The Fourth Workshop on Fluids and PDE was held at the National Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, from Monday 26 May to Friday 30 May 2014. This workshop is held every two to three years in Brazil. The fourth edition of the workshop was the closing event of a Thematic Program on Incompressible Fluids Dynamics, to be held at IMPA next Spring. Hence, the focus of the workshop will be incompressible fluid mechanics .

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. Joint work with: Christian Olivera (Universidade Estadual de Campinas. [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 .