69 research outputs found
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems
A rigorous justification of several well-known mathematical models of incompressible fluid flows can be given in terms of singular limits of the scaled Navier-Stokes-Fourier system, where some of the characteristic numbers become small or large enough. We discuss the problem in the framework of global-in-time solutions for both the primitive and the target system. © 2010 Springer Basel AG
Metric theory of functional solutions of the Cauchy problem for a system of conservation laws
ON EXISTENCE AND UNIQUENESS OF ENTROPY SOLUTIONS TO THE CAUCHY PROBLEM FOR A CONSERVATION LAW WITH DISCONTINUOUS FLUX
EXISTENCE OF STRONG TRACES FOR GENERALIZED SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS
A note on strong solutions to the variational kinetic equation for scalar conservation laws
On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property
Properties of solutions of linear and quasilinear second-order equations with measurable coefficients which are neither strictly nor uniformly parabolic
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