6,521 research outputs found
On Nonlinear Stochastic Balance Laws
We are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in of the approximations, uniformly in the
viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish an
error estimate for the stochastic viscosity method, as well as an explicit
estimate for the continuous dependence of stochastic entropy solutions on the
flux and random source functions. Various further generalizations of the
results are discussed
Scalar conservation laws with rough (stochastic) fluxes
We develop a pathwise theory for scalar conservation laws with quasilinear
multiplicative rough path dependence, a special case being stochastic
conservation laws with quasilinear stochastic dependence. We introduce the
notion of pathwise stochastic entropy solutions, which is closed with the local
uniform limits of paths, and prove that it is well posed, i.e., we establish
existence, uniqueness and continuous dependence, in the form of pathwise
-contraction, as well as some explicit estimates. Our approach is
motivated by the theory of stochastic viscosity solutions, which was introduced
and developed by two of the authors, to study fully nonlinear first- and
second-order stochastic pde with multiplicative noise. This theory relies on
special test functions constructed by inverting locally the flow of the
stochastic characteristics. For conservation laws this is best implemented at
the level of the kinetic formulation which we follow here
Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise
In this article, we are concerned with a multidimensional degenerate
parabolic-hyperbolic equation driven by Levy processes. Using bounded variation
(BV) estimates for vanishing viscosity approximations, we derive an explicit
continuous dependence estimate on the nonlinearities of the entropy solutions
under the assumption that Levy noise depends only on the solution. This result
is used to show the error estimate for the stochastic vanishing viscosity
method. In addition, we establish fractional BV estimate for vanishing
viscosity approximations in case the noise coefficients depend on both the
solution and spatial variable.Comment: 31 Pages. arXiv admin note: text overlap with arXiv:1502.0249
Stochastic isentropic Euler equations
We study the stochastically forced system of isentropic Euler equations of
gas dynamics with a -law for the pressure. We show the existence of
martingale weak entropy solutions; we also discuss the existence and
characterization of invariant measures in the concluding section
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