5,689 research outputs found
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
The Cauchy problem for a multidimensional linear transport equation with
discontinuous coefficient is investigated. Provided the coefficient satisfies a
one-sided Lipschitz condition, existence, uniqueness and weak stability of
solutions are obtained for either the conservative backward problem or the
advective forward problem by duality. Specific uniqueness criteria are
introduced for the backward conservation equation since weak solutions are not
unique. A main point is the introduction of a generalized flow in the sense of
partial differential equations, which is proved to have unique jacobian
determinant, even though it is itself nonunique.Comment: 19-03-200
Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations
Existence and uniqueness of global in time measure solution for a one
dimensional nonlinear aggregation equation is considered. Such a system can be
written as a conservation law with a velocity field computed through a
selfconsistant interaction potential. Blow up of regular solutions is now well
established for such system. In Carrillo et al. (Duke Math J (2011)), a theory
of existence and uniqueness based on the geometric approach of gradient flows
on Wasserstein space has been developped. We propose in this work to establish
the link between this approach and duality solutions. This latter concept of
solutions allows in particular to define a flow associated to the velocity
field. Then an existence and uniqueness theory for duality solutions is
developped in the spirit of James and Vauchelet (NoDEA (2013)). However, since
duality solutions are only known in one dimension, we restrict our study to the
one dimensional case
Numerical methods for one-dimensional aggregation equations
We focus in this work on the numerical discretization of the one dimensional
aggregation equation \pa_t\rho + \pa_x (v\rho)=0, , in the
attractive case. Finite time blow up of smooth initial data occurs for
potential having a Lipschitz singularity at the origin. A numerical
discretization is proposed for which the convergence towards duality solutions
of the aggregation equation is proved. It relies on a careful choice of the
discretized macroscopic velocity in order to give a sense to the product . Moreover, using the same idea, we propose an asymptotic preserving
scheme for a kinetic system in hyperbolic scaling converging towards the
aggregation equation in hydrodynamical limit. Finally numerical simulations are
provided to illustrate the results
A relaxation model for liquid-vapor phase change with metastability
We propose a model that describes phase transition including meta\-stable
states present in the van der Waals Equation of State. From a convex
optimization problem on the Helmoltz free energy of a mixture, we deduce a
dynamical system that is able to depict the mass transfer between two phases,
for which equilibrium states are either metastable states, stable states or {a
coexistent state}. The dynamical system is then used as a relaxation source
term in an isothermal 44 two-phase model. We use a Finite Volume scheme
that treats the convective part and the source term in a fractional step way.
Numerical results illustrate the ability of the model to capture phase
transition and metastable states
The VizieR database of Astronomical Catalogues
VizieR is a database grouping in an homogeneous way thousands of astronomical
catalogues gathered since decades by the Centre de Donnees de Strasbourg (CDS)
and participating institutes. The history and current status of this large
collection is briefly presented, and the way these catalogues are being
standardized to fit in the VizieR system is described. The architecture of the
database is then presented, with emphasis on the management of links and of
accesses to very large catalogues. Several query interfaces are currently
available, making use of the ASU protocol, for browsing purposes or for use by
other data processing systems such as visualisation tools.Comment: 10 pages, 2 Postscript figures; to be published in A&A
A limitation of the hydrostatic reconstruction technique for Shallow Water equations
Because of their capability to preserve steady-states, well-balanced schemes
for Shallow Water equations are becoming popular. Among them, the hydrostatic
reconstruction proposed in Audusse et al. (2004), coupled with a positive
numerical flux, allows to verify important mathematical and physical properties
like the positivity of the water height and, thus, to avoid unstabilities when
dealing with dry zones. In this note, we prove that this method exhibits an
abnormal behavior for some combinations of slope, mesh size and water height.Comment: 7 page
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
Existence and uniqueness of global in time measure solution for the
multidimensional aggregation equation is analyzed. Such a system can be written
as a continuity equation with a velocity field computed through a
self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)),
a well-posedness theory based on the geometric approach of gradient flows in
measure metric spaces has been developed for mildly singular potentials at the
origin under the basic assumption of being lambda-convex. We propose here an
alternative method using classical tools from PDEs. We show the existence of a
characteristic flow based on Filippov's theory of discontinuous dynamical
systems such that the weak measure solution is the pushforward measure with
this flow. Uniqueness is obtained thanks to a contraction argument in transport
distances using the lambda-convexity of the potential. Moreover, we show the
equivalence of this solution with the gradient flow solution. Finally, we show
the convergence of a numerical scheme for general measure solutions in this
framework allowing for the simulation of solutions for initial smooth densities
after their first blow-up time in Lp-norms.Comment: 33 page
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