Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation, and graph solutions

We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves, only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time-regularity of the graph solutions $(X,U)$ introduced by the second author, and propose a geometric version of our scheme; in turn, the spatial component $X$ of a graph solution can be chosen to be continuous in both time and space, while its component $U$ is continuous in space and has bounded variation in time.Comment: 74 page

Conservation Laws with Time Dependent Discontinuous Coefficients

We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal locations. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1 , and that the limits are entropy solutions. Then, using the definition of an entropy solution taken from [K. H. Karlsen, N. H. Risebro, and J. D. Towers, Skr. K. Nor. Vidensk. Selsk., 3 (2003), pp. 1–49], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [S. N. Kruˇzkov, Math. USSR-Sb., 10 (1970), pp. 217–243] and also partially those from Karlsen, Risebro, and Towers

A hyperbolic-elliptic model of two-phase flow in porous media-existence of entropy solutions

We consider the flow of two-phases in a porous medium and propose a modified version of the fractional flow model for incompressible, two-phase flow based on a Helmholtz regularization of the Darcy phase velocities. We show the existence of global-in-time entropy solutions for this model with suitable assumptions on the boundary conditions. Numerical experiments demonstrating the approximation of the classical two-phase flow equations with the new model are presented

Convergence of vanishing viscosity approximations of 2times22times 2 triangular systems of multi-dimensional conservation laws.

We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multi dimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multi-dimensional triangular system

Continuous dependence in the large for some equations of gas dynamics

Work partially supported by TMR project HCL no.ERBFMRXCT960033 of the European UnionConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Stability of a resonant system of conservation laws modeling polymer flow

We prove L"1 uniqueness and stability for a resonant 2 x 2 system of conservation laws that arise as a model for two phase polymer flow in porous media. The analysis uses the equivalence of the Eulerian and Lagrangian formulation of this system, and the results are first established for an auxiliary scalar equation. Our methods are based on front tracking approximations for the auxiliary equation, and the Kruzkov entropy condition for scalar conservation laws. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(99-16) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman