Data evento: 
Giovedì, 5 Giugno, 2014 - 14:00

Aula 1B1 Stochastic Equations in Hydrodynamics

Wladimir Neves Universidade Federal do Rio de Janeiro


We present some results concerning stochastic linear transport equations and 
quasilinear scalar conservation laws, where the additive noise is a perturbation
of the drift. Due to the introduction of the stochastic term,  we may prove for instance well-posedness for continuity equation (divergence-free), Cauchy problem, meanwhile uniqueness may fail for the deterministic case, see [1], [2] and [6]. Also for the transport equation, Dirichlet data, we established a better trace result by the introduction of the noise, see [7].
We introduce the study of stochastic hyperbolic conservation laws, 
in a different direction of [5], applying the kinetic-semigroup theory. 
[1] L.  Ambrosio, Transport equation and
Cauchy problem for $BV$ vector fields,  Invent. Math.,  158,  227--260, 2004.
[2] R. DiPerna,  P. L. Lions,  Ordinary differential
equations, transport theory and Sobolev spaces, Invent. Math.,  98, 
511--547, 1989.
[3]F Fedrizzi , F. Flandoli, Noise prevents singularities in linear transport equations,  Journal of Functional Analysis, 264,  1329--1354, 2013.
[4]  F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation Invent. Math., 180, 1-53, 2010.
[5] P. L. Lions , P.  Benoit,  P. E. Souganidis,  Scalar conservation laws with rough (stochastic) fluxes, Stochastic Partial Differential Equations: Analysis and Computations , 1,  4, 664-686, 2013. 
[6]  W. Neves, C .Olivera,  Wellposedness for stochastic continuity equations 
with Ladyzhenskaya-Prodi-Serrin condition, arXiv:1307.6484v1, 2013. 
[7] W. Neves, C .Olivera, Stochastic transport equation
in bounded domains, in preparation. 
Joint work with: Christian Olivera (Universidade Estadual de Campinas)

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