Radon measure-valued solutions of first order scalar conservation laws

Author:

Bertsch Michiel1,Smarrazzo Flavia2,Terracina Andrea3,Tesei Alberto4

Affiliation:

1. Dipartimento di Matematica , Università di Roma “Tor Vergata” , Via della Ricerca Scientifica, 00133; and Istituto per le Applicazioni del Calcolo “M. Picone”, CNR , Roma , Italy

2. Università Campus Bio-Medico di Roma , Via Alvaro del Portillo 21, 00128 Roma , Italy

3. Dipartimento di Matematica “G. Castelnuovo” , Università “Sapienza” di Roma , P.le A. Moro 5, 00185 Roma , Italy

4. Dipartimento di Matematica “G. Castelnuovo” , Università “Sapienza” di Roma , P.le A. Moro 5, 00185; and Istituto per le Applicazioni del Calcolo “M. Picone”, CNR , Roma , Italy

Abstract

Abstract We study nonnegative solutions of the Cauchy problem { t u + x [ φ ( u ) ] = 0 in  × ( 0 , T ) , u = u 0 0 in  × { 0 } , \left\{\begin{aligned} &\displaystyle\partial_{t}u+\partial_{x}[\varphi(u)]=0&% &\displaystyle\phantom{}\text{in }\mathbb{R}\times(0,T),\\ &\displaystyle u=u_{0}\geq 0&&\displaystyle\phantom{}\text{in }\mathbb{R}% \times\{0\},\end{aligned}\right. where u 0 {u_{0}} is a Radon measure and φ : [ 0 , ) {\varphi\colon[0,\infty)\mapsto\mathbb{R}} is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u 0 {u_{0}} is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ ( u ) = u {\varphi(u)=u} this happens for all times). In the latter case, we describe the evolution of the singular parts.

Publisher

Walter de Gruyter GmbH

Subject

Analysis

Reference28 articles.

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2. J. M. Ball, A version of the fundamental theorem for Young measures, Partial Differential Equations and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, Springer, Berlin (1989), 207–215.

3. M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., to appear.

4. M. Bertsch, F. Smarrazzo and A. Tesei, Pseudo-parabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Anal. PDE 6 (2013), 1719–1754.

5. M. Bertsch, F. Smarrazzo and A. Tesei, On a pseudoparabolic regularization of a forward-backward-forward equation, Nonlinear Anal. 129 (2015), 217–257.

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