11 research outputs found
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 introduced by the second author,
and propose a geometric version of our scheme; in turn, the spatial component
of a graph solution can be chosen to be continuous in both time and space,
while its component 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 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