459 research outputs found
Minimization Solutions to Conservation Laws with Non-smooth and Non-strictly Convex Flux
Conservation laws are usually studied in the context of sufficient regularity
conditions imposed on the flux function, usually and uniform convexity.
Some results are proven with the aid of variational methods and a unique
minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these
classical results can be extended to a flux function that is not necessarily
smooth or uniformly or strictly convex. Although uniqueness a.e. of the
minimizer will generally no longer hold, by considering the greatest (or
supremum, where applicable) of all possible minimizers, we can successfully
extend the results. One specific nonlinear case is that of a piecewise linear
flux function, for which we prove existence and uniqueness results. We also
approximate it by a smoothed, superlinearized version parameterized by
and consider the characterization of the minimizers for the
smooth version and limiting behavior as to that of the
sharp, polygonal problem. In proving a key result for the solution in terms of
the value of the initial condition, we provide a stepping stone to analyzing
the system under stochastic processes, which will be explored further in a
future paper.Comment: 27 pages, 5 figure
Front motion for phase transitions in systems with memory
We consider the Allen-Cahn equations with memory (a partial
integro-differential convolution equation). The prototype kernels are
exponentially decreasing functions of time and they reduce the
integrodifferential equation to a hyperbolic one, the damped Klein-Gordon
equation. By means of a formal asymptotic analysis we show that to the leading
order and under suitable assumptions on the kernels, the integro-differential
equation behave like a hyperbolic partial differential equation obtained by
considering prototype kernels: the evolution of fronts is governed by the
extended, damped Born-Infeld equation. We also apply our method to a system of
partial integro-differential equations which generalize the classical phase
field equations with a non-conserved order parameter and describe the process
of phase transitions where memory effects are present
A Minimization Approach to Conservation Laws With Random Initial Conditions and Non-smooth, Non-strictly Convex Flux
We obtain solutions to conservation laws under any random initial conditions
that are described by Gaussian stochastic processes (in some cases
discretized). We analyze the generalization of Burgers' equation for a smooth
flux function for
under random initial data. We then consider a piecewise linear, non-smooth and
non-convex flux function paired with general discretized Gaussian stochastic
process initial data. By partitioning the real line into a finite number of
points, we obtain an exact expression for the solution of this problem. From
this we can also find exact and approximate formulae for the density of shocks
in the solution profile at a given time and spatial coordinate . We
discuss the simplification of these results in specific cases, including
Brownian motion and Brownian bridge, for which the inverse covariance matrix
and corresponding eigenvalue spectrum have some special properties. We
calculate the transition probabilities between various cases and examine the
variance of the solution in both and . We also
describe how results may be obtained for a non-discretized version of a
Gaussian stochastic process by taking the continuum limit as the partition
becomes more fine.Comment: 36 pages, 5 figures, small update from published versio
Asset Price Volatility and Price Extrema
The relationship between price volatility and expected price market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a general setting of changing fundamentals, the volatility is maximum when expected prices are changing most rapidly, with the maximum of volatility reached prior to the maximum of expected price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation governing the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset
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