The Experts below are selected from a list of 13710 Experts worldwide ranked by ideXlab platform
Monica Torres - One of the best experts on this subject based on the ideXlab platform.
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Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces
arXiv: Analysis of PDEs, 2020Co-Authors: Gui-qiang Chen, Monica TorresAbstract:The classical Gauss-Green Formula for the multidimensional case is generally stated for $C^{1}$ vector fields and domains with $C^{1}$ boundaries. However, motivated by the physical solutions with discontinuity/singularity for Partial Differential Equations (PDEs) and Calculus of Variations, such as nonlinear hyperbolic conservation laws and Euler-Lagrange equations, the following fundamental issue arises: Does the Gauss-Green Formula still hold for vector fields with discontinuity/singularity (such as divergence-measure fields) and domains with rough boundaries? The objective of this paper is to provide an answer to this issue and to present a short historical review of the contributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss-Green Formula possible.
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Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws
Archive for Rational Mechanics and Analysis, 2005Co-Authors: Gui-qiang Chen, Monica TorresAbstract:Divergence-measure fields in L ^∞ over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green Formula over sets of finite perimeter is established for divergence-measure fields in L ^∞. The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in Chen & Frid [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence-measure fields in L ^∞ over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.
Jie Xiao - One of the best experts on this subject based on the ideXlab platform.
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Ringel-Hall Algebras Beyond their Quantum Groups I: Restriction Functor and Green Formula
Algebras and Representation Theory, 2018Co-Authors: Jie Xiao, Minghui ZhaoAbstract:In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green Formula and the restriction functor. By a geometric way to prove the Green Formula, we show that the compatibility of multiplication and comultiplication of a Ringel-Hall algebra can be categorified under Lusztig’s framework.
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Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green's Formula
arXiv: Representation Theory, 2016Co-Authors: Jie Xiao, Minghui ZhaoAbstract:In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green Formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green Formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.
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Green Formula in Hall algebras and cluster algebras
arXiv: Representation Theory, 2008Co-Authors: Jie XiaoAbstract:The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Green's Formula. In this paper, three forms of Green's Formula are highlighted, (1) the original form of Green's Formula \cite{Green}\cite{RingelGreen}, (2) the degeneration form of Green's Formula \cite{DXX} and (3) the projective form of Green's Formula \cite{XX2007a} i.e. Green Formula with a $\bbc^{*}$-action.
Minghui Zhao - One of the best experts on this subject based on the ideXlab platform.
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Ringel-Hall Algebras Beyond their Quantum Groups I: Restriction Functor and Green Formula
Algebras and Representation Theory, 2018Co-Authors: Jie Xiao, Minghui ZhaoAbstract:In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green Formula and the restriction functor. By a geometric way to prove the Green Formula, we show that the compatibility of multiplication and comultiplication of a Ringel-Hall algebra can be categorified under Lusztig’s framework.
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Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green's Formula
arXiv: Representation Theory, 2016Co-Authors: Jie Xiao, Minghui ZhaoAbstract:In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green Formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove the Green Formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.
Gui-qiang Chen - One of the best experts on this subject based on the ideXlab platform.
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Divergence-Measure Fields: Gauss-Green Formulas and Normal Traces
arXiv: Analysis of PDEs, 2020Co-Authors: Gui-qiang Chen, Monica TorresAbstract:The classical Gauss-Green Formula for the multidimensional case is generally stated for $C^{1}$ vector fields and domains with $C^{1}$ boundaries. However, motivated by the physical solutions with discontinuity/singularity for Partial Differential Equations (PDEs) and Calculus of Variations, such as nonlinear hyperbolic conservation laws and Euler-Lagrange equations, the following fundamental issue arises: Does the Gauss-Green Formula still hold for vector fields with discontinuity/singularity (such as divergence-measure fields) and domains with rough boundaries? The objective of this paper is to provide an answer to this issue and to present a short historical review of the contributions by many mathematicians spanning more than two centuries, which have made the discovery of the Gauss-Green Formula possible.
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Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws
Archive for Rational Mechanics and Analysis, 2005Co-Authors: Gui-qiang Chen, Monica TorresAbstract:Divergence-measure fields in L ^∞ over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green Formula over sets of finite perimeter is established for divergence-measure fields in L ^∞. The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in Chen & Frid [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence-measure fields in L ^∞ over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.
D. K. Yablochkin - One of the best experts on this subject based on the ideXlab platform.
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Potential Theory for a Nonlinear Equation of the Benjamin-Bona-Mahoney-Burgers Type
Computational Mathematics and Mathematical Physics, 2019Co-Authors: M. O. Korpusov, D. K. YablochkinAbstract:For the linear part of a nonlinear equation related to the well-known Benjamin–Bona–Mahoney–Burgers (BBMB) equation, a fundamental solution is constructed, which is combined with the second Green Formula to obtain a third Green Formula in a bounded domain. Then a third Green Formula in the entire space is derived by passage to the limit in some class of functions. The properties of the potentials entering the Green Formula in the entire space are examined. The Cauchy problem for a nonlinear BBMB-type equation is considered. It is proved that finding its classical solution is equivalent to solving a nonlinear integral equation derived from the third Green Formula. The unique local-in-time solvability of this integral equation is proved by applying the contraction mapping principle. Next, the local-in-time classical solvability of the Cauchy problem is proved using the properties of potentials. Finally, the nonlinear capacity method is used to obtain a global-in-time a priori estimate for classical solutions of the Cauchy problem.
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Potential Theory for a Nonlinear Equation of the Benjamin–Bona–Mahoney–Burgers Type
Computational Mathematics and Mathematical Physics, 2019Co-Authors: M. O. Korpusov, D. K. YablochkinAbstract:For the linear part of a nonlinear equation related to the well-known Benjamin–Bona–Mahoney–Burgers (BBMB) equation, a fundamental solution is constructed, which is combined with the second Green Formula to obtain a third Green Formula in a bounded domain. Then a third Green Formula in the entire space is derived by passage to the limit in some class of functions. The properties of the potentials entering the Green Formula in the entire space are examined. The Cauchy problem for a nonlinear BBMB-type equation is considered. It is proved that finding its classical solution is equivalent to solving a nonlinear integral equation derived from the third Green Formula. The unique local-in-time solvability of this integral equation is proved by applying the contraction mapping principle. Next, the local-in-time classical solvability of the Cauchy problem is proved using the properties of potentials. Finally, the nonlinear capacity method is used to obtain a global-in-time a priori estimate for classical solutions of the Cauchy problem.
