26,541 research outputs found
Riemann Problem for a limiting system in elastodynamics
In this article, we discuss about the resolution of the Riemann problem for a
2x2 system in nonconservative form exhibiting parabolic degeneracy. The system
can be perceived as the limiting equation (depending on a parameter tending to
0) of a 2x2 strictly hyperbolic, genuinely nonlinear, non-conservative system
arising in context of a model in elastodynamics.Comment: A revised version with considerable change in content. The title of
the article has been changed to 'Riemann Problem for a limiting system in
elastodynamics'. The contents of the earlier version in arxiv is Section 2 in
this revised version. The abstract has been changed as wel
The Dirac delta function in two settings of Reverse Mathematics
The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property integral(R) f(x)delta(x)dx = f (0) of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak Konig's Lemma (see Yu and Simpson in Arch Math Log 30(3): 171-180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the 'Big' systems of Reverse Mathematics. In the context of ERNA's Reverse Mathematics (Sanders in J Symb Log 76(2): 637-664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA's Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA's Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength
Hyperbolic systems of conservation laws in one space dimension
Aim of this paper is to review some basic ideas and recent developments in
the theory of strictly hyperbolic systems of conservation laws in one space
dimension. The main focus will be on the uniqueness and stability of entropy
weak solutions and on the convergence of vanishing viscosity approximations
On the mathematical and foundational significance of the uncountable
We study the logical and computational properties of basic theorems of
uncountable mathematics, including the Cousin and Lindel\"of lemma published in
1895 and 1903. Historically, these lemmas were among the first formulations of
open-cover compactness and the Lindel\"of property, respectively. These notions
are of great conceptual importance: the former is commonly viewed as a way of
treating uncountable sets like e.g. as 'almost finite', while the
latter allows one to treat uncountable sets like e.g. as 'almost
countable'. This reduction of the uncountable to the finite/countable turns out
to have a considerable logical and computational cost: we show that the
aforementioned lemmas, and many related theorems, are extremely hard to prove,
while the associated sub-covers are extremely hard to compute. Indeed, in terms
of the standard scale (based on comprehension axioms), a proof of these lemmas
requires at least the full extent of second-order arithmetic, a system
originating from Hilbert-Bernays' Grundlagen der Mathematik. This observation
has far-reaching implications for the Grundlagen's spiritual successor, the
program of Reverse Mathematics, and the associated G\"odel hierachy. We also
show that the Cousin lemma is essential for the development of the gauge
integral, a generalisation of the Lebesgue and improper Riemann integrals that
also uniquely provides a direct formalisation of Feynman's path integral.Comment: 35 pages with one figure. The content of this version extends the
published version in that Sections 3.3.4 and 3.4 below are new. Small
corrections/additions have also been made to reflect new development
Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation
In this paper, firstly, by solving the Riemann problem of the zero-pressure
flow in gas dynamics with a flux approximation, we construct parameterized
delta-shock and constant density solutions, then we show that, as the flux
perturbation vanishes, they converge to the delta-shock and vacuum state
solutions of the zero-pressure flow, respectively. Secondly, we solve the
Riemann problem of the Euler equations of isentropic gas dynamics with a double
parameter flux approximation including pressure. Further we rigorously prove
that, as the two-parameter flux perturbation vanishes, any Riemann solution
containing two shock waves tends to a delta shock solution to the zero-pressure
flow; any Riemann solution containing two rarefaction waves tends to a
two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum
intermediate state in between tends to a vacuum state.Comment: 17 pages, 4 figures, accepted for publication in SCIENCE CHINA
Mathematic
Singular limits in phase dynamics with physical viscosity and capillarity
Following pioneering work by Fan and Slemrod who studied the effect of
artificial viscosity terms, we consider the system of conservation laws arising
in liquid-vapor phase dynamics with {\sl physical} viscosity and capillarity
effects taken into account. Following Dafermos we consider self-similar
solutions to the Riemann problem and establish uniform total variation bounds,
allowing us to deduce new existence results. Our analysis cover both the
hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large
Riemann data.
The proofs rely on a new technique of reduction to two coupled scalar
equations associated with the two wave fans of the system. Strong
convergence to a weak solution of bounded variation is established in the
hyperbolic regime, while in the hyperbolic-elliptic regime a stationary
singularity near the axis separating the two wave fans, or more generally an
almost-stationary oscillating wave pattern (of thickness depending upon the
capillarity-viscosity ratio) are observed which prevent the solution to have
globally bounded variation.Comment: 30 page
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
On the plane wave Riemann Problem in Fluid Dynamics
The paper contains a stability analysis of the plane-wave Riemann problem for
the two-dimensional hyperbolic conservation laws for an ideal compressible gas.
It is proved that the contact discontinuity in the plane-wave Riemann problem
is unstable under perturbations. The implications for Godunovs method are
discussed and it is shown that numerical post shock noise can set of a contact
instability. A relation to carbuncle instabilities is established.Comment: 27 pages, 1 figur
Analysis in weak systems
The authors survey and comment their work on weak analysis. They describe the basic set-up of analysis in a feasible second-order theory and consider the impact of adding to it various forms of weak Konig's lemma. A brief discussion of the Baire categoricity theorem follows. It is then considered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integration in the stronger system. The paper finishes with three questions in weak analysis.info:eu-repo/semantics/publishedVersio
- …