made with several FTCSs. The results show that the performance of the proposed algorithm is superior to previous works both in transients and upon the failures.

Spatiotemporal Distribution of Precipitates and Mineral Phase Transition During Biomineralization Affect Porosity–Permeability Relationships

Enzymatically induced calcium carbonate precipitation is a promising geotechnique with the potential, for example, to seal leakage pathways in the subsurface or to stabilize soils. Precipitation of calcium carbonate in a porous medium reduces the porosity and, consequently, the permeability. With pseudo-2D microfluidic experiments, including pressure monitoring and, for visualization, optical microscopy and X-ray computed tomography, pore-space alterations were reliably related to corresponding hydraulic responses. The study comprises six experiments with two different pore structures, a simple, quasi-1D structure, and a 2D structure. Using a continuous injection strategy with either constant or step-wise reduced flow rates, we identified key mechanisms that significantly influence the relationship between porosity and permeability. In the quasi-1D structure, the location of precipitates is more relevant to the hydraulic response (pressure gradients) than the overall porosity change. In the quasi-2D structure, this is different, because flow can bypass locally clogged regions, thus leading to steadier porosity–permeability relationships. Moreover, in quasi-2D systems, during continuous injection, preferential flow paths can evolve and remain open. Classical porosity–permeability power-law relationships with constant exponents cannot adequately describe this phenomenon. We furthermore observed coexistence and transformation of different polymorphs of calcium carbonate, namely amorphous calcium carbonate, vaterite, and calcite and discuss their influence on the observed development of preferential flow paths. This has so far not been accounted for in the state-of-the-art approaches for porosity–permeability relationships during calcium carbonate precipitation in porous media.

Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials

Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient matrix. Secondly, we apply such an asymptotic theory for the analysis of Floquet metamaterials. We provide a characterization of asymptotic exceptional points for a pair of subwavelength resonators with time-dependent material parameters. We prove that asymptotic exceptional points are obtained if the frequency components of the perturbations fulfill a certain ratio, which is determined by the geometry of the dimer of subwavelength resonators.

On asymptotic behavior for a class of diffusion equations involving the fractional $$\wp (\cdot )$$-Laplacian as $$\wp (\cdot )$$ goes to $$\infty $$

In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $$\wp (\cdot )$$ -Laplacian operator involving constant/variable exponent, with $$\wp ^{-}:=\min _{(x,y) \in {\overline{\Omega }}\times {\overline{\Omega }}} \wp (x,y)\geqslant \max \left\{ 2N/(N+2s),1\right\} $$ and $$s\in (0,1).$$ In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $$\wp $$ goes to $$\infty $$ . Already, for case the weighted fractional $$\wp (\cdot )$$ -Laplacian operator, we will also study the asymptotic behavior of the problem solution when $$\wp (\cdot )$$ goes to $$\infty $$ , in the whole or in a subset of the domain (the problem involving the fractional $$\wp (\cdot )$$ -Laplacian presents a discontinuous exponent). To obtain the results of the asymptotic behavior in both problems it will be via Mosco convergence.

Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study

<abstract><p>In this work, we consider a viscoelastic wave equation with boundary damping and variable exponents source term. The damping terms and variable exponents are localized on a portion of the boundary. We first, prove the existence of global solutions and then we establish optimal and general decay estimates depending on the relaxation function and the nature of the variable exponent nonlinearity. Finally, we run two numerical tests to demonstrate our theoretical decay results. This study generalizes and enhances existing literature results, and the acquired results are thus of significant importance when compared to previous literature results with constant or variable exponents in the domain.</p></abstract>

Anisotropic parabolic problem with variable exponent and regular data

In this paper, we study the existence of weak solutions for a class of nonlinear parabolic equations with regular data in the setting of variable exponent Sobolev spaces. We prove a "version" of a weak Lebesgue space estimate that goes back to "Lions J. L. Quelques méthodes de résolution des problèmes aux limites. Dunod, Paris (1969)" for parabolic equations with anisotropic constant exponents (pi(⋅)=pi).

Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

We study a characterization of the precompactness of sets in variable exponent Morrey spaces on bounded metric measure spaces. Totally bounded sets are characterized from several points of view for the case of variable exponent Morrey spaces over metric measure spaces. This characterization is new in the case of constant exponents.

Effects of temperature on metabolic scaling in silver carp.

The association between temperature and metabolic scaling varies among species, which could be due to variation in the surface area and its scaling. This study aims to examine the effect of temperature on metabolic scaling and to verify the links between metabolic scaling and surface area scaling at both the whole body and the cell levels. The routine metabolic rate (RMR), gill surface area (GSA), ventilation frequency (VF), red blood cell surface area (SRBC ), and metabolic rate (MRRBC ) were determined in silver carp, and their mass-scaling exponents were analyzed at 10 and 25°C. These results showed that body mass and temperature independently affected the RMR, GSA, and VF, suggesting constant scaling exponents of RMR (0.772), GSA (0.912), and VF (-0.282) with changing temperature. The RMR at 25°C was 2.29 times higher than that at 10°C, suggesting increased metabolic demand at a higher temperature. The results showed that the RMR increased, while the scaling exponents of RMR, GSA, and VF remained unchanged with increasing temperature. These results support the view that the scaling of oxygen supply capacity importantly affects metabolic scaling. The SRBC did not change with either temperature or body mass. However, the MRRBC increased by 5.48 times from 10to 25°C but did not change with body mass. As the scaling exponents of RMR did not change between temperatures, the results indicate that no obvious link exists between the scaling of both the cell size and cell metabolic rate and the metabolic scaling of silver carp.

Numerical Study on Tip Vortex Cavitation Inception on a Foil

In this paper, the inception of tip vortex cavitation in weak water has been predicted using a numerical simulation, and a new scaling concept with variable exponent has also been suggested for cavitation inception index. The numerical simulations of the cavitating flows over an elliptic planform hydrofoil were performed by using the RANS approach with a Eulerian cavitation model. To ensure the accuracy of the present simulations, the effects of the turbulence model and grid resolution on the tip vortex flows were investigated. The turbulence models behaved differently in the boundary layer of the tip region where the tip vortex is developed, which resulted in different pressure and velocity fields in the vortex region. Furthermore, the Reynolds stress model for the finest grid showed a better agreement with the experimental data. The tip vortex cavitation inception numbers for the foil, predicted by using both wetted and cavitating flow simulation approaches, were compared with the measured cavitation index values, showing a good correlation. The current cavitation scaling study also suggested new empirical relations as a function of the Reynolds number substitutable for the two classic constant scaling exponents. This scaling concept showed how the scaling law changes with the Reynolds number and provided a proper scaling value for any given Reynolds numbers under turbulent flow conditions.

Development of Intensity-Duration-Depth Relationships for Western Ghats in Karnataka, South India: A Pragmatic Approach for Utilizing Short Records

Popular models currently available for estimation of design intensities in the wet mountainous Western Ghat areas of South India are not those based on analysis of local rainfall records. Possibilities of development of i-d-f curves applicable in the region are also remote due to the constraint of short length of records. The present work, taken up in this background, presents the novel idea of substituting daily rainfall depth for frequency in the popular i-d-f curves and furnishes a reliable model for the Western Ghat regions of the state of Karnataka. The procedure advocated for the development of the model relies on the fact that in areas characterized by nearly continuous falls and long spells of rain, as here, the intensities are well associated with total rainfall and hence the i-d relationships are functions of daily depth. This work proposes to use all the available subhourly data from different stations across supposedly homogenous regions and to use envelop curves to reduce sampling variations. i-d curves for individual days falling within specified ranges of daily depth are grouped together, and each group is represented by an envelop curve. Power functions, forced through constant exponents, are developed for the different envelop curves. The association between the varying parameter of such curves and the daily depth (D) is modeled by a power function, resulting in i-d-D relationships of the form i=a(Db/dc). The model applicable for the study area has been derived using intensity—duration data of 4–5 years from 33 stations spread across the study area, divided into four regions based on normal annual rainfall. The model has been validated and found reliable. The model results are discussed and used to test the validity of the method presently being followed in the region, the one advocated by the Central Water Commission (CWC). Pitfalls in the CWC method have been brought out. It is concluded that the model presented forms a reliable tool for estimating design intensities until a more authentic one is developed using further extensive data.

Affinity laws for impellers trimmed in two partial emission pumps with very low specific speed

Partial emission pumps or open impeller pumps or tangent pumps with very low specific speed at either fast or conventional speed have found extensive applications in aircraft, liquid rockets, cryogenic fluid systems, chemical and petroleum-chemical industries, energy and food processing systems, etc. Usually, impeller trimming of rotodynamic pumps serves as an effective tool to meet required hydraulic performance of a liquid transport system and also to improve the entire system energy utilization efficiency. However, the affinity laws for the impeller trimming in a partial emission pump have been unavailable so far. In the paper, such affinity laws in terms of constant and variable exponents were established for flow rate, head, efficiency, flow coefficient and head coefficient based on the existing experimental data of two partial emission pumps at best efficiency points. It was shown that the affinity laws differ from the counterparts for centrifugal pumps, especially the exponent of around 2 for flow rate and the exponent of approximately 1.5 for head in comparison with nearly 1.5 for flow rate and about 2 for head in centrifugal pumps. The underlying mechanism for this effect was disclosed in terms of the ratios of the hydraulic, volumetric and mechanical efficiencies after trimming to the efficiencies before trimming analytically. The hydraulic loss, leakage flow rate and recirculation flow rate in two pumps were estimated according to the elements of fluid mechanics. It was identified that the hydraulic loss in the volute is more dominant than in the impeller and responsible for the rise of hydraulic efficiency with the trimming in progress. Moreover, the significant increase of recirculation flow rate between the volute tongue and the impeller outlet contributes to the substantial reduction in flow rate after impeller trimming.

Certain results for a class of nonlinear functional spaces

In this article, we study properties of a class of functional spaces, so-called pn-spaces, which arise from investigation of nonlinear differential equations. We establish some integral inequalities to analyse the structures of the pn-spaces with the constant and variable exponent. We prove embedding theorems, which indicate the relation of these spaces with the well known classical Lebesgue and Sobolev spaces with the constant and variable exponents.

Permeability Prediction in Rocks Experiencing Mineral Precipitation and Dissolution: A Numerical Study

AbstractIn this study, we focus on the electrical tortuosity‐based permeability model k = reff2/8F (reff is an effective pore size, and F is the formation factor) and analyze its applicability to rocks experiencing mineral precipitation and dissolution. Two limiting cases of advection‐dominated water‐rock reactions are simulated, that is, the reaction‐limited and transport‐limited cases. At the pore scale, the two precipitation/dissolution patterns are simulated with a geometrical model and a phenomenological model. The fluid and electric flows in the rocks are simulated by directly solving the linear Stokes equation and Laplace equation on the representative elementary volume of the samples. The numerical results show that evolutions of k and F differ significantly in the two limiting cases. In general, the reaction‐limited precipitation/dissolution would result in a smooth variation of k and F, which can be roughly modeled with a power function of porosity ϕ with a constant exponent. In contrast, the transport‐limited precipitation/dissolution mostly occurs near the pore throats where the fluid velocity is high. This induces a sharp change in k and F despite a minor variation in ϕ. The commonly used power laws with constant exponents are not able to describe such variations. The results also reveal that the electrical tortuosity‐based permeability prediction generally works well for rocks experiencing precipitation/dissolution if reff can be appropriately estimated, for example, with the electrical field normalized pore size Λ. The associated prediction errors are mainly due to the use of electrical tortuosity, which might be considerably larger than the true hydraulic tortuosity.