427 research outputs found
Applications of the DFLU flux to systems of conservation laws
The DFLU numerical flux was introduced in order to solve hyperbolic scalar
conservation laws with a flux function discontinuous in space. We show how this
flux can be used to solve systems of conservation laws. The obtained numerical
flux is very close to a Godunov flux. As an example we consider a system
modeling polymer flooding in oil reservoir engineering
On the Brezis-Lieb Lemma without pointwise convergence
Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of
the gap between the integral for a sequence and the integral for its pointwise
limit. This note studies the question if such gap can be evaluated when there
is no a.e. convergence. In particular, it gives the same lower bound for the
gap in L^p as the gap in the Brezis-Lieb lemma (including the case
vector-valued functions) provided that p is greater or equal than 3 and the
sequence converges both weakly and weakly in the sense of a duality map. It
also shows that the statement is false if p<3. An application is given in form
of a Brezis-Lieb lemma for gradients
On a version of Trudinger-Moser inequality with M\"obius shift invariance
The paper raises a question about the optimal critical nonlinearity for the
Sobolev space in two dimensions, connected to loss of compactness, and
discusses the pertinent concentration compactness framework. We study
properties of the improved version of the Trudinger-Moser inequality on the
open unit disk , recently proved by G. Mancini and K. Sandeep.
Unlike the original Trudinger-Moser inequality, this inequality is invariant
with respect to M\"obius automorphisms of the unit disk, and as such is a
closer analogy of the critical nonlinearity in the higher
dimension than the original Trudinger-Moser nonlinearity.Comment: This version gives the credit to an independently proved result,
missed in the early version, and corrects an error in one of the proof
On the best constant of Hardy-Sobolev Inequalities
We obtain the sharp constant for the Hardy-Sobolev inequality involving the
distance to the origin. This inequality is equivalent to a limiting
Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the
sharp constant coincides with the best Sobolev constant
On the upstream mobility scheme for two-phase flow in porous media
When neglecting capillarity, two-phase incompressible flow in porous media is
modelled as a scalar nonlinear hyperbolic conservation law. A change in the
rock type results in a change of the flux function. Discretizing in
one-dimensional with a finite volume method, we investigate two numerical
fluxes, an extension of the Godunov flux and the upstream mobility flux, the
latter being widely used in hydrogeology and petroleum engineering. Then, in
the case of a changing rock type, one can give examples when the upstream
mobility flux does not give the right answer.Comment: A preprint to be published in Computational Geoscience
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