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4 papers
Short Communications
$d$-dimensional pressureless
Gas equations
A. Dermoune University of Sciences and Technologies
Abstract:
Let
$x\in R^d\to u(x,0)$ be a continuous bounded function
and
$\rho(dx,0)$ a probability measure
on
$R^d$. For all random variables
$X_0$ with probability distribution
$\rho(dx,0)$,
we show that the
stochastic differential equation (SDE)
$$
X_t = X_0 + \int_0^t
E\big[u(X_0,0)\,|\, X_s\big]\,ds,\qquad t\ge 0,
$$
has a solution which is a
$\sigma(X_0)$-measurable
Markov process.
We derive a weak solution for the pressureless gas equation for
$d \ge 1$,
with initial distribution of masses
$\rho(dx,0)$ and initial
velocity
$u(\cdot,0)$.
We show for
$d = 1$
the existence of a unique Markov process
$(X_t)$ solution of our SDE.
Keywords:
pressureless gas equations, variational principles. Received: 10.10.2001
Language: English
DOI:
10.4213/tvp212
Fulltext:
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