By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space $ mathbb R_1timesmathbb P^n_f $ endowed with a weight function $ f $ and whose Riemannian base $ mathbb P^n $ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either $ mathbb R_1timesmathbb G^n $, where $ mathbb G^n $ stands for the so-called $ n $-dimensional Gaussian space (which is the Euclidean space $ mathbb R^n $ endowed with the Gaussian probability measure) or $ mathbb R_1timesmathbb H_f^n $, where $ mathbb H^n $ denotes the standard $ n $-dimensional hyperbolic space and $ f $ is the square of the distance function to a fixed point of $ mathbb H^n $, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over $ mathbb P^n $.
利用适当的liouvile型结果、适当的抛物性判据和漂移拉普拉斯算子的Omori-Yau极大值原理的一个版本，我们推导了具有权函数f的洛伦兹积空间$ mathbb R_1乘以$ mathbb P^n_f $中平均曲率流的完全类空平移孤子的唯一性和不存在性，该空间的黎曼底$ mathbb P^n $被假定为完备且具有非负Bakry-Émery-Ricci张量。当周围的空间是美元 mathbb R_1 * mathbb G ^ n,美元在 mathbb G ^ n代表美元所谓的n维高斯空间美元(这是欧几里得空间$ mathbb R ^ n具有高斯概率测度)美元或美元 mathbb R_1 * mathbb H_f ^ n美元 mathbb H ^ n表示美元的标准n维双曲空间和$ f $美元是距离的平方函数的不动点 mathbb H ^ n,美元我们得到了唯一性和非存在性结果的一些有趣的结果。特别地，我们得到了在$ mathbb P^n $上构造的整个类空间平移图的不存在性结果。
{"title":"Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density","authors":"M. Batista, Giovanni Molica Bisci, H. D. de Lima","doi":"10.3934/mine.2023054","DOIUrl":"https://doi.org/10.3934/mine.2023054","url":null,"abstract":"By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space $ mathbb R_1timesmathbb P^n_f $ endowed with a weight function $ f $ and whose Riemannian base $ mathbb P^n $ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either $ mathbb R_1timesmathbb G^n $, where $ mathbb G^n $ stands for the so-called $ n $-dimensional Gaussian space (which is the Euclidean space $ mathbb R^n $ endowed with the Gaussian probability measure) or $ mathbb R_1timesmathbb H_f^n $, where $ mathbb H^n $ denotes the standard $ n $-dimensional hyperbolic space and $ f $ is the square of the distance function to a fixed point of $ mathbb H^n $, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over $ mathbb P^n $.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in [13], where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.
我们证明了在加权Lebesgue空间$ L^q_w $中$ p $- laplace型退化抛物方程或奇异抛物方程的弱解梯度的局部Calderón-Zygmund型估计，该方程具有$ p > frac{2n}{n+2} $。引入了一个关于权值w的新条件，该条件依赖于抛物线型p -拉普拉斯问题的固有几何性质。我们的条件弱于2010年的条件，在那里得到了类似的估计。特别地，在p = 2 $的情况下，它与通常的抛物线$ A_q $权重的条件相同。
{"title":"Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces","authors":"Mikyoung Lee, J. Ok","doi":"10.3934/mine.2023062","DOIUrl":"https://doi.org/10.3934/mine.2023062","url":null,"abstract":"<abstract><p>We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in <sup>[<xref ref-type=\"bibr\" rid=\"b13\">13</xref>]</sup>, where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.
{"title":"On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces","authors":"P. Pucci, Letizia Temperini","doi":"10.3934/mine.2023007","DOIUrl":"https://doi.org/10.3934/mine.2023007","url":null,"abstract":"In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Giorgia Franchini, V. Ruggiero, F. Porta, L. Zanni
In the context of deep learning, the more expensive computational phase is the full training of the learning methodology. Indeed, its effectiveness depends on the choice of proper values for the so-called hyperparameters, namely the parameters that are not trained during the learning process, and such a selection typically requires an extensive numerical investigation with the execution of a significant number of experimental trials. The aim of the paper is to investigate how to choose the hyperparameters related to both the architecture of a Convolutional Neural Network (CNN), such as the number of filters and the kernel size at each convolutional layer, and the optimisation algorithm employed to train the CNN itself, such as the steplength, the mini-batch size and the potential adoption of variance reduction techniques. The main contribution of the paper consists in introducing an automatic Machine Learning technique to set these hyperparameters in such a way that a measure of the CNN performance can be optimised. In particular, given a set of values for the hyperparameters, we propose a low-cost strategy to predict the performance of the corresponding CNN, based on its behavior after only few steps of the training process. To achieve this goal, we generate a dataset whose input samples are provided by a limited number of hyperparameter configurations together with the corresponding CNN measures of performance obtained with only few steps of the CNN training process, while the label of each input sample is the performance corresponding to a complete training of the CNN. Such dataset is used as training set for a Support Vector Machines for Regression and/or Random Forest techniques to predict the performance of the considered learning methodology, given its performance at the initial iterations of its learning process. Furthermore, by a probabilistic exploration of the hyperparameter space, we are able to find, at a quite low cost, the setting of a CNN hyperparameters which provides the optimal performance. The results of an extensive numerical experimentation, carried out on CNNs, together with the use of our performance predictor with NAS-Bench-101, highlight how the proposed methodology for the hyperparameter setting appears very promising.
{"title":"Neural architecture search via standard machine learning methodologies","authors":"Giorgia Franchini, V. Ruggiero, F. Porta, L. Zanni","doi":"10.3934/mine.2023012","DOIUrl":"https://doi.org/10.3934/mine.2023012","url":null,"abstract":"In the context of deep learning, the more expensive computational phase is the full training of the learning methodology. Indeed, its effectiveness depends on the choice of proper values for the so-called hyperparameters, namely the parameters that are not trained during the learning process, and such a selection typically requires an extensive numerical investigation with the execution of a significant number of experimental trials. The aim of the paper is to investigate how to choose the hyperparameters related to both the architecture of a Convolutional Neural Network (CNN), such as the number of filters and the kernel size at each convolutional layer, and the optimisation algorithm employed to train the CNN itself, such as the steplength, the mini-batch size and the potential adoption of variance reduction techniques. The main contribution of the paper consists in introducing an automatic Machine Learning technique to set these hyperparameters in such a way that a measure of the CNN performance can be optimised. In particular, given a set of values for the hyperparameters, we propose a low-cost strategy to predict the performance of the corresponding CNN, based on its behavior after only few steps of the training process. To achieve this goal, we generate a dataset whose input samples are provided by a limited number of hyperparameter configurations together with the corresponding CNN measures of performance obtained with only few steps of the CNN training process, while the label of each input sample is the performance corresponding to a complete training of the CNN. Such dataset is used as training set for a Support Vector Machines for Regression and/or Random Forest techniques to predict the performance of the considered learning methodology, given its performance at the initial iterations of its learning process. Furthermore, by a probabilistic exploration of the hyperparameter space, we are able to find, at a quite low cost, the setting of a CNN hyperparameters which provides the optimal performance. The results of an extensive numerical experimentation, carried out on CNNs, together with the use of our performance predictor with NAS-Bench-101, highlight how the proposed methodology for the hyperparameter setting appears very promising.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.
{"title":"On the low Mach number limit for 2D Navier–Stokes–Korteweg systems","authors":"L. Hientzsch","doi":"10.3934/mine.2023023","DOIUrl":"https://doi.org/10.3934/mine.2023023","url":null,"abstract":"This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case respectively. One group of weak solutions is Aleksandrov solution, Brenier solution and $ C $-viscosity solution. The other group of weak solutions is Trudinger solution and $ L^p $-viscosity solution.
{"title":"Weak solutions of generated Jacobian equations","authors":"F. Jiang","doi":"10.3934/mine.2023064","DOIUrl":"https://doi.org/10.3934/mine.2023064","url":null,"abstract":"We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case respectively. One group of weak solutions is Aleksandrov solution, Brenier solution and $ C $-viscosity solution. The other group of weak solutions is Trudinger solution and $ L^p $-viscosity solution.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
where $ 1 < lambda leq p $, $ r > 1 $ and $ f in L^1(Omega) $.
In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE begin{document}$ begin{equation*} { } -{{{rm{;div}}}}big(a(x),nabla u|nabla u|^{p-2}big) -{{{rm{;div}}}}big( |u|^{(r-1)lambda+1}nabla u|nabla u|^{lambda-2}big) = f end{equation*} $end{document} where $ 1 < lambda leq p $, $ r > 1 $ and $ f in L^1(Omega) $.
{"title":"Regularizing effect in some double phase problems","authors":"L. Boccardo, G. R. Cirmi","doi":"10.3934/mine.2023069","DOIUrl":"https://doi.org/10.3934/mine.2023069","url":null,"abstract":"<abstract><p>In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE</p> <p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} { } -{{{rm{;div}}}}big(a(x),nabla u|nabla u|^{p-2}big) -{{{rm{;div}}}}big( |u|^{(r-1)lambda+1}nabla u|nabla u|^{lambda-2}big) = f end{equation*} $end{document} </tex-math></disp-formula></p> <p>where $ 1 < lambda leq p $, $ r > 1 $ and $ f in L^1(Omega) $.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. Placidi, E. Barchiesi, F. dell’Isola, V. Maksimov, A. Misra, N. Rezaei, Angelo Scrofani, D. Timofeev
We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.
{"title":"On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure","authors":"L. Placidi, E. Barchiesi, F. dell’Isola, V. Maksimov, A. Misra, N. Rezaei, Angelo Scrofani, D. Timofeev","doi":"10.3934/mine.2023021","DOIUrl":"https://doi.org/10.3934/mine.2023021","url":null,"abstract":"We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a variational problem for hypersurfaces in a wedge in the Euclidean space. Our wedge is bounded by a finitely many hyperplanes passing a common point. The total energy of each hypersurface is the sum of its anisotropic surface energy and the wetting energy of the planar domain bounded by the boundary of the considered hypersurface. An anisotropic surface energy is a generalization of the surface area which was introduced to model the surface tension of a small crystal. We show an existence and uniqueness result of local minimizers of the total energy among hypersurfaces enclosing the same volume. Our result is new even when the special case where the surface energy is the surface area.
{"title":"Stable anisotropic capillary hypersurfaces in a wedge","authors":"Miyuki Koiso","doi":"10.3934/mine.2023029","DOIUrl":"https://doi.org/10.3934/mine.2023029","url":null,"abstract":"We study a variational problem for hypersurfaces in a wedge in the Euclidean space. Our wedge is bounded by a finitely many hyperplanes passing a common point. The total energy of each hypersurface is the sum of its anisotropic surface energy and the wetting energy of the planar domain bounded by the boundary of the considered hypersurface. An anisotropic surface energy is a generalization of the surface area which was introduced to model the surface tension of a small crystal. We show an existence and uniqueness result of local minimizers of the total energy among hypersurfaces enclosing the same volume. Our result is new even when the special case where the surface energy is the surface area.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops the bitensorial formulation of the system of singularities associated with unbounded and bounded Stokes flows. The motivation for this extension is that Stokesian singularities and hydrodynamic fundamental solutions are multi-point functions, and bitensor calculus provides either the proper geometrical setting, in order to avoid inconsistencies and misunderstandings on the role of the different tensorial indices, or a way for compactly deriving hydrodynamic properties. A first relevant result is to provide a clear definition of the singularities (both bounded and unbounded) in Stokes flow, specifying the associated differential equations and boundary conditions. Using this formalism for bounded flows, we show the existence of an integro-differential operator providing the whole system of hydrodynamic singularities by acting on the unbounded Green function (Stokeslet) at its pole and we derive its explicit representation in terms of moments. In the case of an immersed body in a unbounded fluid, we show that, the operator furnishing the disturbance field of a purely $ n $-th order ambient flow, is a generalized $ n $-th order Faxén operator, i.e., it yields the $ n $-th moment on the body if applied to a generic ambient flow, and that a generic disturbance field can be expressed by a summation of the generalized $ n $-th order Faxén operators. Furthermore, we find that the operator providing the disturbance of an ambient flow coincides with the reflection operator for the Stokes solutions in the same flow geometry. We apply this result to the paradigmatic case of fundamental singularities for the Stokes flow bounded by a plane. In this way, we obtain in an alternative and easy way the image system for the Sourcelet and the Rotlet (already derived in the literature) and for the Source Doublet and the Strainlet (presented here for the first time).
{"title":"Bitensorial formulation of the singularity method for Stokes flows","authors":"Giuseppe Procopio, M. Giona","doi":"10.3934/mine.2023046","DOIUrl":"https://doi.org/10.3934/mine.2023046","url":null,"abstract":"This paper develops the bitensorial formulation of the system of singularities associated with unbounded and bounded Stokes flows. The motivation for this extension is that Stokesian singularities and hydrodynamic fundamental solutions are multi-point functions, and bitensor calculus provides either the proper geometrical setting, in order to avoid inconsistencies and misunderstandings on the role of the different tensorial indices, or a way for compactly deriving hydrodynamic properties. A first relevant result is to provide a clear definition of the singularities (both bounded and unbounded) in Stokes flow, specifying the associated differential equations and boundary conditions. Using this formalism for bounded flows, we show the existence of an integro-differential operator providing the whole system of hydrodynamic singularities by acting on the unbounded Green function (Stokeslet) at its pole and we derive its explicit representation in terms of moments. In the case of an immersed body in a unbounded fluid, we show that, the operator furnishing the disturbance field of a purely $ n $-th order ambient flow, is a generalized $ n $-th order Faxén operator, i.e., it yields the $ n $-th moment on the body if applied to a generic ambient flow, and that a generic disturbance field can be expressed by a summation of the generalized $ n $-th order Faxén operators. Furthermore, we find that the operator providing the disturbance of an ambient flow coincides with the reflection operator for the Stokes solutions in the same flow geometry. We apply this result to the paradigmatic case of fundamental singularities for the Stokes flow bounded by a plane. In this way, we obtain in an alternative and easy way the image system for the Sourcelet and the Rotlet (already derived in the literature) and for the Source Doublet and the Strainlet (presented here for the first time).","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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