A new proof of indefinite propagation of singularities for a Hamilton–Jacobi equation

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• 作者：Thomas Strömberg
• 关键词：Hamilton–Jacobi equation ; Generalized characteristic ; Propagation of singularities
• 刊名：Journal of Evolution Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：16
• 期：4
• 页码：895-903
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1424-3202
• 卷排序：16

文摘

We study propagation of singularities for the Hamilton–Jacobi equation$$S_t+H(\nabla S) = 0,\quad (t,x) \in (0,T) \times \mathbb{R}^n,$$where \({H(p)=\frac{1}{2}\langle p,Ap\rangle}\) is a positive definite quadratic form. Each viscosity solution \({S}\) is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015), namely that the singularities propagate along generalized characteristics indefinitely forward in time.KeywordsHamilton–Jacobi equationGeneralized characteristicPropagation of singularitiesMathematics Subject Classification35F2135A2035D4049L25References1.Albano P.: Propagation of singularities for solutions of Hamilton-Jacobi equations. J. Math. Anal. Appl. 411, 684–687 (2014)MathSciNetCrossRefMATHGoogle Scholar2.P. Albano, P. Cannarsa, Propagation of singularities for concave solutions of Hamilton-Jacobi equations. International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 583–588, World Sci. Publ., River Edge, NJ, 2000.3.Albano P., Cannarsa P.: Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch. Ration. Mech. Anal. 162, 1–23 (2002)MathSciNetCrossRefMATHGoogle Scholar4.V.I. Arnold, Mathematical methods of classical mechanics, in: Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.5.Cannarsa P., Mazzola M., Sinestrari C.: Global propagation of singularities for time dependent Hamilton–Jacobi equations. Discrete Contin. Dyn. Syst. 35, 4225–4239 (2015)MathSciNetCrossRefMATHGoogle Scholar6.P. Cannarsa, C. Sinestrari, Semiconcave functions, Hamilton–Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA (2004).7.Cannarsa P., Yu Y.: Singular dynamics for semiconcave functions. J. Eur. Math. Soc. (JEMS) 11, 999–1024 (2009)MathSciNetCrossRefMATHGoogle Scholar8.Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67 (1992)MathSciNetCrossRefMATHGoogle Scholar9.Dafermos C.M.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26, 1097–1119 (1977)MathSciNetCrossRefMATHGoogle Scholar10.Fleming W.H.: The Cauchy problem for a nonlinear first order partial differential equation. J. Differential Equations 5, 515–530 (1969)MathSciNetCrossRefMATHGoogle Scholar11.K. Khanin, A. Sobolevski, On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Rational Mech. Anal. (2015). doi:10.1007/s00205-015-0910-x.12.P.-L. Lions, Generalized solutions of Hamilton–Jacobi equations. Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston–London, 1982.13.Rockafellar R.T.: Convex analysis. Princeton Mathematical Series. Princeton University Press, Princeton, N.J. (1970)Google Scholar14.Strömberg T.: The Hopf-Lax formula gives the unique viscosity solution. Differential Integral Equations 15, 47–52 (2002)MathSciNetMATHGoogle Scholar15.Strömberg T.: A system of the Hamilton–Jacobi and the continuity equations in the vanishing viscosity limit. Commun. Pure Appl. Anal. 10, 479–506 (2011)MathSciNetCrossRefMATHGoogle Scholar16.Strömberg T.: Propagation of singularities along broken characteristics. Nonlinear Anal. 85, 93–109 (2013)MathSciNetCrossRefMATHGoogle Scholar17.Strömberg T., Ahmadzadeh F.: Excess action and broken characteristics for Hamilton-Jacobi equations. Nonlinear Anal. 110, 113–129 (2014)MathSciNetCrossRefMATHGoogle Scholar18.Yu Y.: A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5, 439–444 (2006)MathSciNetMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsThomas Strömberg1Email author1.Department of Engineering Sciences and MathematicsLuleå University of TechnologyLuleåSweden About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1424-3199 Online ISSN 1424-3202 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; 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