Pub Date : 2023-03-01DOI: 10.1142/s0219891623500030
Jianwei Dong, YI-JIE Meng
In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line [Formula: see text] for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line [Formula: see text] by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval [Formula: see text] by introducing some new average quantities.
{"title":"Blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations","authors":"Jianwei Dong, YI-JIE Meng","doi":"10.1142/s0219891623500030","DOIUrl":"https://doi.org/10.1142/s0219891623500030","url":null,"abstract":"In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line [Formula: see text] for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line [Formula: see text] by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval [Formula: see text] by introducing some new average quantities.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42054329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.1142/s0219891623500054
F. Berthelin
Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension [Formula: see text] and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties.
{"title":"Solutions of kinetic equations related to non-local conservation laws","authors":"F. Berthelin","doi":"10.1142/s0219891623500054","DOIUrl":"https://doi.org/10.1142/s0219891623500054","url":null,"abstract":"Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension [Formula: see text] and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.1142/s0219891623500066
Hans Lindblad, Volker Schlue
We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.
{"title":"Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition","authors":"Hans Lindblad, Volker Schlue","doi":"10.1142/s0219891623500066","DOIUrl":"https://doi.org/10.1142/s0219891623500066","url":null,"abstract":"We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47814235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.1142/s0219891623500029
Nobu Kishimoto, Y. Tsutsumi
We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.
{"title":"Well-posedness of the Cauchy problem for the kinetic DNLS on T","authors":"Nobu Kishimoto, Y. Tsutsumi","doi":"10.1142/s0219891623500029","DOIUrl":"https://doi.org/10.1142/s0219891623500029","url":null,"abstract":"We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42363529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.1142/s0219891622500187
Arthur Touati
In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying [Formula: see text] symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the [Formula: see text] level. In addition, we prove a blow-up criterium at the [Formula: see text] level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized [Formula: see text] symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.
{"title":"Einstein vacuum equations with 𝕌(1) symmetry in an elliptic gauge: Local well-posedness and blow-up criterium","authors":"Arthur Touati","doi":"10.1142/s0219891622500187","DOIUrl":"https://doi.org/10.1142/s0219891622500187","url":null,"abstract":"In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying [Formula: see text] symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the [Formula: see text] level. In addition, we prove a blow-up criterium at the [Formula: see text] level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized [Formula: see text] symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43745076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.1142/s021989162250014x
Hongjun Cheng, Hanchun Yang
This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations.
{"title":"Riemann problems for a hyperbolic system of nonlinear conservation laws from the Liou–Steffen pressure system","authors":"Hongjun Cheng, Hanchun Yang","doi":"10.1142/s021989162250014x","DOIUrl":"https://doi.org/10.1142/s021989162250014x","url":null,"abstract":"This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44164990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.1142/s0219891622500114
N. Mori, M. A. Jorge Silva
We address here a viscoelastic Timoshenko model on the (one-dimensional) real line with memory damping coupled on a shear force. Our main results concern a complete decay structure of the system under the so-called equal wave speeds assumption, as well as without this condition. This is the first result of this type for partially dissipative beam systems with memory-type damping on the shear force. Our method is based on expanded structural conditions such as the so-called SK condition. In addition, we give a characterization of the dissipative structure of the system by using a spectral analysis method, which confirms that our decay structure is optimal.
{"title":"Decay property for a novel partially dissipative viscoelastic beam system on the real line","authors":"N. Mori, M. A. Jorge Silva","doi":"10.1142/s0219891622500114","DOIUrl":"https://doi.org/10.1142/s0219891622500114","url":null,"abstract":"We address here a viscoelastic Timoshenko model on the (one-dimensional) real line with memory damping coupled on a shear force. Our main results concern a complete decay structure of the system under the so-called equal wave speeds assumption, as well as without this condition. This is the first result of this type for partially dissipative beam systems with memory-type damping on the shear force. Our method is based on expanded structural conditions such as the so-called SK condition. In addition, we give a characterization of the dissipative structure of the system by using a spectral analysis method, which confirms that our decay structure is optimal.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43837290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.1142/s0219891622500151
Libin Wang, Ke Wang
In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise [Formula: see text] solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small [Formula: see text] norm, we obtain the existence and uniqueness of global piecewise [Formula: see text] solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to [Formula: see text], then the corresponding global piecewise [Formula: see text] solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.
{"title":"Global piecewise classical solutions to quasilinear hyperbolic systems on a tree-like network","authors":"Libin Wang, Ke Wang","doi":"10.1142/s0219891622500151","DOIUrl":"https://doi.org/10.1142/s0219891622500151","url":null,"abstract":"In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise [Formula: see text] solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small [Formula: see text] norm, we obtain the existence and uniqueness of global piecewise [Formula: see text] solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to [Formula: see text], then the corresponding global piecewise [Formula: see text] solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48898133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.1142/s0219891622500084
Wen-Jian Peng, Tian-Yi Wang
The vanishing pressure limit of continuous solutions isentropic Euler equations is analyzed, which is formulated as small parameter [Formula: see text] goes to [Formula: see text]. Due to the characteristics being degenerated in the limiting process, the resonance may cause the mass concentration. It is shown that in the pressure vanishing process, for the isentropic Euler equations, the continuous solutions with compressive initial data converge to the mass concentration solution of pressureless Euler equations, and with rarefaction initial data converge to the continuous solutions globally. It is worth to point out: [Formula: see text] converges in [Formula: see text], while [Formula: see text] converges in [Formula: see text], due to the structure of pressureless Euler equations. To handle the blow-up of density [Formula: see text] and spatial derivatives of velocity [Formula: see text], a new level set argument is introduced. Furthermore, we consider the convergence rate with respect to [Formula: see text], both [Formula: see text] and the area of characteristic triangle are [Formula: see text] order, while the rates of [Formula: see text] and [Formula: see text] depend on the further regularity of the initial data of [Formula: see text].
{"title":"On vanishing pressure limit of continuous solutions to the isentropic Euler equations","authors":"Wen-Jian Peng, Tian-Yi Wang","doi":"10.1142/s0219891622500084","DOIUrl":"https://doi.org/10.1142/s0219891622500084","url":null,"abstract":"The vanishing pressure limit of continuous solutions isentropic Euler equations is analyzed, which is formulated as small parameter [Formula: see text] goes to [Formula: see text]. Due to the characteristics being degenerated in the limiting process, the resonance may cause the mass concentration. It is shown that in the pressure vanishing process, for the isentropic Euler equations, the continuous solutions with compressive initial data converge to the mass concentration solution of pressureless Euler equations, and with rarefaction initial data converge to the continuous solutions globally. It is worth to point out: [Formula: see text] converges in [Formula: see text], while [Formula: see text] converges in [Formula: see text], due to the structure of pressureless Euler equations. To handle the blow-up of density [Formula: see text] and spatial derivatives of velocity [Formula: see text], a new level set argument is introduced. Furthermore, we consider the convergence rate with respect to [Formula: see text], both [Formula: see text] and the area of characteristic triangle are [Formula: see text] order, while the rates of [Formula: see text] and [Formula: see text] depend on the further regularity of the initial data of [Formula: see text].","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45647062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We estimate the [Formula: see text] distance between piecewise constant solutions to the Cauchy problem of scalar conservation laws and propose a sufficient condition for having an [Formula: see text] contraction of such solutions. Moreover, we prove that there exist [Formula: see text] contractive solutions on a set of all monotone bounded initial functions to the Cauchy problem of scalar conservation laws with convex or concave flux functions.
{"title":"Lp Contractive solutions for scalar conservation laws","authors":"Kihito Hinohara, Natsuki Minagawa, Hiroki Ohwa, Hiroya Suzuki, Shou Ukita","doi":"10.1142/s0219891622500059","DOIUrl":"https://doi.org/10.1142/s0219891622500059","url":null,"abstract":"We estimate the [Formula: see text] distance between piecewise constant solutions to the Cauchy problem of scalar conservation laws and propose a sufficient condition for having an [Formula: see text] contraction of such solutions. Moreover, we prove that there exist [Formula: see text] contractive solutions on a set of all monotone bounded initial functions to the Cauchy problem of scalar conservation laws with convex or concave flux functions.","PeriodicalId":50182,"journal":{"name":"Journal of Hyperbolic Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45259205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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