SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4375-4414, August 2024. Abstract. We investigate physical states of spin-1 Bose–Einstein condensate in [math] with mean-field interaction constant [math] and spin-exchange interaction constant [math], two conserved quantities, the number of atoms [math], and the total magnetization [math] are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among [math], [math], [math], and [math]. Furthermore, we show that the corresponding standing wave is strongly unstable. When the atoms are trapped in a harmonic potential, we prove the existence of ground states and excited states along with some precisely asymptotics. Besides, we get that the set of ground states is stable under the associated Cauchy flow while the excited state corresponds to a strongly unstable standing wave. Our results not only show some characteristics of three-dimensional spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field, but also support some experimental observations as well as numerical results on spin-1 BEC.
{"title":"Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in [math]","authors":"Menghui Li, Xiao Luo, Juncheng Wei, Maoding Zhen","doi":"10.1137/23m1572222","DOIUrl":"https://doi.org/10.1137/23m1572222","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4375-4414, August 2024. <br/> Abstract. We investigate physical states of spin-1 Bose–Einstein condensate in [math] with mean-field interaction constant [math] and spin-exchange interaction constant [math], two conserved quantities, the number of atoms [math], and the total magnetization [math] are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among [math], [math], [math], and [math]. Furthermore, we show that the corresponding standing wave is strongly unstable. When the atoms are trapped in a harmonic potential, we prove the existence of ground states and excited states along with some precisely asymptotics. Besides, we get that the set of ground states is stable under the associated Cauchy flow while the excited state corresponds to a strongly unstable standing wave. Our results not only show some characteristics of three-dimensional spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field, but also support some experimental observations as well as numerical results on spin-1 BEC.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4296-4323, August 2024. Abstract. We shall prove the global existence of weak solutions to an initial boundary value problem for a novel phase-field model which is an elliptic-parabolic coupled system. This model is proposed as an attempt to describe the motion of grain boundaries, a type of interface motion by interface diffusion driven by bulk free energy in elastically deformable solids. Its applications include important processes arising in materials science, e.g., sintering. In this model the evolution equation for an order parameter is a nonuniformly, degenerate parabolic equation of fourth order, which differs from the Cahn–Hilliard equation by a nonsmooth term of the gradient of the unknown.
{"title":"Global Solutions to an Initial-Boundary Value Problem of a Phase-Field Model for Motion of Grain Boundaries","authors":"Xingzhi Bian, Peicheng Zhu, Boling Guo, Ying Zhang","doi":"10.1137/22m1477775","DOIUrl":"https://doi.org/10.1137/22m1477775","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4296-4323, August 2024. <br/> Abstract. We shall prove the global existence of weak solutions to an initial boundary value problem for a novel phase-field model which is an elliptic-parabolic coupled system. This model is proposed as an attempt to describe the motion of grain boundaries, a type of interface motion by interface diffusion driven by bulk free energy in elastically deformable solids. Its applications include important processes arising in materials science, e.g., sintering. In this model the evolution equation for an order parameter is a nonuniformly, degenerate parabolic equation of fourth order, which differs from the Cahn–Hilliard equation by a nonsmooth term of the gradient of the unknown.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anna Dall’Acqua, Gaspard Jankowiak, Leonie Langer, Fabian Rupp
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4494-4529, August 2024. Abstract. The elastic energy of a bending-resistant interface depends on both its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham–Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal [math]-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.
{"title":"Conservation, Convergence, and Computation for Evolving Heterogeneous Elastic Wires","authors":"Anna Dall’Acqua, Gaspard Jankowiak, Leonie Langer, Fabian Rupp","doi":"10.1137/23m159086x","DOIUrl":"https://doi.org/10.1137/23m159086x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4494-4529, August 2024. <br/> Abstract. The elastic energy of a bending-resistant interface depends on both its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham–Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal [math]-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4530-4593, August 2024. Abstract. We study the well-posedness of an infinite-dimensional Hamilton–Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton–Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton–Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf–Lax variational representation of the solution.
{"title":"Infinite-Dimensional Hamilton–Jacobi Equations for Statistical Inference on Sparse Graphs","authors":"Tomas Dominguez, Jean-Christophe Mourrat","doi":"10.1137/22m1527696","DOIUrl":"https://doi.org/10.1137/22m1527696","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4530-4593, August 2024. <br/> Abstract. We study the well-posedness of an infinite-dimensional Hamilton–Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton–Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton–Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf–Lax variational representation of the solution.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform Far-Field Asymptotics of the Two-Layered Green Function in Two Dimensions and Application to Wave Scattering in a Two-Layered Medium","authors":"Long Li, Jiansheng Yang, Bo Zhang, Haiwen Zhang","doi":"10.1137/22m1525910","DOIUrl":"https://doi.org/10.1137/22m1525910","url":null,"abstract":"","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141361676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4223-4251, June 2024. Abstract. We consider a Laplace-type problem with a generalized impedance boundary condition of the form [math] on a flat part [math] of the boundary of a domain [math]. Here, [math] is the outward unit normal vector to [math], [math] is the impedance function, and [math] is the coordinate along [math]. Such problems appear, for example, in the modeling of small perturbations of the boundary. In the literature, the cases [math] or [math] have been investigated. In this work, we address situations where [math] contains the origin and [math] or [math] with [math]. In other words, we study cases where [math] vanishes at the origin and changes its sign. The main message is that the well-posedness (in the Fredholm sense) of the corresponding problems depends on the value of [math]. For [math], we show that the associated operators are Fredholm of index zero, while it is not the case when [math]. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl’s sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments that seem to corroborate the theorems.
{"title":"Generalized Impedance Boundary Conditions with Vanishing or Sign-Changing Impedance","authors":"Laurent Bourgeois, Lucas Chesnel","doi":"10.1137/23m1604217","DOIUrl":"https://doi.org/10.1137/23m1604217","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4223-4251, June 2024. <br/> Abstract. We consider a Laplace-type problem with a generalized impedance boundary condition of the form [math] on a flat part [math] of the boundary of a domain [math]. Here, [math] is the outward unit normal vector to [math], [math] is the impedance function, and [math] is the coordinate along [math]. Such problems appear, for example, in the modeling of small perturbations of the boundary. In the literature, the cases [math] or [math] have been investigated. In this work, we address situations where [math] contains the origin and [math] or [math] with [math]. In other words, we study cases where [math] vanishes at the origin and changes its sign. The main message is that the well-posedness (in the Fredholm sense) of the corresponding problems depends on the value of [math]. For [math], we show that the associated operators are Fredholm of index zero, while it is not the case when [math]. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl’s sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments that seem to corroborate the theorems.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4185-4222, June 2024. Abstract. We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. We show the existence of minimizers for nonlocal variational problems with classically defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; we demonstrate properties such as a Meyers–Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators.
{"title":"Nonlocal Problems with Local Boundary Conditions I: Function Spaces and Variational Principles","authors":"James M. Scott, Qiang Du","doi":"10.1137/23m1588111","DOIUrl":"https://doi.org/10.1137/23m1588111","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4185-4222, June 2024. <br/> Abstract. We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. We show the existence of minimizers for nonlocal variational problems with classically defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; we demonstrate properties such as a Meyers–Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3968-4005, June 2024. Abstract. In this paper, we prove the singularity formation for Poiseuille laminar flow of full Ericksen–Leslie system modeling nematic liquid crystal flows in dimension two. The singularity is due to the geometric effect at the origin.
{"title":"Singularity Formation for Full Ericksen–Leslie System of Nematic Liquid Crystal Flows in Dimension Two","authors":"Geng Chen, Tao Huang, Xiang Xu","doi":"10.1137/23m1571046","DOIUrl":"https://doi.org/10.1137/23m1571046","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3968-4005, June 2024. <br/> Abstract. In this paper, we prove the singularity formation for Poiseuille laminar flow of full Ericksen–Leslie system modeling nematic liquid crystal flows in dimension two. The singularity is due to the geometric effect at the origin.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4104-4142, June 2024. Abstract. We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near [math], and (iii) the solution exists globally. In a recent work, we identified 10 main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term. In this paper, we give a proof for the formula corresponding to the sector [math].
{"title":"Boussinesq’s Equation for Water Waves: Asymptotics in Sector V","authors":"C. Charlier, J. Lenells","doi":"10.1137/23m1587671","DOIUrl":"https://doi.org/10.1137/23m1587671","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4104-4142, June 2024. <br/> Abstract. We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near [math], and (iii) the solution exists globally. In a recent work, we identified 10 main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term. In this paper, we give a proof for the formula corresponding to the sector [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3924-3967, June 2024. Abstract. The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of one-dimensional compressible Euler equations is proved in the inviscid limit of compressible Navier–Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in [M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp. 55–146]. The proof contains two new ingredients: (1) a cut-off technique and the expanding property of rarefaction are used to overcome the errors generated by the viscosity related to inviscid rarefaction; (2) the error terms concerning the interactions between shock and rarefaction are controlled by the compressibility of shock, the decay of derivative of rarefaction, and the separation of shock and rarefaction as time increases.
SIAM 数学分析期刊》，第 56 卷第 3 期，第 3924-3967 页，2024 年 6 月。 摘要。可压缩欧拉方程熵解的唯一性是一个基本问题，也是一个具有挑战性的问题。本文在可压缩 Navier-Stokes 方程的不粘性极限中证明了一维可压缩欧拉方程的冲击波和稀释波复合解的唯一性。此外，通过[M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp.］证明包含两个新要素：(1) 使用截止技术和稀释的膨胀特性来克服与无粘性稀释有关的粘度所产生的误差；(2) 有关冲击和稀释之间相互作用的误差项受冲击的可压缩性、稀释导数的衰减以及随着时间的增加冲击和稀释的分离所控制。
{"title":"Uniqueness of Composite Wave of Shock and Rarefaction in the Inviscid Limit of Navier–Stokes Equations","authors":"Feimin Huang, Weiqiang Wang, Yi Wang, Yong Wang","doi":"10.1137/23m156584x","DOIUrl":"https://doi.org/10.1137/23m156584x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3924-3967, June 2024. <br/> Abstract. The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of one-dimensional compressible Euler equations is proved in the inviscid limit of compressible Navier–Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in [M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp. 55–146]. The proof contains two new ingredients: (1) a cut-off technique and the expanding property of rarefaction are used to overcome the errors generated by the viscosity related to inviscid rarefaction; (2) the error terms concerning the interactions between shock and rarefaction are controlled by the compressibility of shock, the decay of derivative of rarefaction, and the separation of shock and rarefaction as time increases.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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