Well Posedness for Pressureless Flow

Abstract: We study the uniqueness problem for pressureless gases. Previous results on this topic are only known for the case when the initial data is assumed to be a bounded measurable function. This assumption is unnatural because the solution is in general a Radon measure. In this paper, the uniqueness of the weak solution is proved for the case when the initial data is a Radon measure. We show that, besides the Oleinik entropy condition, it is also important to require the energy to be weakly continuous initially. Our uniqueness result is obtained in the same functional space as the existence theorem.

[1]  Existence and uniqueness of discontinuous solutions defined by Lebesgue-Stieltjes integral , 1996 .

[2]  Xiaqi Ding,et al.  On the Cauchy problem of transportation equations , 1997 .

[3]  Laurent Boudin,et al.  A Solution with Bounded Expansion Rate to the Model of Viscous Pressureless Gases , 2000, SIAM J. Math. Anal..

[4]  B. François,et al.  Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness , 1999 .

[5]  F. Huang,et al.  Solutions containing delta-waves of Cauchy problems for a nonstrictly hyperbolic system , 1995 .

[6]  Y. Zel’dovich.  Gravitational instability: An Approximate theory for large density perturbations , 1969 .

[7]  Jiaxin Hu.  The Riemann Problem for Pressureless Fluid Dynamics with Distribution Solutions in Colombeau's Sense , 1998 .

[8]  Y. Brenier,et al.  Sticky Particles and Scalar Conservation Laws , 1998 .

[9]  F. Huang.  Existence and uniqueness of discontinuous solutions for a hyperbolic system , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  C. Dafermos.  Generalized Characteristics and the Structure of Solutions of Hyperbolic Conservation Laws , 1976 .

[11]  Zhen Wang,et al.  UNIQUENESS OF GENERALIZED SOLUTION FOR THE CAUCHY PROBLEM OF TRANSPORTATION EQUATIONS , 1997 .

[12]  E Weinan,et al.  Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics , 1996 .