1,330 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
Exponential decay towards equilibrium and global classical solutions for nonlinear reaction-diffusion systems
We consider a system of reaction-diffusion equations describing the
reversible reaction of two species forming a third
species and vice versa according to mass action law kinetics with
arbitrary stochiometric coefficients (equal or larger than one).
Firstly, we prove existence of global classical solutions via improved
duality estimates under the assumption that one of the diffusion coefficients
of or is sufficiently close to the diffusion
coefficient of .
Secondly, we derive an entropy entropy-dissipation estimate, that is a
functional inequality, which applied to global solutions of these
reaction-diffusion system proves exponential convergence to equilibrium with
explicit rates and constants.Comment: 24 page
Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation
equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate
diffusion. The focus of this paper is the unification and generalization of the
well-posedness theory of these models. We prove local well-posedness on bounded
domains for dimensions and in all of space for , the
uniqueness being a result previously not known for PKS with degenerate
diffusion. We generalize the notion of criticality for PKS and show that
subcritical problems are globally well-posed. For a fairly general class of
problems, we prove the existence of a critical mass which sharply divides the
possibility of finite time blow up and global existence. Moreover, we compute
the critical mass for fully general problems and show that solutions with
smaller mass exists globally. For a class of supercritical problems we prove
finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page
Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling
We consider a model system consisting of two reaction-diffusion equations,
where one species diffuses in a volume while the other species diffuses on the
surface which surrounds the volume. The two equations are coupled via a
nonlinear reversible Robin-type boundary condition for the volume species and a
matching reversible source term for the boundary species. As a consequence of
the coupling, the total mass of the two species is conserved. The considered
system is motivated for instance by models for asymmetric stem cell division.
Firstly we prove the existence of a unique weak solution via an iterative
method of converging upper and lower solutions to overcome the difficulties of
the nonlinear boundary terms. Secondly, our main result shows explicit
exponential convergence to equilibrium via an entropy method after deriving a
suitable entropy entropy-dissipation estimate for the considered nonlinear
volume-surface reaction-diffusion system.Comment: 31 page
Global Entropy Solutions to the Gas Flow in General Nozzle
We are concerned with the global existence of entropy solutions for the
compressible Euler equations describing the gas flow in a nozzle with general
cross-sectional area, for both isentropic and isothermal fluids. New
viscosities are delicately designed to obtain the uniform bound of approximate
solutions. The vanishing viscosity method and compensated compactness framework
are used to prove the convergence of approximate solutions. Moreover, the
entropy solutions for both cases are uniformly bounded independent of time. No
smallness condition is assumed on initial data. The techniques developed here
can be applied to compressible Euler equations with general source terms
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