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Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities
Let \(H\) be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the fo...
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creator | Bertsch, M Smarrazzo, F Terracina, A Tesei, A |
description | Let \(H\) be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law. |
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We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Conservation laws ; Continuity (mathematics) ; Discontinuity ; Hamilton-Jacobi equation ; Linear equations ; Radon ; Singularities</subject><ispartof>arXiv.org, 2020-07</ispartof><rights>2020. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. 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We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record> |
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subjects | Conservation laws Continuity (mathematics) Discontinuity Hamilton-Jacobi equation Linear equations Radon Singularities |
title | Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities |
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